Containment Algorithms for Nonconvex Polygons with Applications to Layout
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Citation McIntosh Daniels, Karen. 1995. Containment Algorithms for Nonconvex Polygons with Applications to Layout. Harvard Computer Science Group Technical Report TR-12-95.
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Containment Algorithms for Nonconvex
Polygons with Applications to Layout
Karen McIntosh Daniels
TR
May
Center for Research in Computing Technology
Harvard University
Cambridge Massachusetts
Containment Algorithms for Nonconvex Polygons
with Applications to Layout
A thesis presented
by
Karen McIntosh Daniels
to
The Division of Applied Sciences
in partial ful llment of the requirement for the degree of
Do ctor of Philosophy
in the sub ject of
Computer Science
Harvard University
Cambridge Massachusetts
May
c
by Karen McIntosh Daniels
All rights reserved
Table of Contents
List of Figures
List of Tables
Introduction
Background
Containment Packing and Layout
Thesis Statement and Overview of Chapter
Overview of Containment Work
Related Work
Our Containment Work
Overview of Layout Work
Related Work
Convex Items
Nonconvex Items
Assignment
Our Layout Work
Pro ject Background
Compaction
Panel Placement
Trim Placement
Benchmarks
Contribution
Overview
I Translational Containment
Preliminaries
Introduction
Overview of Chapter
Notation
Hardness
The Restrict Evaluate Subdivid e Paradigm
Approach to Running Time Analysis
Related Work
Containment Characterization
Overview of Part One
Con guration Space Restrictions
Introduction
Overview
Notation
Prop erties of Valid Restrictions
Boundary Intersection
Lo ose Fits
Annular Con guration Spaces
Size and s Analysis
Identical Items
Steady State Geometric Restriction
Avnaim and Boissonnat s Solutions to NN and NP
Steady State Restriction
Running Time
Example
E ectiveness
Steady State Linear Programming Restriction
Example
E ectiveness
Subset Restriction
Union Restriction
Characterization Based Restriction
Symmetry Breaking and Subset Substitution Restrictions
Closed Op erators
Summary and Conclusions
Evaluation and Sub division
Introduction
Evaluation
Sub division
Overview
Greedy Evaluation
Algorithm
Heuristics for Selecting u
ij
Comparing Heuristics for Selecting u
ij
Overlap Reduction Evaluation
Evaluation of a Restricted U
Sub division
De nition and Goals
Size Based Metho d
Combinatorially Based Metho ds
Distance Based Metho d
Summary and Conclusions
Algorithms
Introduction
Overview
An Approximate Containment Algorithm
Introduction
Analysis
Results
A Hybrid Containment Algorithm
Introduction
Results
Conclusion
Summary and Conclusions
II Applications of Containment to Layout
Preliminaries
Introduction
Overview of Chapter
Overview of Container Transformation Work
Overview of Minimal Container Work
Related Work
Our Minimal Container Work
Overview of Maximal Item Work
Our Maximal Item Work
Overview of Part Two
Limited Gaps
Introduction
Decomp osing the Container into Gaps
Overview
The Challenge of Constructing Gap Sets
Compaction
Partitioning Squares
Other Metho ds
Limited Gaps
Reachable Regions
De ning Limited Gaps using Probability of Interference
E cient Computation of Limited Gaps
Breaking Polygons
Inking Polygons
MAAPRs
Summary of MAAPR Results
MAAPR Characterization and Naive Algorithm
Summary and Conclusions
Minimal Container Problems
Introduction
Overview
Strip Packing
Minimal Square Enclosures
Minimal Rectangular Enclosures
Tight Convex and Nonconvex Enclosures
Summary and Conclusions
Maximal Item Problems
Introduction
Overview
Maximally Scaled Items
An Elimination Technique
Randomized Subset Construction
Pre packing
Assignment Strategy
Results
Summary and Conclusions
III Conclusion and App endix
Conclusion and Future Work
Conclusion
Future Work
Containment
Applications of Containment to Layout
A Containment File Format
Bibliography
List of Figures
Translational containment problem for six items
The marker making task in the apparel industry
B t A if and only if t A B
B t A if and only if t A B
Boundary intersection example the containment problem
Boundary intersection example two one comp onent U s with U s
j
j
Boundary intersection example two two component U s with U s
ij
ij
Boundary intersection example the containment problem
Boundary intersection example a one comp onent U with U s
j
j
s Boundary intersection example a two component U with U
ij
ij
An annular con guration space
Geometric restriction
SSGR containment problem place convex items
for the SSGR of U A one comp onent U with U
j
j
A two component U with U for the SSGR of U
ij
ij
A NN example with SSGR restriction
L shap ed container and items
A NN example with average SSGR restriction
An infeasible CN example
A CC example with an unrestrictive container
A NC example with average SSGR restriction
A CC example with average SSGR restriction
U U U
A CC example with strong SSGR restriction
A highly restrictive NN example
A NN example e ect of SSGR on two one comp onent U s
j
Counterexample for A A A
Selected U s for NN example
ij
SSLPR containment problem place nonconvex items
Two two component U s with U s for SSLPR of U
j
j
A one comp onent U with U s for SSLPR of U
ij
ij
E ect of SSLPR on U s when a single comp onent of a U is selected
ij ij
Example for which SSLPR is more p owerful than SSGR
Regularization example
Two rectangles with degenerate intersection
CN example which do es not yield a common intersection of translated U s
j
No common intersection of translated U s
j
No common intersection after maximizing intersection
Common intersection after maximizing convexity of the union
Distance based sub division
Common situation for p t t
j i
NN example time seconds
NN example time seconds
NN example time seconds
NN example time seconds
NN feasible example solved using partial characterization
NN example time seconds
NN example time seconds
Trim placement task
A partitioning square which fragments the container
A neck which is usable although items cannot pass through it
Algorithmic results for a variety of p olygon types
Edge contact types for a determining set
Slop e lemma
Parameter problems with two and three dep endent sliding contacts
Determining sets of contacts for the MAAPR
Limited gaps example
Incremental layout task
Lo ose layout
Final layout
Compaction solution
Minimal square enclosure for nonconvex items
Minimal square enclosure for items
Minimal square enclosure for nonconvex items
Tight rectangular enclosure for convex items
Tight rectangular enclosure for nearly identical nonconvex items
Tight rectangular enclosure for nonconvex items
Tight nonconvex enclosure for items
Scaling circles
Scaling up items in a greedy fashion
Scaling up items in an enhanced balanced fashion
Scaling up two items
Find a large subset which ts
A large subset which ts
Items and container for randomized construction task
Randomly constructed item group
Randomly constructed item group
Trim placement task for c mrk
First Fit for c mrk
Assignment for c mrk
Assignment plus First Fit for c mrk
Trim placement task for b mrk
First Fit for b mrk
Assignment for b mrk
Assignment plus First Fit for b mrk
List of Tables
SSGR applied to infeasible examples
Number of solutions found without sub division
Running times in seconds denotes infeasible example
Running time in seconds and number of hypotheses examined by approxi
mate algorithm
Number of hypotheses examined by sequential algorithm denotes infea
sible example
Number of hypotheses examined for identical items denotes infeasible
example
Exact vs approximate running times for CN examples calls each
Running times for feasible examples in seconds
Running times for infeasible examples in seconds
Number of hypotheses examined denotes infeasible example
Number of hypotheses evaluated for restriction options denotes infeasible
example
Number of hypotheses evaluated for infeasible problems no hypothesis re
striction
Pre packing results
Abstract
Layout and packing are NP hard geometric optimization problems of practical imp or
tance for which nding a globally optimal solution is intractable if P NP Such problems
app ear in industries such as aerospace ship building apparel and sho e manufacturing
furniture pro duction and steel construction At their core layout and packing problems
have the common geometric feasibility problem of containment nd a way of placing a set
of items into a container In this thesis we fo cus on containment and its applications to
layout and packing problems We demonstrate that although containment is NP hard it
is fruitful to develop algorithms for containment as opp osed to heuristics design
containment algorithms so that they say no almost as fast as they say yes use ge
ometric techniques not just mathematical programming techniques and maximize the
number of items for which the algorithms are practical
Our approach to containment is based on a new restrict evaluate subdivide paradigm
We develop new theory and practical techniques for the op erations within the paradigm
The techniques are appropriate for two dimensional containment problems in which the
items and container may b e irregular i e nonconvex p olygons and have multiple com
p onents and in which the items may b e translated but not rotated Our techniques can
b e combined to form a variety of two dimensional translational containment algorithms
The paradigm is designed so that unlike existing iteration based algorithms containment
algorithms based on the paradigm are adept at saying no even for slightly infeasible prob
lems Infeasible problems o ccur frequently in practice We present two algorithms based on
our paradigm We obtain the rst practical running times for NP complete two dimensional
translational containment problems for up to ten nonconvex items in a nonconvex container
Most of our examples are from apparel manufacturing Typically each item has from to
vertices and the container has from to vertices
We demonstrate that viewing containment as a feasibility problem has many b ene ts
for packing and layout problems For example we present an e ective metho d for nding
minimal enclosures which uses containment to p erform binary search on a parameter Com
paction techniques can accelerate the search We also use containment to develop the rst
practical pre packing strategy for a multi stage pattern layout problem in apparel manu
facturing Pre packing is a layout metho d which packs items into a collection of containers
by rst generating groups of items which t into each container and then assigning groups
to containers
Acknowledgments
My research has b een carried out under the sup ervision of my advisor Victor Milenkovic
Assistant Professor of Computer Science at Harvard who is now an Asso ciate Professor at
the University of Miami His guidance and supp ort of my research have b een sup erb I
gratefully acknowledge the advice provided by my entire thesis committee Professor Victor
Milenkovic Professor Fred Ab ernathy and Professor Harry Lewis In addition I thank
Professor Les Valiant who served on my quali er committee
This thesis research is part of a pro ject funded by the Sloan Foundation within the
Harvard Center for Textile and Apparel Research The principal investigator is Professor
Fred Ab ernathy who has help ed us forge relationships with industry Don Walker of the
Harvard O ce of Technology and Licensing has help ed us formalize our relationships with
industry Micro dynamics Inc and Gerb er Garment Technologies have help ed us transfer
our work to the apparel manufacturing industry
Sp ecial thanks are due to two former Harvard graduate students Dr Zhenyu Li of
GTE Labs and Dr Dan Roth Zhenyu Li was an integral member of our research team
while working on his dissertation at Harvard His comments on drafts of the thesis were
very helpful I am grateful to Dan Roth for his collab oration on nding the maximum area
axis parallel rectangle in a p olygon
Our research team has involved a number of Harvard undergraduates over the years
Ra jarshi Bhattachyarya Lee Wexler Jacqueline Huang Jackie Chang Sanjoy Dasgupta
Venkatesh Reddy Shivashish Chatterjee Kirat Singh Eric Wilfrid Sanjay Madan Matt
LaMantia John Immel Ben Westbrook Ed Han and Iraklis Kourtidis I also thank future
Harvard student Umesh Shankar for his work while at the MIT Summer Research Institute
Administrative assistance from Carol Harlow Sheila Toomey Baiba Menke Pauline
Mitchell and the rest of the sta has b een wonderful The constant supp ort of the Aiken
computer systems sta has kept our machines up and running through years of research and
many demonstrations I convey long distance thanks to Irina Kaliman at the University of
Miami for her supp ort of our software and for her translations of Russian layout work
On a p ersonal note I thank my family for their encouragement and patience my daugh
ters Theresa and Michelle and my husband Murray Daniels My mother Jane McIntosh
has also provided constant supp ort I dedicate this thesis to my husband Murray whose
user interface co de is an integral part of our pro ject s software and whose love and supp ort
have made it p ossible for me to pursue this research
C h a p t e r
Introduction
Background
Many industrial problems involve placing ob jects into containers so that no two ob jects
overlap each other The general goal is to either minimize the size of the container or nd
a maximal collection of placed ob jects These problems are known by a variety of names
such as layout packing loading placement and nesting A number of industries apply
layout techniques when cutting new parts from sto ck material In apparel manufacturing
for instance pattern pieces are arranged on cloth The goal is to nd a non overlapping
arrangement which uses the least amount of cloth In sheet metal layout ob jects are cut
from metal sto ck sheets The goal here is typically to cut as many ob jects as p ossible from
a given sheet In sho e manufacturing each hide is a di erent shap e and the goal is to cut
as many ob jects from each hide as p ossible
Layout applications are not limited to cutting op erations nor to pro cesses where the
goal is to minimize the waste of material In VLSI layout for example rectangular mo dules
are arranged on a chip in such a way as to b est meet the two comp eting goals of minimizing
chip area and minimizing the length of interconnections b etween mo dules on the chip In
furniture layout an arrangement of furniture usually must satisfy a set of aesthetic criteria
When putting together a jigsaw puzzle the goal is simply to nd a feasible solution
Placement problems also app ear in rob otics The simplest problem is to nd a p osition
for a rob ot amidst a collection of obstacles This is generalized to planning a motion for
the rob ot through the obstacles This can b e further generalized to problems in assembly
planning in which motions must b e planned not only for the rob ot but also for parts which
the rob ot manipulates
People typically do not think of all of these problems as similar as they corresp ond to
entirely di erent elds of research However some work in rob ot motion planning and com
putational geometry has noted the corresp ondence b etween placement and motion planning
problems Cha CK AB Avn For example Avnaim Avn solves placement
and displacement motion planning problems noting that the former type is useful in
layout applications and the latter in rob otics The relationship b etween cutting packing
1
One p ossible maximality criterion is the number of ob jects Another is the total area of the ob jects
and motion planning was also observed by Cuninghame Green in a survey pap er CG
At their core all these problems have the common problem of containment nd a way of
placing a set of ob jects into a container
In this thesis we fo cus on containment and its applications to layout and packing prob
lems In Section we de ne containment packing and layout and show how these form
a hierarchy of problems We also brie y discuss the state of the art with resp ect to these
problems Section states our main thesis and provides an overview of the remainder
of this chapter
Containment Packing and Layout
In the rst part of Section we de ne containment and show how it ts into a
hierarchy of layout and packing problems We then discuss how di cult these problems are
to solve their hardness and the current state of the art in solving them in the second
part of Section
Hierarchy of Problems
We de ne containment as follows given a container a set of items and a set of al
lowable transformations of the items nd transformations of the items which result in a
non overlapping arrangement of the items within the container Figure shows an ex
ample of a two dimensional containment problem for six nonconvex p olygonal items and a
nonconvex p olygonal container consisting of the complement of some other already placed
items within the rectangle The items are pattern pieces from a layout problem in ap
parel manufacturing The pieces are allowed to translate but not rotate and so this is a translational containment problem
22 17 28 15
18 31 16 7 8 1 21 2
3 30 12 9 5 26 19 20 29 10 6
4 25 13
0 14 27 24
11 23
Figure Translational containment problem for six items
A packing problem is an extension of a containment problem Its solution is a non
overlapping arrangement that also optimizes some criteria applied to either the container
or the items Typically the goal is to either nd the minimal container into which a given
set of items t or to nd the maximal collection of items which t into a given container
A packing problem is therefore an optimization problem over a range of containers or
collections of items A containment problem considers only the feasibility of placing a given
collection into a given container Thus containment is the feasibility problem corresp onding
to the packing optimization problem
One example of a minimal container type of packing problem is the classical one
dimensional bin packing problem People often think of this as a sp ecialized two dimensional
packing problem for which a set of rectangular axis parallel bins and a set of rectangular
items are given The set of bins is in nite the set of items is nite Bins and items all
have unit width Each item has height c where c is the common bin height The task
is to minimize the number of bins required to pack all the items using only translation
An example of the maximal collection of items type of packing problem is the three
dimensional task of cargo load planning for aircraft where the goal is to e ciently utilize
the available three dimensional cargo space
The terms packing and layout are sometimes used interchangeably in the literature
Often however a layout problem refers to a packing problem which either has additional
constraints or multiple goals For example as mentioned earlier a VLSI layout problem
has the two comp eting goals of minimizing chip area and minimizing the length of intercon
nections b etween mo dules This type of problem also has constraints such as a requirement
to maintain a certain minimum separation b etween mo dules For other types of layout
problems the constraints are sometimes in the form of aesthetic criteria as in the case of
laying out furniture in a ro om
De nitions vary but we consider containment packing and layout to b e a hierarchy of
problems Loading generally refers to a sp eci c type of three dimensional layout problem
that arises in shipping In VLSI design placement is that asp ect of the layout task concerned
with the p ositions of mo dules whereas routing deals with interconnections b etween mo dules
The term nesting is asso ciated with packing tasks for irregularly shap ed items in particular
ones which can b e placed inside of others
State of the Art
Work on containment problems for small numbers of items three or fewer app ears
o ccasionally in the literature on computational geometry and has primarily b een motivated
by problems in rob otics Section gives a detailed account of this work
By comparison the literature on packing and layout problems is vast This work app ears
in the literature on management sciences engineering sciences information and computer
science mathematics and op erations research Section reviews this work which
employs a broad range of techniques
For rectangular shap es the layout and packing literature and algorithms are well
developed For example some sp ecial types of rectangular problems lend themselves to
linear and or dynamic programming in particular when guillotine cuts are used exclu
sively DD Ground breaking work in this area was done in the s by Gilmore and
2
One p ossible minimality criterion is area
3
A guillotine cut is a horizontal or vertical cut which completely partitions the remaining material
Gomory GG There also exists a rich literature on bin packing for rectangles that con
tains much theoretical work on the e ectiveness of simple heuristics JGJ
For irregular shap es packing and cutting algorithms have yet to match human p erfor
mance In industry such problems are usually solved either by humans or by computer
based heuristics Exp ert humans construct nearly optimal layouts in some domains One
very large clothing company has humans create layouts p er week averaging ab out
hours of lab or for a layout of items The automatic generation of pro duction quality
layouts of irregular items is an active area of research
What is primarily missing from work on packing and layout is the ability to quickly
detect an infeasible problem that is to say no to the containment question Infeasible
problems arise frequently in practice and this shortcoming signi cantly limits the types of
techniques which can b e used in layout and packing
Saying no is di cult Except in the simplest of cases heuristics and even humans
cannot detect infeasibility For instance the example in Figure is not obviously in
feasible A human could try for a long time to place the items but would never succeed
Furthermore this failure would not constitute a pro of of infeasibility until every p ossible
p osition had b een tried
Containment in its most general form is NP hard and therefore computationally in
tractable if P NP Current containment algorithms have worst case running times which
are exp onential in the number of items b eing placed Furthermore current containment
algorithms attain their worst case running times when faced with any infeasible problem
Multiple resolution techniques for approximating p olygonal b oundaries can b e used to im
prove the ability to say no However algorithms using these techniques still p erform
slowly when faced with problems which are just barely infeasible Figure is an example
of such a problem This type of problem arises frequently in practice b ecause a tight packing
is the goal of a packing problem
Another shortcoming of current practical containment algorithms is that they place at
most three items Why has the computational geometry community not pro duced practical
containment algorithms for larger numbers of items Since containment is NP hard p erhaps
their view is that computational geometry techniques are not p owerful enough for such
problems and that mathematical programming techniques o er the only chance for success
Thesis Statement and Overview of Chapter
This thesis provides two dimensional translational containment algorithms which over
come b oth of the disadvantages of existing containment algorithms Our algorithms are
designed to place an arbitrary number of items The algorithms can say b oth no and
yes quickly They do not necessarily p erform more slowly on slightly infeasible inputs
One of our algorithms is practical for up to ten items The implementation of the other
algorithm is currently fast for up to four items The slowness of our current implementation
of p olygon set op erations makes this algorithm slow for more than four items The ability
to solve containment for up to ten items is p owerful Previously unsolved packing and lay
out problems for hundreds of items can now b e solved using containment as the feasibility
4
In this case the problem is infeasible and the global overlap minimum is small but nonzero
5
Our examples are drawn primarily from the apparel industry Each item is represented as a p olygon
which typically has from to vertices The container is also represented as a p olygon it typically has
from to vertices The items and container may b e nonconvex
oracle inside other metho ds
Our ability to say no well comes primarily from our new geometric techniques Our
ability to say yes for so many items is partly due to our geometric techniques and partly
due to our application of and improvements to existing linear programming based tech
niques Thus we demonstrate that a computational geometry approach can contribute
signi cantly to practical algorithms for NP hard geometric problems
Our main thesis is that although containment is NP hard it is fruitful to develop
algorithms for containment as opp osed to heuristics design containment algorithms so
that they say no almost as fast as they say yes use geometric techniques not just
mathematical programming techniques and maximize the number of items for which the
algorithms are practical
The remainder of the thesis is divided into two main parts Part I describ es our work
on containment algorithms Part II presents applications of containment to layout The
applications validate our approach to containment
The remainder of this chapter is organized as follows Section discusses work from
the containment literature and then gives an overview of our own work on containment
Section describ es work from the layout literature and discusses our applications of con
tainment to layout problems Section describ es the research pro ject which has motivated
the work in this thesis Section discusses b enchmarks Section highlights the main
contributions of the thesis Finally Section provides a road map for the remainder of
the thesis
Overview of Containment Work
Containment problems are in general NP hard As we show in Section containment
is NP complete if the items are allowed to translate but not rotate Part I of the thesis
discusses our work on the NP complete two dimensional translational containment problem
for p olygons The p olygons and container may b e nonconvex We present new theory
leading to practical algorithms for up to ten p olygons
The following two subsections provide an overview of related work on containment and of
our work on containment Section surveys the literature on containment Section
summarizes our work on containment
Related Work
The term containment app ears as early as in a seminal pap er by Chazelle on two
dimensional p olygon containment Cha Subsequent work on containment has app eared
in literature on rob ot motion planning computational geometry and op erations research
The layout and packing communities do not seem to use this containment work Either they
are unaware of it or they have encountered di culties when implementing it In layout
the equivalent problem is sometimes called arrangement
The next two parts of this subsection survey related work on containment The rst part
discusses work on rotational containment The second part treats work on translational
6
Some of the containment research has fo cused more on combinatorial issues than on implementation
details and so this is not surprising
7
The term arrangement has a very di erent meaning in the computational geometry community
containment For these discussions we assume that the container is a p olygon with n
vertices and that a p olygonal item to b e placed has at most m vertices
Rotational Containment
Chazelle Cha considers tting a single p olygon P into another p olygon C where
b oth translation and rotation of P are allowed He observes that if one only seeks a single
solution it su ces to examine stable placements of P in C for which P s three degrees of
freedom are removed He gives an O mn time algorithm for solving the problem when C
is convex and an O m n time algorithm if C is convex and only translations are allowed
He gives a naive algorithm for the general case requiring O m n m n log m n time
Avnaim Avn and Avnaim et al AB give an O m n log mn time algorithm for a
single nonconvex p olygon in a nonconvex container Their algorithm yields al l the solutions
Grinde and Cavalier GC use mathematical programming to solve the rotational
containment problem for a convex p olygon in a convex container Their algorithm requires
O bmn time where b is the number of two contact sets obtained as P rotates through
As b O m n when the p olygon and container are convex their algorithm requires
O m n mn time
Translational Containment
In order to discuss related work on translational containment we use the con guration
space concept from rob otics In rob otics a p oint in the con guration space is a con g
uration The comp onents of the con guration are the values of the joint angles and any
other parameters necessary to sp ecify the rob ot p osition and orientation For our approach
as in AB Avn Dev an essential type of con guration space is the set of rela
k
tive positions for a pair of p olygons For k p olygons and one container there are
such con guration spaces A con guration in this type of con guration space is a k
tuple t t t where t is a translation of the ith p olygon and t is a translation of
k i
the container usually set to If applying these translations to the p olygons yields a
non overlapping arrangement the k tuple is a valid con guration
In the following discussion we use a shorthand notation to describ e various types of
translational containment problems The shorthand is based on the two de nitions b elow
De nition The kNN problem is that of nding a valid con guration of k noncon
vex polygons in a nonconvex container Variations replace N with either C convex R
rectangle or P parallelogram
De nition The r k NN problem is that of nding al l subsets of size k out of a set of
r nonconvex polygons such that the k polygons can be simultaneously placed in a nonconvex
container The variations of De nition apply
We know of at least one industrial solution to translational containment and it is used
in an interactive setting That solution app ears to b e based on inner and outer approxi
mations of p olygonal b oundaries and the running time increases with the tightness of the
t Published algorithms for translational containment for small k have asymptotic running
times which tend to dep end on a high p ower of n For arbitrary k and nonconvex p olygons
and container the only known algorithms have running time which is exp onential in k This
is not surprising since k NN is NP complete
k
A naive k NN algorithm requires O k mn log mn time It searches for a valid
con guration by iterating over all choices of k contacting vertex edge pairs among the
p olygons and container More details app ear in Section The naive algorithm has
k
running time in mn b ecause it must iterate over all p ossible choices of contacts
whenever the problem is infeasible
Fortune For gives a solution to CN by computing the Minkowski sum using a gen
eralized Voronoi diagram This O mn log mn result is an improvement over the O n
mn log mn time algorithm for CN given by Baker et al BFM Avnaim and Bois
sonnat AB Avn use the Minkowski sum and convex decomp osition to solve NN and
NN and Avnaim Avn gives an algorithm for NN which iterates over the edges of the
con guration space p olygons Devillers Dev gives faster algorithms for CN and CN
These running times are summarized b elow
k k CN k NN
O mn log mn For O m n log mn AB Avn
O m n log m Dev O m n log mn AB Avn
O m n log m Dev O m n log mn Avn
Avnaim and Boissonnat also give a solution to the NP problem three nonconvex p oly
gons in a parallelogram container using time in O m log m AB Avn Their NP
algorithm is based on p olygon unions intersections and Minkowski sum op erations Even
though the asymptotic running time of the NP algorithm contains a p olynomial whose
degree is high compared to the others in practice the NP algorithm is faster than the NN
algorithm which is in turn faster than the naive algorithm We defer further discussion of
this until Section Unfortunately Avnaim shows that there is no formula for a solution
to NN purely based on p olygon union intersection and Minkowski sums Avn This
means that for k any k NN algorithm must use iteration over edges or sub division
or some technique other than set op erations on p olygonal regions The fact that Avnaim s
algorithm for NN uses iteration gives it the m n lower b ound
Milenkovic et al DML o er three approaches to translational containment The
rst metho d op erates on convex p olygons It iterates over combinations of orientations for
the lines which separate the convex p olygons This approach yields an algorithm for CN
with running time in O m n log mn an improvement of a factor of O n over previous
algorithms in particular Dev The second metho d uses a MIP mixed integer pro
gramming mo del for k NN This approach is describ ed in detail in Li It takes one or
two minutes on a typical workstation for two or three p olygons but is slow for four or more
p olygons it can easily take more than an hour to detect an infeasible CN problem The
third metho d is an algorithm for k NN which nds an approximate con guration using
k
time in O k s log s where s is the maximum number of vertices in a p olygon log
8
For p olygons A and B the Minkowski sum is de ned as A B fa b j a A b B g See Section
for more details
9
For an approximate con guration no p oint of any p olygon is more than inside of the b oundary
of any other p olygon when the p olygons are laid out according to the con guration For some cutting
application s CAD vendors round p olygon vertex co ordinates to integers In such cases an overlap of up to
one unit corresp onding to inches is acceptable
which is created through successive applications of a certain collection of p olygon set op er
ations see the discussion of size analysis in Section This algorithm is introduced in
DM DM and app ears in Chapter of this thesis
Milenkovic Mil DM improves up on the CN algorithm of DML by giving an
algorithm which has the same asymptotic running time but is simpler to implement He also
introduces an exact k NN algorithm based on linear programming and approximation and
k
sub division of the con guration spaces which runs in O mn LP k O k mn k m
time where LP a b is the time to solve a linear program with a variables and b constraints
This algorithm is practical for k
Our Containment Work
Existing two dimensional translational containment algorithms developed by other re
searchers are practical only for k We present new theory leading to algorithms which
handle any k One of our algorithms is practical for k the implementation of the
other algorithm is currently fast for k As stated in Section our examples are
drawn primarily from the apparel industry Each item is represented as a p olygon which
typically has from to vertices The container is also represented as a p olygon it
typically has from to vertices The items and container may b e nonconvex Under
these conditions we can solve containment problems quickly typically in several minutes
or less on a MHz SPARCstation Our work is therefore practical for solving a number
of layout related problems for nonconvex p olygons
Although containment is NP hard in general we b elieve that it is valuable to make the
practical value of k as large as p ossible Thus we continue the b ottom up progression
CC NN CN NN etc of containment work which has taken place in rob otics and
computational geometry Our ability to say no well comes primarily from the use of our
new geometric techniques Our ability to say yes for so many items is partly due to our
geometric techniques and partly due to our application of and improvements to existing
linear programming based techniques Thus we demonstrate that a computational ge
ometry approach can contribute signi cantly to practical algorithms for NP hard geometric
problems
Saying no quickly is imp ortant Iteration based algorithms for containment attain
their worst case running time when the containment problem is infeasible This situation
o ccurs frequently in practice We replace iteration with sub division Our algorithms are
sp eci cally designed to detect infeasibility quickly and we have some theoretical justi cation
for their b ehavior which we discuss in more detail later in the thesis
We present a new paradigm for containment algorithms Our containment algorithms
apply three op erations in order to the con guration spaces restriction which prunes
the spaces and helps detect infeasibili ty evaluation which tries to nd a solution and
sub division which partitions one of the con guration spaces into two parts The algorithm
then recurses on each part while keeping the other con guration spaces constant
We introduce new theory and practical techniques for each of these types of op era
tions for the case of two dimensional containment problems in which the items and con
tainer may b e irregular i e nonconvex and have multiple comp onents and in which
10
SPARCstation is a trademark of SPARC International Inc licensed exclusively to Sun Microsystems
Inc
11
We use the CPLEX linear programming system CPLEX Optimization Inc
the items may b e translated but not rotated Our techniques can b e combined in vari
ous ways to form two dimensional translational containment algorithms which follow the
restrict evaluate subdivide paradigm The paradigm is designed so that unlike existing
iteration based algorithms containment algorithms based on the paradigm are adept at
saying no even for slightly infeasible problems
At present there app ear to b e two approaches to containment within the restrict eval
uate sub divide framework direct and indirect Direct metho ds act on the con guration
spaces directly and these metho ds tend to b e based on computational geometry Indirect
metho ds act on convex approximations to the con guration spaces either inner or outer
approximations Indirect metho ds use linear programming or other types of mathematical
programming which is why they can only act on convex approximations Direct metho ds
have a more accurate view of the world but they can only op erate on a few three in
our case con guration spaces at a time Indirect metho ds can act on all the con gura
tion spaces simultaneously in a single op eration but they use only approximations to the
con guration spaces
The thesis presents two new algorithms based on our paradigm They are brie y de
scrib ed b elow The rst is a direct algorithm We call the second a hybrid algorithm b ecause
it uses an indirect evaluation metho d and b oth direct and indirect metho ds for restriction
Direct Algorithm
This thesis presents a new direct algorithm for containment It is an approximate
algorithm which pro duces an approximate con guration whose overlap is b elow a given
tolerance As mentioned earlier for some cutting applications CAD vendors round p olygon
vertex co ordinates to integers In such cases an overlap of up to one unit corresp onding
to inches is acceptable Our approximate algorithm uses our new direct techniques for
restriction evaluation and sub division It app ears in DM DM as well as in Section
Our direct restrictions are based on p olygon set op erations and so we call them geometric
restrictions They are p owerful to ols for detecting infeasibili ty We de ne two general types
of restrictions valid restrictions which preserve all solutions to a containment problem and
semi valid restrictions which preserve a single solution if one exists We present a variety
of restrictions and several general prop erties of valid restrictions We examine in detail the
e ectiveness of our geometric restrictions which are valid restrictions Our characterization
of the types of containment problems for which these restrictions are e ective is based on
theoretical as well as exp erimental results
The evaluation metho d uses restriction together with a geometric metho d for maximizing
the intersection of translations of the con guration spaces The evaluation metho d often
nds a containment solution without any sub division The sub division metho d is a size
based one Its goal is to partition a con guration space into two parts each smaller than
the unsub divided con guration space
Our approximate algorithm has an extra layer of structure on top of the restrict evaluate
sub divide paradigm This layer has a set of containment subproblems to which course
grained parallelism could b e applied in the future We create the subproblems by charac
terizing a valid con guration for translational containment
The current implementation of this algorithm is fast for up to four items For k
the current implementation of our geometric restrictions based on p olygon set op erations
slows the algorithm down considerably Increasing the sp eed of our p olygon set op erations
is a sub ject of future work see Section In our exp eriments our approximate
algorithm p erforms fewer sub divisions than the faster hybrid algorithm and so we b elieve
the approximate algorithm has the p otential to b e comp etitive with the hybrid algorithm
Hybrid Algorithm
In addition to our direct algorithm we also present signi cant improvements to the
restriction evaluation and sub division techniques of Milenkovic s exact k NN algorithm
Mil which is an indirect algorithm Milenkovic s indirect algorithm in its original form
was slower than our direct algorithm for infeasible problems and its practical upp er b ound
on k was k The improved indirect algorithm is currently our fastest containment
algorithm for k
Our rst improvement to the indirect algorithm is a pro of that restrictions when used
in the indirect algorithm do not increase the algorithm s worst case running time if the
algorithm bases certain decisions on the unrestricted con guration spaces This is imp ortant
b ecause restrictions can add vertices to the con guration spaces and so they might have
increased the worst case running time The pro of allows us to add our new direct geometric
restrictions to the algorithm without increasing the worst case number of sub divisions It
also allows us to apply a new linear programming based restriction metho d developed by
Milenkovic
The second improvement to the indirect algorithm is related to the evaluation metho d
We apply the work of Li and Milenkovic on overlap resolution Li ML LM c LM a
to improve the evaluation step Their overlap resolution algorithm is an indirect algorithm
It uses inner convex approximations an approach which complements the outer approxi
mations of the indirect containment algorithm Our third improvement is our new practical
sub division technique which attempts to maximize the distance from the current evaluation
to the convex approximations of the sub divided con guration spaces
We call the resulting algorithm a hybrid algorithm b ecause it uses indirect metho ds
for evaluation developed by Li and Milenkovic and b oth direct and indirect metho ds for
restriction The hybrid algorithm is describ ed in MD as well as in Section
The hybrid algorithm in addition to b eing a formidable containment algorithm has
the prop erty that a partial containment solution from it is valuable This is b ecause each
evaluation step pro duces either a non overlapping solution or an overlapping con guration
for which a linear program has reached a lo cal minimum As a result the hybrid algorithm
when terminated early is the most p owerful heuristic known for generating a lo cal overlap
minimum
Overview of Layout Work
Part I I of the thesis discusses our application of containment to layout problems Layout
and packing are NP hard problems for which nding a globally optimal solution is gener
ally intractable The more restricted problem of packing squares onto a rectangle without
rotation as in VLSI applications is NP complete Rectangle packing is also NP complete
when rotation is not allowed even for the one dimensional bin packing problem JGJ
In general translational packing is NP complete
In this section we survey related work in layout and then describ e our applications of
containment to layout Section discusses related work at a high level Section
reviews work on layout of convex items Section treats work on nonconvex items
Related Work
Work on layout is spread across a broad range of literature Relevant areas include
Dyc management sciences engineering sciences information and computer science
mathematics and op erations research
Work on layout cutting packing loading and nesting has b een surveyed in Hin
SP Dyc DF DD DD Sweeney and Paternoster give an application oriented
bibliography for cutting and packing problems in SP Dowsland and Dowsland survey
research on packing problems in DD In DD the same authors review work on irregular
nesting problems emphasizing the di erent types of approaches which have b een taken
Dyckho analyses the structure of cutting and packing problems and presents a classi cation
in Dyc this forms the basis for Dyckho and Finke s meta survey for cutting and packing
literature DF
In DF Dyckho and Finke distinguish four ma jor types of cutting and packing
problems based on the way items are assigned to containers and on the assortment of items
They observe that in the layout literature problems of the same type tend to b e solved using
the same technique The four types are bin packing in which heterogeneous items are
assigned to a selection of containers cutting sto ck which is similar to bin packing but the
items can b e partitioned into a small set of equivalence classes based on shap e knapsack
in which all the containers must b e used and the items are heterogeneous and pallet
loading in which all the containers must b e used but the items are homogeneous These
bin packing and knapsack de nitions are more general than typical usage in the literature
Because of their di culty bin packing problems are typically solved sub optimally using
fast simple heuristics such as First Fit The small number of distinct shap es involved in
cutting sto ck problems often allows them to b e solved using rep eating patterns Knapsack
problems are frequently solved using branch and bound or dynamic programming For a
pallet loading problem the containers are often homogeneous In this case a solution for
one container can b e applied to all the others
Layout techniques cover a wide range of metho ds In some cases exact techniques from
mathematical programming such as linear or dynamic programming are appropriate for
all or part of the problem Integer and mixed integer programming are still considered
to o slow for even mo derate sized problems Some researchers are exp erimenting with AI
approaches such as heuristic search knowledge based systems case based reasoning and
neural networks Other researchers use improvement strategies based on meta heuristics
such as simulated annealing genetic algorithms and tabu search Other techniques in
clude database substitution greedy heuristics Monte Carlo placement lattice packings
and clustering metho ds
Our discussion of layout work treats only two dimensional layout problems b ecause the
containment problem we solve is two dimensional We b egin in the rst part of Section
with the simplest of situations in which the items are rectangular Next we consider
arbitrary convex items in the second part of Section Section discusses nonconvex
items The amount of published research on nonconvex items in nonconvex containers is
quite small compared with the amount of research on convex items Finally we discuss in
12
Tabu search is a usually deterministic neighborho o d search technique with mechanisms for managing
restrictions on tab o o regions of the search space and for keeping historical information Ree
Section how the layout literature treats the assignment problem that is the problem
of assigning items to di erent comp onents of a container The assignment problem is a
combinatorial optimization problem which can arise as a subproblem of containment when
the container consists of multiple non identical comp onents
Convex Items
Here we review layout literature related to convex items The rst part of Section
surveys work on rectangular items and the second part of Section discusses work on
convex items which need not b e rectangular
Rectangular Items
Dyckho Dyc observes that most problems in the cutting and packing literature
deal with shap es which can b e sp eci ed using a small number of parameters such as the
rectangle Research into problems for which b oth the container and items are rectangular
is reviewed in DF DD JGJ
Co man et al JGJ survey approximation algorithms for bin packing The survey
fo cuses on approximation algorithms which are e ective in practice and simple enough so
that their worst case or average case p erformance has b een analyzed in the literature These
include for example First Fit Next Fit Best Fit and their variations One dimensional
bin packing was de ned ab ove in Section In two dimensional strip packing a semi
in nite vertical strip of unit width is given along with a set of rectangular items of width
The goal is to pack items on the strip while minimizing the height of the strip The
classical two dimensional bin packing problem JL is a variant on strip packing in which
horizontal b oundaries are added to the strip at integer heights Few published analyses
exist for the case where the bins are not identical
Israni and Sanders IS present heuristics for packing non identical rectangles One
uses decreasing length p erp endicular strip placement DLPER and another uses decreas
ing height p erp endicul ar strip placement DHPER These heuristics essentially place the
rectangles in nested L shap ed strips In IS Israni and Sanders compare the p erfor
mance of DLPER and DHPER with other bin packing heuristics
Some sp ecial cases of rectangular two dimensional packing for cutting sto ck problems
lend themselves to linear and or dynamic programming for example when guillotine cuts
are used exclusively DD Gilmore and Gomory GG solve some restricted types of
guillotine cutting problems using a linear programming formulation whose sp eed relies on
solving knapsack problems the knapsack problems can b e solved using dynamic program
ming Da gliand Tato glu DT observe that if each item has a weight and if at least one
dimension of the items and containers is equal the two dimensional problem reduces to a
one dimensional single container problem which can b e solved with dynamic programming
13
An approximation algorithm for an optimization minimization problem has the prop erty that the
value of its output is times the optimal minimum value This is di erent from the de nition of an
approximate algorithm in which the solution generated by the algorithm is accurate to within a tolerance
For example a containment algorithm which pro duces an approximate con guration is an approximate
algorithm b ecause the con guration may contain small overlaps
14
We remind the reader that a guilloti ne cut is a horizontal or vertical cut which completely partitions
the remaining material
In AA Adamowicz and Albano lay out rectangles by rst forming candidate strips
and then selecting a subset of candidate strips using dynamic programming Haims and
Freeman HF also use dynamic programming to place rectangles they use linear pro
gramming and two dimensional dynamic programming and take a multi stage approach
Convex Items
Non rectangular packing results have app eared primarily for problems that pack identi
cal copies of a convex p olygon It is well known that the densest packing by translates of a
convex p olygon is given by a lattice packing Rog In a lattice packing the p olygons are
placed at p oints on a lattice A lattice is determined by two linearly indep endent vectors a
p oint on the lattice is an integral linear combination of the two vectors The densest lattice
packing of an n vertex convex p olygon can b e found in O n time MS b A double lattice
packing is the union of two lattice packings such that a rotation of either of the two lattice
packings by pro duces the other lattice packing The densest double lattice packing can
b e found in O n time for any convex p olygon that is allowed to rotate by Mou The
p
density of such a packing is KK Dori and Ben Bassat DBB show how to
nd an appropriate circumscribing hexagon for a convex p olygon in order to nd a lattice
packing
Nonconvex Items
Dowsland and Dowsland DD survey work on irregular nesting problems highlighting
the di erent types of approaches which have b een taken No fast high quality solutions
have emerged to date for layout problems involving a variety of nonconvex items Many
algorithms for packing nonconvex ob jects are unpublished QS suggests this may b e
due to commercial con dentiality Published work with nonconvex items app ears to b e
narrowly fo cused on the particular application under consideration
Published metho ds for packing nonconvex ob jects can b e classi ed as in AS as
indirect or two stage where nonconvex items are rst enclosed in convex p olygons and
techniques for packing convex ob jects are then applied and direct where the nonconvex
p olygons themselves are manipulated In the following discussion we restate more gen
erally in terms of clustering an item or collection of items is replaced by an approximation
and placement is p erformed on the approximations We further sub divide as follows a
prepro cessing clustering o ccurs only as a prepro cessing step and b integration clustering
is integrated into the layout algorithm
Clustering
Prepro cessing One p opular prepro cessing metho d encloses p olygons in rectangles The
advantage of this is the abundance of literature on packing rectangles the disadvantage is
that the resultant waste typically makes this approach unacceptable in practice For rect
angular containers Adamowicz and Albano AA Haims and Freeman HF Amaral
et al ABJ Da gli and Tato glu DT and Bailleul et al BTS pack nonconvex
items into rectangles to form comp osite items and then place the rectangles In AA
Adamowicz and Albano give an algorithm for pairwise clustering of irregular shap es in a
15
Packing density is the ratio of used area to total area
rectangle and describ e limited clustering options for more than two items They assume the
shap es can rotate The rectangles are placed in strips using dynamic programming Linear
and or dynamic programming are not p erceived as feasible options unless nonconvex items
are enclosed rst within rectangles Amaral et al ABJ adopt the approach of AA
for clustering items
B ohme and Graham BG enclose each item within another p olygon and then use
Monte Carlo metho ds to place each p olygon onto rectangular metal plates It has b een
suggested that a nonconvex p olygon b e approximated by its convex hull if the waste is not
to o large CG DBB If the waste is large Dori and Ben Bassat DBB suggest that
small groups of nonconvex items b e combined into a convex ob ject and then convex packing
techniques can b e applied This approach is also followed by Karoupi and Loftus KL
However this leaves op en the problem of how to combine nonconvex items automatically
into a convex or nearly convex one Heistermann and Lengauer HL observe that it is
not known what conditions a comp osite part must satisfy in order to b e useful in nesting
If the comp osite convex ob jects are di erent from each other is not clear that algorithms
for packing non identical convex ob jects have advanced enough to make such an approach
practical DD
If a large set of nonconvex items to b e placed on a rectangle or rectangular strip
contains many identical items an e cient layout can often b e created using rep eating
patterns For some item shap es lattice or double lattice packings are appropriate However
it is not necessarily true that the densest packing of identical nonconvex p olygons is lattice
based BK
Prasad and Somasundaram PS and Babaev Bab follow the rep eating pattern
approach for nesting sheet metal blanks They form an e cient cluster of items then build
a rep eating pattern Ismail and Hon IH use a genetic algorithm to cluster two similar
shap es together
To reduce waste an indirect metho d can approximate each p olygon using a collection
of p olygons For example Qu and Sanders QS approximate nonconvex items by non
overlapping rectangles then use a greedy layout algorithm to arrange the rectilinear items
into layers of descending staircase patterns They assume the container is a set of rectangular
sto ck sheets
Tanaka and Wachi TW place horizontally convex approximations of garment pieces
on a rectangular strip using heuristic search They rep ort material waste of to
Nakajima and Hayashi NH improve up on the algorithm of TW using several tech
niques they obtain slightly less waste than TW
Heistermann and Lengauer HL use a exible approximate representation for noncon
vex items and a nonconvex container To compute the approximation they rst construct
a set of triangles along the b oundary then iteratively smo oth the contour selecting at each
step the smallest area triangle This technique could b e used in our containment work
Integration Some work rests on the observation that go o d lo cal nesting often pro duces
go o d global nesting Da gli and Tato glu DT cluster pairs of items in a greedy fashion
They rst cluster two items make them a unit cluster this unit with another item and so
on Babaev Bab also places non identical nonconvex items this way Dighe and Jakiela
DJ present a hierarchical genetic algorithm The algorithm uses a binary tree in which
each item together with values for its three parameters x y and constitutes a leaf and
a no de represents a search for the minimal enclosing rectangle of its two children Thus
the genetic algorithm seeks the b est binary tree for a set of items They note that genetic
algorithms rely on the hypothesis that combining two go o d building blo cks leads to another
go o d building blo ck
Milenkovic et al MDL take advantage of the large small item dichotomy and
natural equivalence classes in pants layout problems To place large items they build
column clusters of four items and form a grid like layout consisting of a set of columns An
algorithm based on dynamic programming and allowing backtracking adds one column at
a time to the layout The technique of clustering items into columns also app eared in the
work of Gurel at IBM in the s Gur b Gur a Gur
Heckmann and Lenguaer HL use dynamic approximations for items As temp erature
decreases in their simulated annealing pro cess they use tighter approximations They
rep ort that this approach can reduce running time by up to This idea of using tighter
approximations later in the pro cess is similar to the use of convex hulls in Mil and in
our hybrid containment algorithm as sub division pro ceeds the convex hulls b ecome b etter
approximations to the sub divided p ortions of the con guration spaces
Direct
Dealing directly with an assortment of nonconvex p olygons is an extremely di cult
problem If many items are similar in shap e and size it can b e useful to partition the set
of items into equivalence classes For example if a large small item dichotomy exists it is
common to place all the large items rst and then place the small ones Bab MDL
ABJ
In some applications the problem de nition sp eci es equivalence classes For woo d
OF and sheet metal cutting PS BG Bab there is often more than one copy of
each item and so an equivalence class might contain all identical items Items in apparel
manufacturing are usually similar in shap e and size b ecause a layout typically contains a
mixture of sizes for the same garment style Heistermann and Lengauer HL use topology
classes of nonconvex p olygons to help select the next item to b e placed on a leather hide
using a greedy strategy One of the ways Nakajima and Hayashi NH improve up on the
algorithm of Tanaka and Wachi TW is to apply the Group Technology manufacturing
principle to classify items according to a criterion such as size Group Technology is de ned
by Mo on and Kao MK as a manufacturing principle and practice of identifying simi
larities among diverse parts forming part families classifying new parts into part families
and e ciently storing and retrieving information ab out part families See HW for more
details We note that grouping together similar items can have b ene ts for containment
see the discussion of symmetry breaking and subset substitution restrictions in Section
Some of the earliest work in direct placement of nonconvex items was done by IBM
and used a greedy approach to placement see for example MH An early nongreedy
approach was taken by Adamowicz Ada Albano and Sappup o AS note that the
exp erimental system in Ada involves iterative application of linear programming For
problems of nontrivial size techniques of mathematical programming have b een viewed as
to o slow even in cases where they can b e applied Integer and mixed integer formulations
for exact solutions have to date led to programs whose running time is unacceptably
slow AI and meta heuristic techniques can pro duce layouts with low waste but again at
the cost of much computation time e g several hours even for or items
Some researchers are exp erimenting with AI approaches such as heuristic search case
based reasoning knowledge based systems and neural networks For example the Clavier
system describ ed in HT uses cased based reasoning to arrange comp osite parts in a
convection auto clave oven for a curing pro cess Yuzu et al YLW present an exp ert
system for placing nonconvex p olygons in the garment industry Albano and Sappup o
AS pack assortments of nonconvex p olygons using heuristic search techniques They
allow nonconvex items to rotate by and they pack them onto a rectangle They control
the number of no des expanded in their search tree by for example introducing a waste
parameter They present results for the value of waste for some very small apparel
examples containing no more than items They assume the container is a single rectangle
large enough to contain all the items
Other researchers use improvement strategies for nonconvex p olygons based on meta
heuristics such as simulated annealing genetic algorithms and tabu search A central
question for such metho ds is how to reconcile the two comp eting goals of minimizing waste
and eliminating overlaps Some researchers explicitly forbid overlap others allow it but
assign a p enalty for it For example the Dowsland survey discusses the tabu search algo
rithm of Blazewicz et al BHW this algorithm forbids overlaps Both approaches rely
on fast overlap detection The latter also requires p owerful overlap reduction which until
the recent work of Li and Milenkovic Li ML LM c LM a has not b een available
This thesis o ers an even more p owerful overlap reduction heuristic recall our discussion
of the overlap reduction evaluation metho d within the hybrid containment algorithm in
Section
The Dowsland survey DD discusses some simulated annealing work such as that of
Oliveira and Ferreira OF We augment the Dowsland list with two more examples
Mangen and Lasudry use simulated annealing for marker making in the garment industry
ML They forbid overlaps Heckmann and Lengauer HL apply simulated annealing
to textile manufacturing They allow overlaps and use a p enalty term In b oth of these
cases items are nonconvex and the container is a rectangular strip In the latter case
the nonconvex p olygons which are laid out are nonconvex approximations of the original
items Heckmann and Lengauer obtain waste levels comparable to human levels for small
numbers of items Achieving this requires multiple runs of their algorithm and from
one to ve hours of computing time Heckmann and Lengauer also use simulated annealing
to solve a fteen piece puzzle in seven hours in this case the container is a rectangle
Heckmann and Lengauer compare their simulated annealing results to results they obtain
using search techniques closely related to simulated annealing such as Threshold Accepting
TA The primary di erences b etween TA and simulated annealing are TA has only
one parameter and it accepts every new con guration which is not much worse than
the old one Due
Dighe and Jakiela DJ survey the published work on genetic algorithms for layout
and introduce genetic algorithms capable of placing irregular shap es They forbid overlaps
b etween items In addition to the hierarchical clustering algorithm describ ed earlier Dighe
and Jakiela give an alternate genetic algorithm which is based on ordering the items and
adding them one at a time to the layout without overlap using a leftmost heuristic In this
case the genetic algorithm seeks the b est ordering of the items
Li and Milenkovic Li ML LM c LM a introduce a substitution based approach
to pants marker making for the garment industry Given a database of markers a new
marker can b e created by nding a template marker in the database whose items are similar
16
Marker making is the pro cess of laying out pattern pieces on cloth
to the new ones arranging the new items in the same pattern as those of the template and
then eliminating overlaps This metho d is appropriate for applications in which there is
a large collection of previous layouts and the items and containers are su ciently similar
from one layout to another It is not appropriate in general for sheet metal work where
the layouts are di erent enough from each other that humans nd it di cult to develop
exp ertise It is also not appropriate for leather nesting problems b ecause each hide is
di erent with resp ect to b oth shap e and the lo cation of defects HL
Assignment
The assignment problem is a combinatorial optimization problem which can arise as
a subproblem of containment when the container consists of multiple comp onents which
are not identical and are considered separately If the layout algorithm treats comp onents
individuall y it must decide which items to assign to each comp onent of the container Arb el
Arb identi es two di erent assignment strategies pre packing where feasible groups
for each container are generated and then assignment is p erformed and post packing
where groups are assigned based on some estimate of e ciency without determining their
feasibility
There is a rich literature on combinatorial optimization See CMTS PS Law
LLKS for discussions of exact metho ds and Ree for a treatment of mo dern heuristics
In spite of this discussions of assignment in the literature on nonconvex layout are rare
Often the issue is either ignored or else resolved using a greedy strategy Da gliand Tato glu
DT observe that most researchers do not handle allo cation of items among multiple
containers Arb el Arb dismisses pre packing as prohibitively exp ensive and therefore
uses p ost packing to pack nonconvex items on rectangular sto ck sheets His p ost packing
metho d uses integer programming B ohme and Graham BG who use Monte Carlo
metho ds to place items onto metal plates state that the set of candidate plates is selected
by a prepro cessor neither this algorithm nor the metho d of selecting the next plate to
pack is describ ed Qu and Sanders QS who approximate nonconvex items by non
overlapping rectangles do not indicate how their algorithm selects the next sheet to b e
packed Heistermann and Lengauer HL for whom a nonconvex container is partitioned
into quality zones use a greedy algorithm which treats one region at a time
To our knowledge no successful pre packing strategies app ear in the literature
Our Layout Work
We claim that the ability to quickly solve containment problems leads to improved
metho ds for solving a variety of layout problems Recall that our containment approach
is a b ottom up one in which we increase the practical value of k Clustering techniques
for layout also use a b ottom up approach Some of them even allow the items to rotate
However the metho ds which cluster more than two arbitrary nonconvex p olygons are greedy
heuristics Our containment based solutions to minimal container problems do not allow
the items to rotate but in contrast to the heuristics we establish an actual lower bound
on the size of the container for multiple items
In addition we b elieve that some p otentially promising layout techniques have b een
abandoned by the layout community b ecause they lack strong b ottom up metho ds The
primary example of this is pre packing If the layout world had a practical metho d for
generating groups of items which t into each container this plus an appropriate assignment
strategy would make pre packing viable We demonstrate through our containment based
solutions to maximal item problems that pre packing can work
Before describing our applications in more detail we pause to argue that in general
feasibility problems can help to solve optimization problems We also show how to construct
geometric optimization problems from a containment problem
One need only lo ok as far as the Russian ellipsoid metho d in linear programming Kha
to nd a p owerful example of how solving feasibility problems can facilitate solving opti
mization problems In addition a feasibility problem is a decision problem which can
b e used as an oracle in order to solve problems involving a parameter For example one
can p erform binary search on the value of the parameter or in some cases one can use
Meggido s parametric search technique Meg The binary search approach has b een sug
gested by Chazelle Cha in the computational geometry literature as a way to nd the
maximum scaling factor which allows a p olygon to t inside another p olygon In the litera
ture on combinatorial optimization Reeves Ree discusses binary search and comments
However we would not usually try to solve an instance of an optimization problem by
this means Our b elief is consistent with b oth of these views Binary search is valuable
but pure binary search alone is insu cient The b est approach in practice accelerates the
binary search by using an improvement strategy typically a heuristic to help up date the
upp er b ound
This metho d provides one way to establish a lower b ound for a solution to a minimization
problem The b ound can b e compared with a value obtained by a heuristic The ability to
solve a feasibility problem thus makes it p ossible to evaluate the e ectiveness of a heuristic
whether or not the heuristic is automated Evaluating the e ectiveness of heuristics is a
challenging and imp ortant task Other techniques exist for establishing lower b ounds for
some types of optimization problems For example in the area of combinatorial optimiza
tion lower b ounds for minimization problems can b e found using relaxation techniques
Ree
We construct two di erent types of geometric optimization problems from a containment
problem The rst type is a minimal container problem the second is a maximal item
problem Given a collection of items and a particular type of container a minimal container
problem asks for the minimal container of that type which contains the given items
Similarly given a container and a particular collection of item characteristics a maximal
item problem asks for the maximal collection of items which p ossesses those characteristics
and which also ts into the container
The thesis gives examples of layout problems of each type which can b e solved using
two dimensional translational containment for irregular p olygons We give four examples
of minimal container problems strip packing nding minimal square enclosures for
items nding minimal rectangular enclosures and nding tight convex and nonconvex
enclosures
From the discussion of related work in layout in Section we see that solutions to
17
The Russian ellipsoi d metho d is not currently regarded as practical but it was the rst p olynomial time
algorithm for linear programming
18
Reeves et al Ree discuss two types of relaxation metho ds linear programming relaxation for
integer or mixed integer programming which can either b e solved heuristicall y using dual ascent or exactly
and Lagrangean relaxation
19
One p ossible minimality criterion is area
minimal enclosure problems are often required for the rst clustering stage of a multi stage
layout problem Minimal container problems also arise when nding the b est sto ck sheet
size for a given set of items or the b est cloth width in garment manufacturing Section
discusses computational geometry work on various types of enclosure problems for single
items Current solutions to minimal enclosure problems for more than two items are rare and
heuristic This work app ears in the layout literature see Section and Section
The problem is typically solved either incrementally by adding one item at a time DT
or by placing small items in gaps inside the b ounding b ox of two large items AA
Our general approach to nding minimal enclosures is to nd an optimal value for a
parameter via binary search combined with containment The binary search can b e ac
celerated by using compaction techniques such as those developed by Li and Milenkovic
Li ML LM c LM a to help up date the upp er b ound i e to improve a feasible solu
tion found by containment Our hybrid containment algorithm is appropriate for minimal
enclosure problems It is extremely fast for lo ose ts and so the initial stages of the search
are fast
For the problem of nding tight convex and nonconvex enclosures we show that re
moving unreachable p ortions of a tight rectangular enclosure can yield tight convex and
nonconvex enclosures which preserve all the containment solutions for the rectangle
We also give four examples of maximal item problems For this discussion let k
pr actical
b e the largest number of p olygons for which current containment algorithms are practical
Our problems are for a feasible containment problem nding maximally scaled copies
of items which t in the container given a container and a collection of k items
pr actical
for which the sum of the item areas do es not exceed the area of the container nding a
large sub collection which ts in the container given a container and a collection of
items for which either the sum of the item areas exceeds the area of the container or the
number of items is k nding a large sub collection of k items which
pr actical pr actical
ts in the container and problem for a collection of containers
For problem we use nested binary search combined with containment The key to
solving problem is to carefully choose which item to eliminate from an infeasible subset
We reject the most troublesome item which is identi ed using the results of the overlap
elimination evaluation metho d of the hybrid containment algorithm For problem we
randomly construct a set of items and solve problem This yields a rich collection of
n
sets which t into the container For each set of size n which ts we automatically have
subsets which t and which need not b e individual ly represented This work provides the
rst practical metho d for constructing sets of up to ten p olygons which t in a container
For problem we solve problem for each container and then apply a lo ok ahead
assignment strategy This is a pre packing strategy Containment thus allows us to trans
form a geometric optimization problem into a combinatorial optimization problem Our
results challenge the existing b elief that pre packing metho ds are impractical for layout
Arb
In a multi stage layout problem problem might involve a large highly nonconvex
container Highly nonconvex containers also app ear in industries in which defects in material
cause the material to b e partitioned into quality zones see Heistermann and Lengauer s
work with leather HL We show how to decomp ose such a container in a natural fashion
transforming this problem into problem The transformation removes p ortions of the
container which are clearly not reachable by any item Our decomp osition metho d uses
an algorithm we developed for nding the maximum area axis parallel rectangle MAAPR
in a p olygon Our theoretical work on this problem gives the fastest known asymptotic
running time for a MAAPR algorithm for p olygons
Pro ject Background
The work in this thesis is part of a pro ject whose goal is to automate the marker making
pro cess in the U S apparel industry Marker making is the pro cess of laying out pattern
pieces on cloth so as to minimize cloth waste As an example Figure depicts a multi
stage marker making problem from the apparel industry At the top is a collection of large
pants panels and a set of smaller trim pieces The marker making task is a strip packing
problem arrange these items on a semi in nite strip of material while minimizing the length
of the strip
Our marker making research has b een carried out under the direction of Victor J
Milenkovic Assistant Professor of Computer Science at Harvard University The research
is part of a larger pro ject funded by the Sloan Foundation The principal investigator
of this pro ject is Frederick H Ab ernathy Professor of Mechanical Engineering at Harvard
University and director of the Harvard Center for Apparel and Textile Research
Humans generally place the large panels rst and then place the trim in the gaps formed
by neighboring panels Thus they rst solve a panel placement problem followed by a trim
placement problem A layout of panels is shown in the gure followed by the complete
layout pro duced by a human exp ert In this example the items may translate and may also
rotate by a discrete set of angles
The goal of our pro ject is to meet or exceed human p erformance for the pants marker
making problem One way to do this is to apply an improvement strategy to an existing
layout pro duced by a human Another way is to generate a new layout from scratch using
for example a two stage approach such as the one adopted by human marker makers
Accordingly we initially partitioned our pro ject into three subpro jects compaction
panel placement and trim placement Below we brie y describ e these three parts of our
pro ject
Compaction
Li and Milenkovic s work on compaction has b een a signi cant part of our pro ject Li
ML LM c LM a Given a layout their algorithms use linear programming to improve
the layout through continuous motion of the pieces A p osition based linear programming
mo del is at the heart of the algorithms The algorithms accomplish a variety of tasks One
task is to reduce the length of a marker This type of compaction can b e used to accelerate
binary search in our work on minimal enclosures Another task is to reduce overlaps among
pieces This is an integral part of the substitution based approach to pants marker making
describ ed in Section Thus compaction can serve b oth as an improvement metho d for
existing layouts and as the basis of a metho d for constructing layouts The overlap reduction
evaluation metho d we apply within our hybrid containment algorithm is an extension by
Milenkovic of the overlap reduction work
20
Victor Milenkovic is now an Asso ciate Professor at the University of Miami Set of polygonal parts
Strip of fixed width and unknown length
Placement of large pieces
Name: 37457c Width: 59.75 in Length: 269.04 in Pieces: 108 Efficiency: 89.54%
Human layout with 180 degree rotations and xy-flips allowed
Figure The marker making task in the apparel industry
Panel Placement
Section summarized the work of Milenkovic MDL on placing large items by
building column clusters To place large items he builds column clusters of four items and
forms a grid like layout consisting of a set of columns An algorithm based on dynamic
programming adds one column at a time to the layout Backtracking is allowed
Trim Placement
The trim placement task falls within the two dimensional bin packing category accord
ing to Dyckho and Finke s typology DF One might argue that it also b elongs in the
cutting sto ck category b ecause the items can b e partitioned into equivalence classes How
ever the items in each class are not congruent they are graded copies of each other Trim
placement is not solved using rep eating patterns and so it do es not t Dyckho and Finke s
cutting sto ck type However approaches to trim placement can certainly take advantage of
the similarity b etween pieces in the same equivalence class
Trim placement can b e viewed as a containment problem for which the container is the
complement of the placed panels within the original container The container is highly
nonconvex and has multiple comp onents The number of items far exceeds the number
which can currently b e handled by any practical containment algorithm However practical
algorithms for solving the asso ciated packing problem i e place as many pieces as p ossible
can b e developed if we rst decomp ose the container into a collection of containers where
each container s capacity is appropriate for a containment algorithm
This trim placement problem provided the original motivation for our work on contain
ment Most of the containment examples in this thesis are drawn from various marker
making problems in the apparel industry
Surprisingly we found that the problem of decomp osing a complex container is a chal
lenge in its own right We therefore discuss container decomp osition in a separate chapter
Chapter within the applications part of the thesis
Benchmarks
As we have already discussed our test data comes primarily from apparel manufacturing
We welcome data from other sources We prop ose that a set of b enchmarks b e constructed
so that researchers may compare the p erformance of their translational containment algo
rithms We o er to provide data for our containment examples to interested researchers
providing that the data is not used for commercial b ene t To facilitate this exchange while
protecting the source of the data we o er our data using a simple le format which is given
in App endix A
Contribution
The primary contribution of this thesis is our demonstration that although containment
is NP hard it is fruitful to develop algorithms for containment as opp osed to heuristics
design containment algorithms so that they say no almost as fast as they say yes
21
In apparel manufacturing grading is a combination of scaling and morphing
use geometric techniques not just mathematical programming techniques and maximize
the number of items for which the algorithms are practical
Our approach to containment has a number of imp ortant features First we intro
duce a restrict evaluate subdivide paradigm Second we develop new theory and practical
techniques for the op erations within the paradigm The techniques are appropriate for two
dimensional containment problems in which the items and container may b e irregular i e
nonconvex and have multiple comp onents and in which the items may b e translated but
not rotated Our techniques can b e combined to form a variety of two dimensional transla
tional containment algorithms which follow the restrict evaluate subdivide paradigm The
paradigm is designed so that unlike existing iteration based algorithms containment algo
rithms based on the paradigm are adept at saying no even for slightly infeasible problems
Infeasible problems o ccur frequently in practice
We develop geometric restrictions which are p owerful con guration space pruning tech
niques They often detect infeasibility without any evaluation or sub division We give two
algorithms which use geometric restrictions and are based on the restrict evaluate subdivide
paradigm We obtain the rst practical running times for NP complete two dimensional
translational containment problems for up to ten nonconvex items in a nonconvex container
Our examples are drawn primarily from the apparel industry Each item is represented as
a p olygon which typically has from to vertices The container is also represented as
a p olygon it typically has from to vertices
We demonstrate that viewing containment as a feasibility problem has many b ene ts
for layout optimization problems We present an e ective metho d for nding minimal
enclosures which uses containment to p erform binary search on a parameter We note that
compaction can b e used to accelerate the search This work represents the rst practical
approach to nding minimal enclosures for multiple nonconvex items Clustering is used
often in layout but current clustering techniques are weak Although they often allow
items to rotate they typically use greedy heuristics and therefore cannot provide a lower
b ound on the size of the container Our minimal enclosure work should lead to more
p owerful clustering metho ds and is an example of how strong b ottom up techniques
from computational geometry can help the layout community
We b elieve that some p otentially promising layout techniques have b een abandoned by
the layout community b ecause they lack strong b ottom up metho ds The primary example
of this is pre packing We challenge the view that pre packing is impractical for multiple
container packing problems by generating a rich collection of groups for each container
using containment and then applying an assignment strategy
Our advances in containment will help automate layout for industrial pro cesses that rely
on two dimensional packing of ob jects Our results are particularly relevant to industries
which deal with irregularly shap ed items This includes industries such as aerospace ship
building apparel and sho e manufacturing furniture pro duction and steel construction
Overview
The remainder of the thesis is divided into two main parts Part I describ es our
work on translational containment for nonconvex items in a nonconvex container This
part contains four chapters The rst chapter introduces our restrict evaluate subdivide
paradigm for containment and provides the notational and technical background required by
our containment work The second chapter describ es our con guration space restrictions
The third chapter discusses evaluation and sub division techniques The fourth chapter
presents two di erent algorithms which can b e formed via sp eci c choices of restriction
evaluation and sub division techniques Portions of our containment work also app ear in
DML DM DM MD
Part I I of the thesis gives examples of how containment can b e applied to solve various
layout problems This part contains four chapters The rst chapter discusses preliminaries
The second shows how to transform a container so that regions not reachable by any item
are eliminated and the remaining reachable regions are decomp osed in a natural way Our
container decomp osition work also app ears in DM Our theoretical work on nding the
maximum area axis parallel rectangle in a p olygon which is used for the decomp osition is
summarized in that chapter and also app ears in DMR DMR The third chapter solves
four minimal container problems The fourth solves four maximal item problems
Conclusions future work and an app endix containing a le format for containment
examples app ear in Part I I I of the thesis
Part I
Translational Containment
C h a p t e r
Preliminaries
Introduction
In this part of the thesis we consider the two dimensional translational containment
problem for p olygons The problem is stated as follows given a container and a set of
items nd translations of the items which result in a non overlapping arrangement of the
items within the container The items and container may b e nonconvex
Existing two dimensional translational containment algorithms developed by other re
searchers are practical only for k We present new theory leading to the rst practical
running times for two dimensional translational containment problems for up to ten non
convex items in a nonconvex container For k one of our algorithms typically
runs in several minutes or less on a MHz SPARCstation and it is therefore practical
for solving a number of layout related problems Most of our examples are from apparel
manufacturing Usually each item has from to vertices and the container has from
to vertices
Although containment is NP hard in general we b elieve that it is valuable to increase
the practical value of k as much as p ossible Thus we continue the b ottom up progres
sion CC NN CN NN etc of containment work which has taken place in rob otics
and computational geometry Since containment is NP hard and all known containment
algorithms have running time which is exp onential in k an increase in the practical value
of k from three to ten is signi cant
Our ability to say no well comes primarily from the use of our new geometric tech
niques Our ability to say yes for so many items is partly due to our geometric techniques
and partly due to our application of and improvements to existing linear programming based
techniques Thus we demonstrate that a computational geometry approach can contribute
signi cantly to practical algorithms for NP hard geometric problems
Saying no quickly is imp ortant Current translational containment algorithms are
iteration based Iteration based algorithms for containment attain their worst case run
ning time when the containment problem is infeasible This situation o ccurs frequently
in practice We replace iteration with sub division Our sub division b ased algorithms are
sp eci cally designed to detect infeasibility quickly
We present a new paradigm for containment algorithms The paradigm is based on
op erations on two dimensional con guration spaces As discussed in Section each of
these spaces represents the set of relative p ositions for a pair of p olygons The algorithm
applies three types of op erations in order to the con guration spaces restriction which
prunes the spaces and helps detect infeasibility evaluation which tries to nd a solution
and subdivision Sub division partitions one of the con guration spaces into two parts
The algorithm then recurses on each part while keeping the other con guration spaces
constant
We introduce new theory and techniques for each of these types of op erations for the case
of two dimensional containment problems in which the items and container may b e irregu
lar i e nonconvex and have multiple comp onents and the items may b e translated but
not rotated Our techniques can b e combined to form a variety of two dimensional transla
tional containment algorithms which follow the restrict evaluate subdivide paradigm The
paradigm is designed so that unlike existing iteration based algorithms containment algo
rithms based on the paradigm are adept at saying no even for slightly infeasible problems
At present there app ear to b e two approaches to containment within the restrict eval
uate sub divide framework direct and indirect Direct metho ds act on the con guration
spaces directly and these metho ds tend to b e based on computational geometry Indirect
metho ds act on convex approximations to the con guration spaces either inner or outer
approximations Indirect metho ds use linear programming or other types of mathematical
programming which is why they can only act on convex approximations Direct metho ds
have a more accurate view of the world but they can only op erate on a few con guration
spaces at a time three in our case Indirect metho ds can act on all the con guration spaces
simultaneously in a single op eration but they use only approximations to the con guration
spaces
This thesis presents two containment algorithms which follow our paradigm The rst
op erates directly on the con guration spaces and is therefore a direct algorithm We call the
second a hybrid algorithm b ecause it uses an indirect evaluation metho d and b oth direct
and indirect metho ds for restriction Our direct algorithm is also an approximate algorithm
b ecause it pro duces a con guration whose overlap is b elow a given tolerance Ironically
our hybrid algorithm whose indirect evaluation metho d op erates on approximations of the
con guration spaces is an exact algorithm b ecause it pro duces a con guration with no
overlap We brie y discuss each algorithm individuall y b elow
Direct Algorithm
Our new direct algorithm for containment is an approximate algorithm which pro duces
a solution whose overlap is b elow a given tolerance As mentioned in Section for some
cutting applications CAD vendors round p olygon vertex co ordinates to integers In such
cases an overlap of less than one unit corresp onding to inches is acceptable For the
applications we consider the ratio of the largest value to the smallest nonzero value is usually
under this makes an approximate algorithm a practical choice Our approximate
algorithm uses our new direct techniques for restriction evaluation and sub division The
algorithm app ears in DM DM as well as in Section
Our direct restrictions are based on p olygon set op erations and so we call them geometric
restrictions They are p owerful to ols for detecting infeasibili ty We de ne two general types
of restrictions valid restrictions which preserve all solutions to a containment problem and
semi valid restrictions which preserve a single solution if one exists We present a variety
of restrictions and several general prop erties of valid restrictions We examine in detail the
e ectiveness of our geometric restrictions which are valid restrictions Our characterization
of the types of containment problems for which these restrictions are e ective is based on
theoretical as well as exp erimental results
The evaluation metho d uses restriction together with a geometric metho d for maxi
mizing the intersection of translations of the con guration spaces The evaluation metho d
often nds a containment solution without any sub division The sub division metho d is a
size based one Its goal is to partition a con guration space into two parts such that each
has less area than the con guration space Our approximate algorithm has an extra layer
of structure on top of the restrict evaluate subdivide paradigm This layer has a set of
containment subproblems to which course grained parallelism could b e applied in the fu
ture We create the subproblems by characterizing a valid con guration for translational
containment
The current implementation of this algorithm is fast for up to four items For k
the current implementation of our geometric restrictions based on p olygon set op erations
slows the algorithm down considerably Increasing the sp eed of our p olygon set op erations
is a sub ject of future work see Section In our exp eriments our approximate
algorithm p erforms fewer sub divisions than the faster hybrid algorithm and so we b elieve
the approximate algorithm has the p otential to b e comp etitive with the hybrid algorithm
Hybrid Algorithm
In addition to our direct algorithm we also present signi cant improvements to the
restriction evaluation and sub division techniques of an existing indirect algorithm of
Milenkovic Mil Milenkovic s indirect algorithm in its original form was slower than our
direct algorithm for infeasible problems and its practical upp er b ound on k was k The
improved indirect algorithm is currently our fastest containment algorithm for k
Our rst improvement to the indirect algorithm is a pro of that restrictions when used
in the indirect algorithm do not increase the algorithm s worst case running time if the
algorithm bases certain decisions on the unrestricted con guration spaces This is imp ortant
b ecause restrictions can add vertices to the con guration spaces and so they might have
increased the worst case running time The pro of allows us to add our new direct geometric
restrictions to the algorithm It also allows us to apply a new restriction metho d developed
by Milenkovic which is based on linear programming
The second improvement to the indirect algorithm is related to the evaluation metho d
We apply the work of Li and Milenkovic on overlap resolution Li ML LM c LM a
to improve the evaluation step Their overlap resolution algorithm is an indirect algorithm
It uses inner convex approximations an approach which complements the outer approxi
mations of the indirect containment algorithm Our third improvement is our new practical
sub division technique which attempts to maximize the distance from the current evaluation
to the convex approximations of the sub divided con guration spaces
We call the resulting algorithm a hybrid algorithm b ecause it uses indirect metho ds
for evaluation developed by Li and Milenkovic and b oth direct and indirect metho ds for
restriction The hybrid algorithm is describ ed in MD as well as in Section
The hybrid algorithm in addition to b eing a formidable containment algorithm has
the prop erty that a partial containment solution from it is valuable This is b ecause each
evaluation step pro duces either a non overlapping solution or an overlapping con guration
for which a linear program has reached a lo cal minimum As a result the hybrid algorithm
when terminated early is the most p owerful heuristic known for generating a lo cal overlap
minimum
Overview of Chapter
In this chapter we rst give notation and technical background in Section Sec
tion establishes the NP completeness of translational containment Section describ es
our restrict evaluate subdivide paradigm Section introduces the concept of size analy
sis Section supplements the discussion of related work in containment which app ears in
Section Section presents a characterization of translational containment problems
Section outlines the contents of the remaining chapters in this part of the thesis
Notation
The goal of translational containment is to translate k p olygonal regions P P P
k
into a p olygonal container without overlap If we denote by P the complement of the
container region C then containment is equivalent to the placement of k p olygons
P P P P in non overlapping p ositions
k
De nition Min GRS Ser Ser The Minkowski sum of two point
sets of R in our case is
A b A B fa b j a A b B g
b B
The Minkowski di erence is
A B A b
b B
For a p oint set A let A denote the set complement of A and de ne A f a j a Ag
For a vector t de ne A t fa t j a Ag and A t fa t j a Ag Note that
A t A ftg
Theorem Ser A B A B
Theorem states that the Minkowski sum and di erence are duals with resp ect to
complementation
Theorem GRS Ser B t A if and only if t A B
Figure illustrates Theorem Readers familiar with mathematical morphology
will recognize this as the hit transformation Ser There is an analogous miss trans
formation involving which is illustrated in Figure and stated in Theorem
Theorem Ser B t A if and only if t A B B A + (-B) A
-B
Figure B t A if and only if t A B
B
-B
A - (-B)
A
Figure B t A if and only if t A B
De nition Ser The op ening of A with respect to B is A B B
If B ts into A then the op ening of A with resp ect to B consists of all p oints of A which
are reachable by some part of B via translation
The symbol U i j k i j denotes the set of displacements from P to P
ij i j
such that they do not overlap each other For translations t and t P t and P t
i j i i j j
overlap if and only if t t U It follows that U U Since P cannot b e displaced
j i ij ij j i i
with resp ect to itself we de ne U f g Expressed in terms of set op erations and
ii
Minkowski sums
U P P i j k i j
ij i j
Equation follows directly from Theorem The set U is the two dimensional free
ij
con guration space for placing P with resp ect to P Note that U is unbounded unless
j i ij
i or j assuming that the container and P P are b ounded
k
Let P and U denote the set of all P and U resp ectively Strictly sp eaking these
i ij
are lists even if two elements such as U and U are congruent they are maintained as
separate elements
De nition A con guration of P is a k tuple t t t where t is a trans
i k i
lation x y of P and t is arbitrarily set to A valid con guration of P is a
i i i
con guration of P which satis es
t t U i j k
j i ij
A valid con guration is an exact solution to a translational containment problem It
corresp onds to a p oint in k dimensional space The set of p oints corresp onding to all valid
con gurations for a given P is the k dimensional free con guration space for P
De nition An approximate con guration of P is a con guration such that the
distance of t t from U is for i j k
j i ij
In an approximate con guration no p oint of any p olygon in P is more than inside
of the b oundary of any other p olygon when the p olygons are laid out according to the
con guration
De nition Given U U for an instance of translational containment U is the set
ij
ij
of al l values of t t such that t and t are translations of P and P respectively in a
j i i j i j
valid con guration of P
is the pro jection of the set of all k dimensional valid con gurations In other words U
ij
into U Clearly U U Also U U implies U U Let U denote the set of
ij ij ij j i
ij ij j i
all U
ij
In Section we de ned the k NN problem the k r NN problem and their variations
see De nition and De nition We refer to the entire collection of two dimensional
translational containment problems including k NN k CN etc as translational contain
ment problems Note that we do not require the set of al l valid con gurations a single
valid con guration su ces Each problem has an approximate version which is to nd an
approximate con guration Each containment problem is an o line problem b ecause the
entire set P is available at the start of the containment problem
Let the number of vertices of P b e m We de ne m max m We use m
i i i k i
frequently and it tends to b e larger than the other m Hence we distinguish m by
i
denoting it by n In general we express the running time of a containment algorithm in
terms of m n k and r
This is not the most general containment problem since it is restricted to two di
mensions do es not allow rotation and the representation of items is limited to p olygons
However this problem is still of great practical imp ortance Industries dealing with two
dimensional p olygonal items include aerospace ship building apparel sho e and furniture
manufacturing and steel construction In applications such as apparel sho e and furniture
manufacturing only a discrete set of rotations is typically allowed this can b e dealt with
by iterating over the angles In contrast metal pieces are often represented using curves as
b oundaries and continuous rotation is frequently allowed
Hardness
The main result of this section is that k NN is NP complete Before proving this we
establish two lemmas The rst shows that k RR a containment problem for rectangular
items in a rectangular container is NP hard
Lemma The k RR problem is NP hard
Pro of Li Li examines the two dimensional co ordinated motion planning problem of
strip compaction in which the motion of the p olygons is restricted to a strip of xed width
and arbitrary length and the goal is to minimize the length of the strip He shows this
problem is NP hard when all the p olygons are rectangles and only translations are allowed
He reduces the NP complete PARTITION problem to strip compaction We use a similar
reduction here to establish that k RR is NP hard In an instance of PARTITION GJ
we are given a nite set A and a size s a Z for each a A We ask Is there a subset
P P
s a We reduce the NP complete PARTITION s a A A such that
a AnA a A
problem to k RR For each a A we construct a rectangle of unit height and width s a We
P
s a PARTITION has a solution also construct a container of height and width
a A
if and only if there is a valid con guration of the rectangular items inside the rectangular
container This establishes that k RR is NP hard 2
Lemma Two dimensional translational containment is in NP
Pro of Milenkovic et al MDL show that the marker making problem in the absence
of rotations is in NP The main idea of their argument applies to translational contain
ment They show that if a solution exists then there exists a p ossibly di erent solution
which is uniquely determined by its set of vertex edge contacts We observe that there are
only k degrees of freedom Therefore we can nondeterministically select the correct k
vertex edge contacts solve the set of linear equations and verify the solution This is a
p olynomial time pro cess b ecause the linear system has a p olynomial number of equations
and a p olynomial number of linear equations can b e solved in time which is p olynomial in
the number of equations 2
Theorem Two dimensional translational containment is NP complete
Pro of Lemma shows that k RR is NP hard Since all the translational containment
problems are harder than k RR this proves that translational containment is NP hard
Lemma shows that translational containment is in NP which completes the pro of 2
The Restrict Evaluate Subdivide Paradigm
The set U enco des the entire containment problem and one can think of it as the input
to and current state of a containment algorithm We refer to U as the hypothesized solution
space or hypothesis A solution within the current hypothesis consists of a valid con guration
for the current set U
Our paradigm uses three op erations on the hypothesis
Restriction prune the hypothesis by applying restrictions to the current set U A restric
tion is an op eration on U which replaces some or all U s by subsets of themselves
ij
Naturally if any U U is replaced by the empty set then there is no solution
ij
Evaluation try to nd a solution within the given hypothesis The result is that either
yes a valid con guration exists within the hypothesis no a valid con guration
do es not exist within the hypothesis or maybe a valid con guration exists but it
has not yet b een found
Sub division divide the current hypothesis into two sub hypotheses such that if a solution
exists within the current hypothesis then a solution must exist within one of the two
sub hypotheses
The pseudo co de b elow describ es the ow of a general containment algorithm which
abides by our paradigm We present it in the form of a decision pro cedure which returns
YES or NO Of course if a valid con guration is found the con guration can b e stored
in a global data structure The routines RESTRICT EVALUATE and SUBDIVIDE
p erform the functions describ ed in the list of op erations given ab ove RESTRICT returns a
restriction of U EVALUATE returns a status ag indicating whether it decides YES NO
or MAYBE SUBDIVIDE partitions one of the con guration spaces U U into two parts
ij
U and U The algorithm then recurses on each part while keeping the other con guration
ij ij
spaces constant That is it creates U U n fU g fU g and U U n fU g fU g
ij ij
ij ij
and then recurses on U and U
CONTAIN U k
U RESTRICT U k
if some U U return NO
ij
status EVALUATE U k
if status YES return YES
if status NO return NO
U U SUBDIVIDE U k
return CONTAIN U k or CONTAIN U k
A containment algorithm can b e designed by selecting a restriction metho d an evalua
tion metho d and a sub division metho d The correctness of the algorithm dep ends on the
correctness of the chosen metho ds Termination of the algorithm dep ends on the compat
ibility of the metho ds For example supp ose we have an algorithm which has a certain
worst case running time based on a sub division metho d for which the number of re ex ver
tices of U and U is less than the number of re ex vertices of U Now supp ose we add
ij
ij ij
restrictions to this algorithm The restriction pro cess may increase the number of re ex
vertices in the hypothesis This can increase the worst case running time of the algorithm
A mismatch b etween evaluation and sub division metho ds can also b e problematic For
instance consider an algorithm which uses the sub division technique mentioned ab ove for
reducing the number of re ex vertices together with a size based evaluator That is the
evaluator says no if it cannot nd a valid con guration in U and the size of U is b elow a
tolerance Sub division will eventually pro duce hypotheses containing convex p olygons but
those p olygons will not necessarily have size less than the tolerance If a valid con guration
exists within U in this case the algorithm will not terminate unless the evaluator happ ens
to nd it
Course grained parallelism can b e applied to this paradigm and is a sub ject for future
work In order to parallelize the general containment algorithm one would maintain a queue
of hypotheses instead of p erforming recursion Each pro cessor would remove a hypothesis
from the queue and restrict it and evaluate it If no solution were found the hypothesis
would b e sub divided and the two resulting sub hypotheses would b e added to the queue
Each hypothesis represents a self contained containment problem and so pro cessors could
work on hypotheses indep endently Of course the enqueueing and dequeuing of hypotheses
should b e implemented as atomic op erations
One limitation of this approach is that the subproblems would not necessarily all b e
available at the start of the algorithm Some pro cessors might remain idle until a su cient
number of hypotheses were created
Approach to Running Time Analysis
In order to analyze the worst case time complexity of a containment algorithm which
follows our paradigm one must know the time complexity of RESTRICT EVALUATE
and SUBDIVIDE In addition one needs an upp er b ound on the number of times CON
TAIN is invoked this is determined by the conditions for which EVALUATE decides
no Finally an upp er b ound on the total number of vertices of U is required
Calculating the total number of vertices of U is complicated by the p ossibility that
RESTRICT and SUBDIVIDE may introduce new vertices into U in spite of the fact
that each replaces some U s by subsets of themselves One way to treat this is to estimate
ij
the worst case upp er b ound on the number of new vertices However this upp er b ound
may b ear little resemblance to the actual worst case size which is attainable in practice
For example supp ose that RESTRICT and SUBDIVIDE use only p olygon set op er
ations such as intersection and union In theory for p olygons A and B jA B j jA B j
O jAj jB j but in practice jA B j jA B j jAj jB j On the one hand new vertices
are created by edge edge intersections but on the other hand many vertices are discarded
when the union or intersection b oundary is computed
We introduce the concept of size analysis in order to capture the practical running time
of a containment algorithm which follows our paradigm Let s b e the maximum number
1
One measure of the size of U is the total area of all U in U
ij
of vertices pro duced by a p ossibly in nite sequence of p olygon op erations which replaces U
by a subset of U In Chapter we are more sp eci c ab out the op erations In some cases
one can prove that s is nite Although s can b e exp onential in the original size of U in
practice it is usually not more than quadratic Since our analysis is based on s we refer to
size analysis as s analysis in the remainder of the thesis
Related Work
In Section we reviewed the literature on containment Here we ll in some of the
technical details omitted by that discussion
k
A naive k NN algorithm requires O k mn log mn time It searches for a valid
con guration by iterating over all choices of k contacting pairs among the p olygons and
k k
container There are O k mn k m O mn p ossible choices of k vertex edge
contacts For each choice we can solve a set of k linear equations in O k time the cost
of Gaussian elimination to obtain t i k This con guration can b e checked for
i
overlap in O log n time assuming the following prepro cessing Precompute U according
to Equation The number of vertices in each of these p olygons has an upp er b ound of
O m n Prepro cess the p olygons in U so that a p oint in polygon test can b e p erformed in
O log mn time Kir To check a layout p erform a p oint in polygon test for each t t
j i
k
in U This costs O k log mn time The naive algorithm has running time in mn
ij
b ecause it must iterate over all p ossible sets of contacts whenever the answer is no
The following running times were given in Section
k k CN k NN
O mn log mn For O m n log mn AB Avn
O m n log m Dev O m n log mn AB Avn
O m n log m Dev O m n log mn Avn
Avnaim and Boissonnat also give a solution to the NP problem three nonconvex p olygons
in a parallelogram container using time in O m log m AB Avn Even though
the asymptotic running time of the NP algorithm contains a p olynomial whose degree
is high compare to the others the NP algorithm is actually more practical than some
of the other lower degree algorithms Avnaim s NN algorithm in particular iterates
over all choices of edges from U U U U and U Avnaim uses a sweep line
algorithm to avoid iterating over the edges of U The worst case running time of the NN
algorithm is therefore prop ortional to the pro duct of the sizes of these sets To determine
that a con guration is imp ossible the algorithm must iterate over all these choices Its
O m n log mn running time is worse than the O mn log n running time for the naive
algorithm for k For reasons outlined b elow it is faster than the naive algorithm in
practice
The m factor for NP arises from the worst case b ound on the number of vertices
of the outputs of several consecutive Minkowski sums and p olygon unions intersections
In practice the worst case is usually a gross overestimate and unlike the m n lower
b ound for the NN algorithm the factor of m may not b e attainable At each step the
running time of the NP algorithm is sensitive to the output of the previous step and so the
algorithm takes advantage of the fact that the number of vertices almost certainly grows
2
See Section for a description of the NN characterization which allows this iteration
much more slowly than the estimate Recall our discussion of s analysis in Section
An observation similar to the one made ab out p olygon intersection and union holds for
the Minkowski sum Using s analysis the NN algorithm has running time in O s log s
whereas the NP algorithm has running time in O s log s The naive algorithm has running
k
time in O s log s for k this is O s log s With this analysis NP is faster than
NN and b oth are faster than the naive algorithm
Containment Characterization
Here we take advantage of the fact that the goal of our containment problem is to nd
a single valid con guration as opp osed to nding the entire set of valid con gurations We
characterize a containment solution by showing that if a valid con guration exists for
a given containment problem then a valid con guration must exist where can b e
reached from by a non overlapping motion of the pieces and p ossesses characteristics
which facilitate the search for a solution In particular is a vertex of a comp onent of
the free con guration space and this comp onent contains In this case we can solve the
containment problem by nding This type of approach has b een used for containment
by Chazelle to solve NN with rotation Cha by Avnaim to solve NN with translation
only Avn and by Milenkovic et al to solve CN DML
Our characterization of a solution to k NN stems from a small number of simple obser
vations ab out contacts and degrees of freedom First we observe that b ecause k NN allows
translation in x and y each p olygon has two degrees of freedom Second the collection of
k p olygons has k degrees of freedom
There are two types of contacts The rst type involves a vertex and an edge A vertex
edge contact is a single contact b ecause it implies a single linear constraint The second
type involves two vertices A vertex vertex contact is a type of double contact b ecause it
implies two linear constraints A linear constraint reduces the number of degrees of freedom
by one if and only if it is linearly indep endent of the existing system of constraints
In terms of U a single contact of P with P means t t is on the b oundary of U and
i j j i ij
a double contact means t t is at a vertex of U
j i ij
In our characterization b elow let I b e an instance of k NN
Theorem If I has a valid con guration then I has a valid con guration with the
fol lowing characteristics
k contacts
contacts per polygon P i k
i
Pro of Assume I has a valid con guration We rst show that there is a valid con gura
tion with k contacts by arguing as in MDL Less than k linear constraints cannot
determine the value of k variables and therefore there must b e at least one degree of
freedom There must therefore b e some simultaneous motion of the p olygons P i k
i
for which at least some subset of the p olygons move along lines while all p olygons retain
their contacts If some subset of the p olygons are moving along lines then eventually some
new contact must form One of three events must o ccur two p olygons collide neither
of which is P or two p olygons collide one of which is P or a new contact o ccurs
b etween two already contacting p olygons The o ccurrence of one of these events is guar
anteed by the fact that the complement of P is b ounded recall that P is the container
If two p olygons P and P collide then a new vertex edge contact is formed If P and P
i j i j
already have a vertex edge contact then that vertex may slide to the end of the edge and
form a vertex vertex contact We stop the motion when a new contact o ccurs Rep eating
this pro cess forms a set of k contacts
To establish the second part of the claim we observe that if a valid con guration for
I exists and a p olygon has fewer than two contacts then we can translate that p olygon
until it loses b oth degrees of freedom Translating P is equivalent to xing P and
translating the remaining P by the same amount in the opp osite direction This implies a
i
valid con guration in which each p olygon has at least two contacts 2
A containment problem can b e converted to a set of indep endent containment subprob
lems using Theorem A collection of indep endent subproblems can b e solved in parallel
if su cient pro cessors are available In the sp ecial case where some of the P are identical
i
equality of some of the U s reduces the number of subproblems
ij
The cost e ectiveness of iterating over a particular set of subproblems should b e carefully
evaluated when designing a containment algorithm Let contacting pair refer to a pair
of p olygons which is in contact The characterization tells us which contacting pairs can
o ccur The number of p ossible contacting pairs is purely a function of k not m or n Testing
the characterization requires solving a subproblem for each p ossible contacting pair If the
pair is in single contact we solve a boundary restricted subproblem by replacing U by
ij
its b oundary If the pair is in double contact we solve a vertex restricted subproblem by
replacing U by its vertices Since k is a constant the characterization do es not increase
ij
the asymptotic running time in terms of m or n Nevertheless the number of ways to choose
k
contacting pairs is exp onential in k roughly k
For k the implication of the rst part of the characterization k contacts is
particularly nice If there exists a solution to NN then there exists a solution in which a
set of k determining contacts contains either a double contact or all single contacts
Testing for a double contact solution requires solving a vertex restricted subproblem for each
U For a solution where six pairs of p olygons are in contact each P P pair must b e
ij i j
in contact This implies a subproblem in which every p olygon in U is b oundary restricted
Avnaim Avn takes advantage of this to eliminate a factor of m from the running time
of his NN algorithm
k
Unfortunately for k we must choose k out of p ossible contacting pairs The
number of ways to choose k pairs is exp onential in k Both a generalization of Avnaim s
NN algorithm and the naive algorithm must iterate over all ways of choosing k single
contacts Using the second part of the characterization contacts p er p olygon also
generates an exp onential number of subproblems if we apply it to every p olygon Applying
it to only a single p olygon P requires that we check two cases
i
a double contact with at least one P j k i j or
j
a single contact with at least one P j k i j and a single contact with at
j
j k i j j least one P
j
Testing the rst case requires k vertex restricted subproblems one for each U
ij
j k i j Each subproblem is vertex restricted for U The second case requires
ij
k k subproblems Each subproblem is b oundary restricted for two U s This
ij
approach therefore requires solving O k subproblems
Applying part of this characterization leads to a semi valid con guration space restric
tion in Chapter Partial characterization is also used in Chapter for our approximate
containment algorithm
Overview of Part One
The remainder of the containment part of the thesis contains three chapters Chapter
describ es our con guration space restrictions Chapter discusses evaluation and sub divi
sion techniques Chapter gives two di erent algorithms which can b e formed via sp eci c
choices of restriction evaluation and sub division techniques The rst of the two algo
rithms is an approximate and direct algorithm the second is our exact hybrid algorithm
Portions of our containment work also app ear in DML DM DM MD
3
A semi valid restriction is guaranteed to preserve at least one valid con guration if one exists
C h a p t e r
Con guration Space Restrictions
Introduction
Here we show how to restrict certain two dimensional con guration spaces asso ciated
with translational containment Recall from Section that a restriction of U replaces
some or all U s by subsets of themselves Naturally if any U U is replaced by the
ij ij
empty set then there is no solution for the hypothesis U We de ne two general types
of restrictions valid restrictions which preserve all solutions to a containment problem
and semi valid restrictions which preserve a single solution if one exists Restrictions can
prune away invalid regions from the con guration spaces b efore a containment algorithm is
applied They can help an algorithm quickly detect infeasibili ty Restrictions can also b e
integrated into the b o dy of an existing containment algorithm
This chapter makes several general observations ab out restrictions of con guration
spaces for translational containment We present a variety of valid and semi valid restric
tions We examine in detail the e ectiveness of our geometric restrictions which are valid
restrictions Our characterization of the types of containment problems for which these
restrictions are e ective is based on theoretical as well as exp erimental results Most of our
restrictions are based on closed rather than regularized set op erations and so we discuss
this imp ortant implementation issue
Overview
Section provides notation for this chapter Section makes several general ob
servations ab out valid restrictions The next three sections present valid restrictions In
Section we present our geometric restriction give an example and discuss the e ective
ness of this restriction In Section we describ e a linear programming restriction give
an example and discuss the e ectiveness of this restriction Section gives a restriction
which is valid when one item is a subset of another Section gives a parallelizable valid
restriction based on unions Section gives a parallelizable restriction based on the char
acterization in Section Section gives a restriction for identical items Section
1
Regularized op erations replace each p olygon by the closure of its interior Our closed op erations simply
replace each p olygon by its closure See Section
discusses closed op erators on which many of these restrictions are based Section
summarizes this chapter and presents conclusions
Note Figures depicting U p olygons app ear throughout this chapter to illustrate the
ij
di erence b etween U s b efore and after a restriction is applied to U A U for i
ij ij
j k is unbounded in its original form as de ned by Equation so we p erform the
following b ounding op eration b efore displaying a U
ij
U U U U
ij ij i j
This op eration is a valid restriction as established in Section
Notation
De nition A restriction is an operation on U that replaces some or al l U U by
ij
subsets of themselves A valid restriction preserves the invariant U U
ij
ij
is the set of all values of t t which b elong to a valid Recall from Section that U
j i
ij
the replacement of U in a valid restriction of U then con guration If we denote by U
ij
ij
U U U We also denote the set of all U for a particular valid restriction by U
ij
ij ij ij
De nition A semi valid restriction of U is a restriction of U having the property
that if there exists a valid con guration in U then there exists a valid con guration in the
restriction of U
When it is imp ortant to refer to an original U as de ned by Equation we use the
ij
orig orig
orig
notation U and U for the set of all U
ij ij
Prop erties of Valid Restrictions
Here we discuss several general prop erties of any valid restriction for translational con
tainment
Boundary Intersection
Consider the following two restriction examples The rst example shown in Figure
is of four nearly rectangular items to b e placed in a nearly convex container Figure
shows two one comp onent U p olygons for this example with U p olygons shaded in black
j
j
Figure shows two two component U p olygons with U p olygons
ij
ij
The second example shown in Figure is of four oblong items to b e placed in a
nonconvex container Figure shows a one comp onent U p olygon for this example
j
p olygons shaded in black Figure shows a two component U p olygon with with U
ij
j
U p olygons
ij
These examples have the prop erty that each comp onent of U intersects the b oundary
ij
orig
of U That is they share part of the b oundary in common This phenomenon o ccurs
ij
frequently in practice The reason this b ehavior o ccurs often in practice is due to the
characterization of translational containment problems in DM DM summarized by
Theorem The following theorem formalizes the relationship b etween Theorem
and the b oundary intersection b ehavior
2
The algorithms which generate these examples app ear later in this chapter
Figure Boundary intersection example the containment problem
Figure Boundary intersection example two one comp onent U s with U s
j
j
s Figure Boundary intersection example two two component U s with U
ij
ij
Figure Boundary intersection example the containment problem
Figure Boundary intersection example a one comp onent U with U s
j
j
Figure Boundary intersection example a two component U with U s
ij
ij
Theorem If an instance of translational containment has a valid con guration then
for each j j k there exists i j i k such that at least one component of U
ij
orig
shares part of its boundary with U
ij
Pro of Part of Theorem implies the weaker condition that if a solution exists
then a solution exists such that for each j j k there is some i j i k
such that P and P are in at least single contact A contact corresp onds to a p oint on
i j
orig
the b oundary of U this p oint remains after any valid restriction b ecause it is asso ciated
ij
with a valid con guration Thus a comp onent of a replacement of U in a valid restriction
ij
orig
of U must intersect the b oundary of U for some i i k 2
ij
This explains comp onents of U that cling to its b oundary as in Figure and Fig
j
ure It also explains comp onents of U i k that cling to the inner b oundary
ij
which is the b oundary of P P see the right hand two component U of Figure
i j ij
Theorem is an imp ortant consideration in designing a containment algorithm The
b oundary intersection b ehavior o ccurs frequently in practice and so it is unreasonable to
exp ect for example that the convex hull of a replacement p olygon is strictly inside the
original p olygon
Lo ose Fits
It is clear that little restriction can b e exp ected if the containment problem has a lo ose
tting solution In the extreme case of no container no restriction is p ossible at all
Annular Con guration Spaces
For a given set of items as the container b ecomes less restrictive each replacement of
a U p olygon in a restriction of U for i k assumes an annular shap e Here we use
ij
the terms annular and annulus in the top ological sense see Figure
This b ehavior o ccurs b ecause when the t is lo ose enough each p oint on the b oundary
of P P is a valid relative displacement That is each p ossible contact b etween P and
i j i
P is legal An example is shown in Figure In this example the outer b oundary of the
j
annulus is formed by our geometric restriction see Section
The annular shap e presents a challenge for certain types of containment algorithms For
example if the algorithm uses external approximations to the U s then it will consider a
ij
con guration to b e valid if every t t is within the approximation to U However if some
j i ij
t t is within the inner hole of the annulus then P and P overlap and the con guration
j i i j
is not valid
Size and s Analysis
Restriction reduces the area of U but it is not guaranteed to reduce the number of
vertices of U This can cause problems in analyzing the running time of a restriction or a
containment algorithm For this reason for a given set of restriction op erations we choose to
express the running time of such an algorithm in terms of the maximum number of vertices
s generated by applying any sequence of the restriction op erations to U DM Expressing
running time in terms of s is what we call size analysis or s analysis of the algorithm as
Figure An annular con guration space
introduced in Section The use of s analysis gives a measure of the practical running
time of a containment algorithm
Supp ose U has a nite number of vertices Some sets of restriction op erations on U
have nite s even when applied an in nite number of times One example is intersections
using vertical or horizontal half planes Intersections with arbitrary half planes however
cause s to b e in nite The question of whether or not the op erations we use in our valid
restrictions cause s to b e in nite is an op en problem and a sub ject of future work see
Section We show b elow that if al l p ossible valid restrictions are p erformed on U
the resulting number of vertices is nite
Theorem U has a nite number of vertices and a nite number of connected com
ij
ponents
has a nite number of vertices It then follows that the num Pro of We rst show that U
ij
b er of connected comp onents is also nite Recall from Section that m max m
i k i
Here we let M max m We establish a rough upp er b ound on the number of vertices
i k i
in U by rst nding an upp er b ound on the number of vertices of the k dimensional free
ij
con guration space S Each vertex of S is the intersection of k hyper planes Let Q b e
an upp er b ound on the number of hyper planes then the number of vertices of S is no
k
more than Q b ecause it is well known that the size of an arrangement of n hyper planes
d
in d dimensions is n Ede Each hyper plane corresp onds to a contact b etween two
p olygons P and P For each p olygon pair there are at most M contacts and there are
i j
3
Knowing all p ossible valid restrictions is NP hard
k k pairs so Q M k k Thus an upp er b ound on the size of the free
k k k
con guration space is given by O M k k M k k Now U is a
ij
pro jection of S The pro jection step can introduce new vertices and so the number of
vertices in U is b ounded by the size of an arrangement of the hyper planes Thus the
ij
2 2
k k k k k
number of vertices in U is b ounded by O M k k O M k k
ij
Therefore U has a nite number of vertices and connected comp onents 2
ij
We note that tighter b ounds have b een obtained for the size of the free con guration
space in sp ecial cases For example Avnaim and Boissonnat AB show that for two non
convex p olygons and a nonconvex container the free con guration space is of size O M
Recently for example Aronov and Sharir AS showed that for a convex p olyhedron B
translating in space among k convex p olyhedral obstacles A A with disjoint interi
k
ors the free con guration space for B has size O M k log k where M is the total size of
the Minkowski sums P A B for i k
i i
The number of connected comp onents of the restricted U is closely related to the num
b er of vertices We simply remark here that restriction can fragment U into many small
connected comp onents and that this should b e taken into account when designing a con
tainment algorithm
Identical Items
and U If P and P are identical with no rotations allowed then U U so U
i j i j
j i
can replace each other In this case P P is symmetric ab out the origin If any p oint
i j
p in U i k is removed by a restriction then p can also b e removed Thus U is
ij
ij
symmetric with resp ect to the origin
This symmetry re ects the fact that given a valid con guration identical items pro duce
valid equivalent con gurations If an algorithm exp ects to deal often with identical items
it is wise to take advantage of the equivalent con gurations see Section
Steady State Geometric Restriction
In DM DM we present restrictions which op erate directly on U The restrictions use
the Minkowski sum and a closed intersection op erator see Section Each element of U
is replaced using op erations that include other elements of U as input Theorem b elow
encompasses all three restrictions from DM DM Figure illustrates the theorem
Theorem U U U U h i j k h i j is a valid restriction
ij ij ih hj
Pro of The theorem states that a valid displacement from P to P must b e a valid
i j
displacement from P to P plus a valid displacement from P to P 2
i h h j
4
Guibas et al note GRS that if A is convex then A A is symmetric with resp ect to the origin
The extension to arbitrary A is trivial
5
Two con gurations are equivalent if their t s corresp ond to a p ermutation of identical items
i Pj
U tj - ti c- i j t - t c- j h Uh j P i t - h ti c- Ui h P
h
Figure Geometric restriction
Avnaim and Boissonnat s Solutions to NN and NP
We show that the op eration of Theorem can b e viewed as a generalization of
Avnaim and Boissonnat s AB solutions to NN and NP
Avnaim and Boissonnat show for NN that the set of all the valid relative p ositions of
U U U This is an exact P with resp ect to P is given by in our notation U
ij i j j i
ij
solution Each element of the set U corresp onds to a valid con guration We observe that
ij
for k NN each valid displacement for two p olygons must also b e a solution for NN
Avnaim and Boissonnat observe that for NP a valid translation t exists for P only
if
t U U t t U t t
where t t U and t t U Thus U U U U U U is a
valid restriction for NP
For their solution to NP Avnaim and Boissonnat show that any vector in U U
U is guaranteed to have an asso ciated displacement for P with resp ect to P corresp ond
ing to a solution to NP The pro of relies on their application of Helly s Theorem Supp ose
we have three parallelograms which are the same a ne transformation of di erent but
axis parallel rectangles If the three parallelograms intersect pairwise then all three have
a common p oint We observe that in the general case U U U for h i j
ih hj
ij
In our notation Avnaim and Boissonnat s solution to NP applies U U U U
ij ij i j
i j followed by U U U for h i j Each element of U
ij ih hj ij
then corresp onds to a solution to NP
Steady State Restriction
One would hop e that rep eated applications of the restriction in Theorem would
yield the set of U in the limit Actually this is always true only for k For k one
ij
can construct counterexamples but only with di culty For k it is easy to construct
counterexamples We iteratively apply the valid restriction of Theorem until no more
signi cant shrinking o ccurs We denote the resulting restriction by SSGR steady state
geometric restriction In the pseudo co de b elow E mpty U is a set containing k k
copies of the empty set
SSGR U k
do
for i to k
for j i to k
for h to k
if h i j
U U U U
ij ij ih hj
if any U U return E mpty U
ij
else U U
j i ij
S maximum fractional area shrinkage in U
d maximum diameter of a p olygon in U
while S and d
return U
The parameter is sp eci ed by the user It is used to help judge when to terminate
the restriction pro cess The parameter is not sp eci ed by the user It is the relative
area shrinkage For the program achieves a true steady state if the
maximum p olygon diameter is greater than and the maximum area is greater than in
which additional restrictions cannot shrink the p olygons further However this can require
an in nite number of iterations We use and in our implementation these
are satisfactory for our applications Using nonzero values of and gives us a b ound on
the number of restriction iterations see Section
The steady state restriction pro cess is quite p owerful We know that U U U
ih hj
ij
U U U for q h i j In for h i j However it is also true that U
ih hq q j
ij
order to generalize this result we de ne the notion of a list restriction We then show in
Theorem that SSGR satis es all list restrictions
De nition Given a list of indices i j the list restriction
q q
of U corresponding to is the restriction
ij
U U U U U U U
ij ij i j
q q
q 1 2 3 1 2 1
In the following theorem our use of the term steady state refers to the true steady
state describ ed ab ove in which
Theorem If multiple applications of SSGR reach a steady state then al l possible list
restrictions are satis ed
Pro of The pro of is by induction on the length of a list which is the number of indices
in the list The base case for a path of length two is true b ecause SSGR satis es all list
restrictions of length two The induction hypothesis is that the list restriction for every list
of length q is satis ed Consider the list i j of length
a b c q
q We show that the list restriction for is satis ed First by the inductive hypothesis
the list restriction for the path i j of length q is satis ed
a c q
so
U U U U U
ij ij i j
a c q
1
Now b ecause a steady state has b een achieved applying SSGR to U do es not change
a c
U This means
a c
U U U U
a c a c a c
b b
and hence U U U Substitution yields
a c a c
b b
U U U U U U
ij ij i j
a c q
1
b b
which is the list restriction for Thus the list restriction for is satis ed 2
Running Time
Here we calculate the asymptotic running time of SSGR We assume
s is the largest number of vertices of any p olygon generated by a sequence of inter
section and Minkowski sum op erations on p olygons in U
is a real number representing a lower b ound on the maximum diameter of a p olygon
in U
is the fractional shrinkage in area required by each iteration of SSGR
Theorem The running time of SSGR is in O log k s log s
Pro of Let r b e the number of iterations of the while lo op in SSGR We have
O log r O log
We p erform the op eration U U U U inside of three nested lo ops Each
ij ij ih hj
op eration takes O s log s time Each lo op executes k times Therefore the running time
is k r s log s
The total running time is therefore
O log k s log s
2
Example
Figure shows an example from the apparel industry Three trim pieces are to b e
placed The nearly triangular shap e displayed in Figure is a scaled version of the U
j
p olygon for the rectangular trim piece with resp ect to the container on it the U p olygon
j
for the SSGR of U is shown as a black lled region More than of the area is eliminated
by the restriction pro cess all that remains is the tip Figure shows the two component
U p olygon for the rectangular trim piece with resp ect to one of the other two trim pieces
ij
The U p olygon for the SSGR of U is shown as a black lled region Again more than
ij
of the area is removed by our restrictions SSGR takes three seconds to execute this
example on a MHz SPARCstation
Figure SSGR containment problem place convex items
Figure A one comp onent U with U for the SSGR of U
j
j
for the SSGR of U Figure A two component U with U
ij
ij
E ectiveness
In order to gauge the e ectiveness of SSGR for k we applied it to more than
containment problems Some of these were infeasible examples and some were feasible
Most of these examples were drawn from apparel manufacturing Typical ranges for m and
n were m and n For our apparel examples we pro duced each
feasible containment problem by removing a group of neighboring items from an existing
layout To create an infeasible problem we moved some of the remaining items a small
distance into the container In all cases the area of the container was at least as large as
the sum of the item areas In addition each pair of items t into the container so that all
U s were non null b efore applying SSGR
ij
The actual running time of SSGR in practice dep ends on the values of k m n
and the sp eed of the underlying set op erations We currently use and
Although in theory SSGR can continue for many rounds b efore reaching a steady state
we nd that if we terminate SSGR when no p olygon s area decreases by at least one
or two rounds is usually su cient For these values of and k our current
implementation of set op erations and the typical ranges for m and n given ab ove our
SSGR runs approximately ten times more slowly than the steady state linear programming
restriction of Section On a MHz SPARCstation SSGR running times tend to range
from less than one second for small values of k m and n to several minutes for larger values
of k near m near and n near For more details see the running times for our
approximate algorithm in Section this algorithm uses SSGR
Our fo cus here is on how well SSGR restricts U We measure e ectiveness as the
orig
p ercentage of area removed by SSGR averaged over all U s Since the U s are unbounded
ij
ij
and in nite in area for i j we apply Equation b efore measuring p erformance As
exp ected little or no restriction o ccurs for lo ose ts
Infeasible Examples
Table shows results for infeasible examples We use the k CN and k NN notation
introduced in Section For each case where SSGR do es not detect infeasibility we give
the average relative p ercentage area restriction over all the U s In all the failure cases
ij
this gure is less than
In slightly more than of the cases SSGR detects infeasibili ty This suggests that
SSGR can b e valuable as a prepro cessing step for a containment algorithm
Type Detected Type Detected
CN yes CN yes
CN no CN yes
CN no CN no
NN yes NN yes
NN yes NN yes
NN no NN yes
CN no NN no
CN no
Table SSGR applied to infeasible examples
Feasible Examples
For the feasible examples k For of the feasible examples the average
relative p ercentage area restriction over all the U s is b elow We found that
ij
is ab ove and is ab ove This data shows a pattern similar to the infeasible
examples for our test cases it is rare to see a mo dest amount of SSGR restriction of around
Constructing Restrictive Examples
Our goal here is to provide insight into the types of situations in which SSGR p erforms
well Naively one might think that a tight tting example whose only solutions are highly
interlocking would have high restriction However the requirements are more subtle The
only valid con gurations for the NN example in Figure are tight tting and highly
interlocking yet the average SSGR p ercentage for this example is only
In order to create a restrictive example we want U U U to b e false as much
ij ih hj
as p ossible That is we desire restrictive triples as de ned b elow in De nition We
show b elow some ways of creating restrictive triples
De nition If U U U then P P P is a restrictive triple
ij ih hj i j h
Figure A NN example with SSGR restriction
1
2
3
Figure L shap ed container and items
A simple L shap ed container can b e used to induce restriction b ecause the base of the
L can b e used to trap an item Consider the example in Figure Restrictions will
reduce the area of U and U b ecause P can only t in the base of the L this prohibits
P and P from b eing in the base of the L The limited p ossibilities for P will also cause
restrictions to reduce the area of U SSGR for this example reduces the area of U and
U each by more than and the area of U by
Figure shows a NN example which has average SSGR restriction of The
shap e of the container greatly restricts the p ositions for the narrow p ointed item in the
middle of the column of unplaced items This means there are many restrictive triples
Swapping this item with the rectangular item at the b ottom right corner of the layout
pro duces an example with average SSGR restriction of only
Figure A NN example with average SSGR restriction
In b oth Figure and Figure the shap e of the container limits the p ossible
p ositions of one item P this in turn removes placement p ossibiliti es for the remaining
i
items Thus U U U for some i j
j i ij
orig
We de ne a restrictive container as follows Consider U Apply Equation so
that each U is b ounded If for the resulting U there is some U U U then
ij j i ij
the container is restrictive Note that except for applying Equation we do not use
replacement p olygons in this de nition Under these assumptions if there is a restrictive
triple which includes P then the container is restrictive Clearly if a containment problem
has no restrictive triple then the container is not restrictive
Figure shows a di erent type of situation Here the container is complicated and the
containment problem is infeasible SSGR detects the infeasibility However the container
is not restrictive b ecause the placement p ositions for a single item within the container are
not limited by adding a second item to the container In this example it is restrictions of
the form U U U U i j h which are instrumental in detecting the
ij ij ih hj
infeasibility
If we choose an unrestrictive container we must select the items carefully in order to
induce strong restriction For example given a rectangular container of height h and width
w we can form three items of height h and width w see Figure in which the items
orig
are shown placed in the container In this case each U is a vertical line segment of
j
Figure An infeasible CN example
length h Each U i j after applying Equation is a vertical line segment
ij
of length h with a segment of length h removed from its center Restrictions shrink
each U i j down to four p oints which are the endp oints of the two line
ij
segments These are propagated to U j leaving three p oints on its vertical line
j
spaced at intervals of h
1
2 h
3
Figure A CC example with an unrestrictive container
This example do es not work for k b ecause in that case there is no restrictive triple
which excludes P
Theorem Given a rectangular container of height h and width w and an integer
k for the translational containment problem with k rectangular items each of height
h k and width w the U i j k are una ected by restrictions of the form
ij
U U U U for which neither i j nor h is
ij ij ih hj
Pro of The U i j k are identical b ecause the items are identical Hence
ij
each replacement of such a U is the same Recall from Section that U U each
ij ij j i
restriction therefore reduces to U U U U This has no e ect if U U U
ij ij ij ij ij ij ij
Each U after applying Equation is a vertical line segment of length h k k with
ij
a segment of length h k removed from its center This forms two vertical line segments
let a b c and d b e their endp oints from top to b ottom U a covers all of segment
ij
ab except for a segment of length h k extending downwards from a U b covers all of
ij
segment ab except for a segment of length h k extending upwards from b Because each of
the two segments comprising U is of length h k the union of these sums completely
ij
covers segment ab Similarly the union of the sums U c and U d cover segment cd
ij ij
Thus U is a subset of U U and therefore the U s i j k are una ected by
ij ij ij ij
restrictions of the form U U U U for which neither i j nor h is 2
ij ij ih hj
A containment problem which satis es the conditions of Theorem clearly also has
no restrictive triples which include P Thus SSGR applied to such a problem pro duces no
restriction
1 2 3 4 5 6
Figure A NC example with average SSGR restriction
The previous example might suggest that a rectangular container cannot b e restrictive
for k This is not true It is easy to create a restrictive triple involving a rectangular
container for k using nonconvex items Figure shows one example for which the
items are placed in the container SSGR provides average restriction of in this case
with the area of some U s reduced by more than One of the restrictive triples in this
ij
example is P P P
An example for k with a rectangular restrictive container and convex items can b e
constructed using only slightly more e ort It can b e done using rectangles and the idea
of guil lotine cuts An example is given in Figure the items are shown placed in the
container This example has average SSGR restriction Item participates in several
restrictive triples The amount of restriction is controlled by the width of item In order
to prevent items from tting side by side the width of item should b e less than half
the width of the container
Theorem Given a rectangular container of height h and width w and an integer
k for the translational containment problem with k rectangular items one of height h
and width z w and k of height h k and width w z the U i j k
ij
are a ected by restrictions of the form U U U U for which neither i j nor
ij ij ih hj
h is
Pro of Assume w l o g that the item of width z is item number We show that
U U U U which implies that the restriction U U U U
removes area from U Refer to Figure in which the origin is marked o The two
6
A guillotine cut is a common term in the cutting and layout world It refers to a metho d of applying a
sequence of horizontal and vertical cuts to an axis parallel rectangle Each cut go es all the way through the
remaining rectangle See for example DD 2
3 1
4
5
Figure A CC example with average SSGR restriction
shaded rectangles represent U Each rectangle has width z and height k h k
The rectangles are separated by a vertical distance of h k The two vertical dashed
lines represent U U Each line has length h and is at a distance w from the origin
The Minkowski sum U U consists of three vertical lines of length h adjacent lines are
separated by a distance of w Two of them are at a distance of w from the origin and hence
cannot intersect U One of the vertical lines runs through the origin Because it extends
a length of h ab ove and b elow the origin and each rectangle of U ends at a distance of
k h k ab ove or b elow the origin the intersection of U with this vertical line
consists of two vertical line segments shown as thick lines in Figure This intersection
is clearly a prop er subset of U 2
o
Figure U U U
Figure shows another guillotine like layout with the p olygons placed in the con
tainer In such an example the area of some U p olygons is reduced by as much as
ij
For example the area of U is greatly reduced due to items and
Our nal example of how to construct a restrictive containment problem is shown in
Figure Figure depicts a highly restrictive NN example In this case the large
nonconvex item together with the two large ovals form a restrictive triple Figure shows
the e ect of SSGR on two one comp onent U s in this case each has more than of its
j
area removed by the restrictions 2 3 5 6
8 1 4 7
9
Figure A CC example with strong SSGR restriction
Figure A highly restrictive NN example
Figure A NN example e ect of SSGR on two one comp onent U s
j
Barriers to Restriction
Restrictions of the form U U U U are ine ective when U U U
ij ij ih hj ij ih hj
or equivalently when U U U The following theorem provides a condition for
ij ih j h
which this is true
Theorem Let A and B be polygons and let the origin be denoted by O If either
O B or A is connected and no translated copy of A lies entirely within the component of
B that contains O then A A B
Pro of Consider a p oint p A To prove that A A B it su ces to show that
p a b where a A and b B Translate A by p and denote this A If B contains O
p
B which contains O then p p O so we re done Otherwise let B b e the comp onent of
No translated copy of A lies entirely within B this guarantees the existence of q p A
p
such that q p B If q p B then q p b B implies p q b so we re done Now
assume q p B A is connected and so there exists a connected path from O A to
p p
q p A The endp oints of are also in di erent comp onents of the exterior of B so
p
must intersect B at some p oint b B Then b ecause A there exists a p A such
p p
that b a p This implies p a b which completes the pro of 2
The following corollary is immediate from the ab ove theorem
Corollary Let A be a simple bounded connected polygon such that the origin is
contained within the interior of the outer boundary of A Then A A A
Note It is not true in general that A A A even with the additional constraint
that A A Figure presents a counterexample In the gure A consists of two
p oints and A A However A A consists of three p oints which are not part of A
Therefore A A A
a) A b) A + -A
= origin
Figure Counterexample for A A A
The next corollary follows immediately from transitivity
Corollary Let A and B be polygons satisfying the conditions of Theorem and
let C be a subset of A Then C A B
In practice we observe that it is rare for SSGR to remove any area when all items
are identical Even when only some items are identical we observe limited restriction
This makes sense in light of the ab ove results For if all the items are identical then
U U U U If in addition U satis es the conditions of Theorem on
ij ih hj j h ij
A and B then U U U and therefore no restriction will o ccur Although SSGR
ij ih j h
can b e ine ective when items are identical Milenkovic has recently designed a restriction
to deal e ectively with identical items see the brief description of his symmetry breaking
restriction in Section In addition we remind the reader that when items are identical
the number of subproblems created by using the translational containment characterization
of Section is reduced
When the b oundary of a U for i k is a closed curve it always contains the origin
ij
In this case the U satis es a necessary condition for Theorem and its corollaries
ij
We tested the hypothesis that the extent to which the U s i j k are closed
ij
curves is a key factor in the e ectiveness of SSGR For each of the more than feasible
cases we estimated the closure by sho oting rays from the origin at equally spaced angles
ab out the origin Let b e the fraction of rays which hit the U We recorded the average
ij
over all U s as well as the maximum and minimum for each example We then calculated
ij
correlation co e cients for each of these with resp ect to the total restriction p ercentage We
hop ed to nd high negative correlation indicating a linear relationship in which high and
a small amount of restriction were related
We did nd that whenever the average was high SSGR e ectiveness was low For
e ectiveness was and for e fectiveness was The interlocking
NN example of Figure has average and SSGR restriction of less than
Some of its U s are shown in Figure
ij
However e ectiveness is sometimes low even when average is not high This tends to
o ccur when several items are identical The correlation co e cient for average
meaning only of the restriction results can b e predicted using average The correlation
co e cient for maximum and for minimum Thus alone is not a
su cient predictor of SSGR e ectiveness
Figure Selected U s for NN example
ij
Steady State Linear Programming Restriction
A di erent metho d of restriction based on linear programming was developed by
Milenkovic and is describ ed in DM MD This steady state linear programming restric
tion technique SSLPR is an indirect one which uses the convex hulls of the con guration
spaces It works as follows Temporarily replace every element U U by its convex hull
ij
H U The set of constraints
ij
t t H U i j k
j i ij
can b e expressed as a linear program which is called the CLP constraint linear program
A p olynomial time algorithm is given to pro ject the k dimensional feasible space F into
any element U of U If the pro jection is denoted F then the replacement
ij ij
U U F
ij ij ij
is a valid restriction
Example
We present an example of SSLPR from the apparel industry Figure depicts three
v shap ed items which are to b e placed in a nonconvex gap in a clothing marker Fig
ure shows two two component U p olygons for this example Their replacements
j
7
Programming work on the linear programming restriction was p erformed by Umesh Shankar while at
the MIT Summer Research Institute Sha
are shaded in black Figure shows two U p olygons and their replacements The one
ij
comp onent p olygon on the left is una ected by restriction The area of the three comp onent
p olygon on the right however is visibly reduced This example takes less than second to
execute on a MHz SPARCstation
Figure SSLPR containment problem place nonconvex items
s for SSLPR of U Figure Two two component U s with U
j
j
s for SSLPR of U Figure A one comp onent U with U
ij
ij
E ectiveness
Here we discuss the e ectiveness of SSLPR We also compare the e ectiveness of SSLPR
with SSGR Preliminary results on the e ectiveness of the linear programming restriction
app ear in Sha
SSLPR has the prop erty that it cannot increase the number of re ex vertices in U Of
course SSGR has no such guarantee
SSLPR approximates each U by its convex hull If all the U s are convex then SSLPR
ij ij
can give an exact solution to the containment problem If a U is very di erent from
ij
its convex hull then many invalid p oints will b e considered valid by the linear program
Therefore as Shankar observes Sha SSLPR provides b etter restriction when the U
ij
p olygons are similar to their convex hulls than when the U s and their convex hulls are
ij
dissimilar
Unfortunately most of the U s i j k are quite di erent from their convex hulls
ij
immediately after applying Equation b ecause each b ecomes the di erence b etween two
p olygons For this reason we observe very little SSLPR restriction for our collection of
more than test cases Only of the feasible examples have any SSLPR restriction
SSLPR is not able to detect infeasibility in any of the infeasible examples For all the cases
where SSLPR achieves some restriction the SSLPR of U is a sup erset of the SSGR of U
SSLPR clearly cannot b e exp ected to function well for examples having many annular
U s This includes lo ose ts and identical items
ij
The examples in which SSLPR is e ective have restrictive containers that force the U
j
of some item to consist of a small single nearly convex comp onent For example for the
L shap ed container of Figure and its three items SSLPR achieves an average of
restriction SSLPR do es not do well in situations for which the container is unrestrictive
For example Figure where SSGR reduces the area of some U s by up to has no
ij
SSLPR restriction
Based on the success of SSLPR on examples like that of Figure it is reasonable to
exp ect SSLPR to b e e ective in propagating the e ect of selecting a single nearly convex
comp onent of a U for consideration An example is shown in Figure where a single
ij
comp onent of a U on the far right is selected causing the area of the two two component
ij
U s on the left to b e reduced This selection is useful in a containment algorithm which
ij
relies on sub division see Section For our current implementation SSLPR is roughly
ten times faster than SSGR this makes it a practical type of restriction to use within a
sub division containment algorithm
Figure E ect of SSLPR on U s when a single comp onent of a U is selected
ij ij
Although our tests suggest that SSGR is more p owerful than SSLPR there are in theory
cases for which SSLPR can provide more restriction than SSGR Also in our hierarchical
containment algorithm we have o ccasionally seen SSLPR remove area not removed by
SSGR see Section
The following is an example for which SSLPR is more p owerful than SSGR Consider
the container and three items in Figure The container is the union of the b oundaries
of two equilateral triangles Each triangle has sides of length s The p oints a b and c
at alternating re ex vertices of the container form an equilateral triangle Each of the
three items has length s Each also has two holes each hole is of the way from an
endp oint The lo cal origin of each item is indicated in the gure The reader may verify
that U fb cg U fc ag and U fa bg Also U f a bg U f b cg
and U f a cg This example is infeasible b ecause any simultaneous placement of the
three items would force their endp oints to collide SSGR is unable to detect this infeasibili ty
b ecause the U s satisfy U U U However there is also no feasible solution to the
ij ij ih hj
CLP using the convex hulls of the U s and so SSLPR can detect the infeasibility
ij
It is easy to construct an example for which SSGR removes a region which SSLPR
cannot Consider again the containment problem of Theorem In that case each U
j
is a pair of parallel vertical line segments The convex hull of these segments will cause
the CLP to allow solutions in which item overlaps another item This in turn will cause
p oints of the U for an identical pair of rectangles which SSGR removes to b e valid for
ij
the CLP Thus p oints will b e removed by SSGR that are not removed by SSLPR
We conclude that SSGR and SSLPR are complementary restrictions
container local origin P3 P1 c
b P2
a holes
Figure Example for which SSLPR is more p owerful than SSGR
Subset Restriction
If one item P is a subset of another item P then the following theorem can b e used
i j
The theorem assumes that when the lo cal origins of the two items are aligned then P P
i j
P P and U P P Theorem If P P then for P U
i h h i i j h
hj j h
Pro of P P implies that P P P P Since P Q implies that Q P
i j i h j h
U P P P P so U P P For the second part U we have
j h i h i h
hj j h j h
and P P P P applied to U P P yield U P P This
i h h i i h h i
j h hj
immediately gives U P P 2
h i
hj
Corollary If P P then U U P P and U U P P are
i j j h j h i h hj hj h i
valid restrictions
We call the valid restrictions given by Corollary subset restrictions Subset re
strictions can b e useful when initially calculating U If a U is null then U need not b e
hi hj
calculated
Union Restriction
This section introduces a valid restriction which assumes that a particular U is the
ij
union of a collection of p olygons First we argue that if A U then restrictions on A
ij
are well de ned if we replace U in U by A Recall from Section that U is the set of
ij
ij
all values of t t which b elong to a valid con guration and recall from Section that
j i
U is the replacement of U in a valid restriction of U Analogously A is the set of all
ij
ij
values of t t in A which b elong to a valid con guration and A is the replacement of A
j i
in a valid restriction
Lemma If A U then A U and A A U
ij
ij ij
Pro of The rst part follows immediately from A U To obtain the second part by
ij
de nition A A this together with the rst part of the lemma imply A A U To
ij
obtain equality observe that if a p oint t t is in A U and hence in U it is part of
j i
ij ij
a valid con guration therefore t t must also b e in A 2
j i
S S
A A then U Theorem If U
h ij
h h
ij
h
S S
A Now A Pro of By de nition of a valid restriction A A This implies
h h
h h h h
S S
since A U Lemma implies A Substituting this into A U A A
h ij h
ij h h
h h h
S S S S S
yields A U A This implies U A A Since U A this
h h ij h
h h h h h
ij ij
h h
S
simpli es to U A 2
ij h
h
Theorem immediately yields a valid restriction which we call a union restriction
Course grained parallelism can b e used to implement this restriction In the pseudo co de
b elow let A b e the set of all the A p olygons whose union is U and supp ose there are
h ij
h h
is pro cessor h s l such p olygons U represents a copy of U for the hth pro cessor and U
ij
U is a set containing k k copies con guration space for P with resp ect to P E mpty
j i
of the empty set
UNION RESTRICT U k i j A l
for h to l do in parallel
h
U U
h
A U
h
ij
h h
U ANY VALID RESTRICTION U k
U E mpty U
for s to k do
for t to k do
for h to l do
h
U U U
st st
st
return U
Characterization Based Restriction
We present here a restriction which do es not satisfy the criteria for a valid restriction
That is it do es not preserve the invariant U U However the restriction is useful
ij
ij
when only a single valid con guration is required b ecause if a valid con guration exists
a valid con guration also exists after the restriction Thus the restriction is a semi valid
restriction see De nition
The restriction is based on the containment characterization given in Section We
use a partial characterization based on the second part of Theorem which guarantees
that if a valid con guration exists then one also exists for which there are at least two
contacts p er p olygon We apply this to P and use the weaker condition that the p olygon
has at least one contact This means that if a valid con guration exists there must exist
a valid con guration for which some t t is on the b oundary of its U Therefore we
h h
preserve the existence of such a con guration if we replace each U by its b oundary apply
h
any valid restriction and then set U equal to the union of the restricted sets Course grained
parallelism can b e used here
h
As in the previous section in the pseudo co de b elow U represents a copy of U for
h
the hth pro cessor and U is pro cessor h s con guration space for P with resp ect to P
j i
ij
E mpty U is a set containing k k copies of the empty set
BOUNDARY RESTRICT U k
for h to k do in parallel
h
U U
h
BOUNDARY OF U U
h
h
h h
U ANY VALID RESTRICTION U k
U E mpty U
for i to k do
for j to k do
for h to k do
h
U U U
ij ij
ij
return U
Symmetry Breaking and Subset Substitution Restric
tions
Section observes that identical items pro duce equivalent con gurations These
p ose a sp ecial challenge for SSGR as discussed in Section In recent work Mil b
Milenkovic has developed a new form of restriction to overcome the di culty that SSGR
and SSLPR have with identical items His idea is to arbitrarily choose a vector v and
then imp ose the constraint that t v t v if items P and P are identical This has
i j i j
the e ect of a reduction in the area of U A resymmetrizing step is ultimately
ij
necessary to make the symmetry breaking restriction a valid restriction without this it is
only a semi valid restriction
The symmetry breaking restriction can also handle non identical items through a valid
restriction called a subset substitution restriction The main idea of this restriction is as
follows If a p olygon A is a subset of each of a collection of p olygons then we can substitute
a copy of A for each of the p olygons in the collection calculate U using these copies of A
p erform restrictions on U for the copies of A and then use these restrictions as valid
restrictions for the U involving the original items Note that this is not the same as the
subset restriction of Section in that case b oth P and P are to b e placed simultaneously
i j
in the same layout and P is not substituted for P In the most general form of the subset
i j
substitution restriction one can replace each p olygon P by a subset P The P s can all
i
i i
b e the same as ab ove or they can all b e di erent or some the same and some di erent
Closed Op erators
Here we discuss an imp ortant design issue related to the type of set op erations used to
implement restrictions
Consider a general p olygon i e a collection of p olygonal comp onents where each com
p onent is connected
De nition A general polygon P is regularized if it is equal to the closure of its
interior
Figure b depicts a regularization of the non regularized general p olygon of Fig
ure a Regularized p olygons are not su cient for our application Using only regular
ized set op erations can cause the algorithm to incorrectly conclude during a restriction that
no solution exists This can o ccur if the intersection of two p olygons is either a line segment
or a p oint In fact this situation can o ccur often for tight ts One example is a stack of
rectangles inside a rectangular container where each rectangle to b e placed has the same height as the container see Figure
(a) Non-regularized general polygon (b) Regularized general polygon
Figure Regularization example
A non regularized p olygon do es allow isolated p oints and line segments see Figure a
Unfortunately it also allows isolated p oints or line segments to b e removed from its interior
Thus a non regularized p olygon need not b e closed i e equal to its closure its b oundary
plus the closure of its interior Since the con guration space we op erate on is closed we
must deal with closed sets We therefore need closed p olygons Union and intersection
op erations on closed sets are de ned in the normal way b ecause the union and intersection
of two closed sets is closed However the complement of a closed set is op en so we take
the closure of the complement of a closed p olygon
Correct implementation of the algorithm requires the use of robust closed set op erations
The set op erations must b e implemented robustly in order to minimize error accumulation
during the steady state restriction pro cess Closed op erators are imp ortant to b oth the geo
metric and linear programming restrictions They are more crucial to geometric restrictions
since these rely more heavily on set op erations
AB A U B
Figure Two rectangles with degenerate intersection
For our rst implementation of SSGR we used robust intersection union and comple
ment op erators which handled only regularized general p olygons Our robust set op erators
are based on geometric rounding LM b MN Mil a Mil b In order to shrink p oly
gons to their b oundaries we added op erators which can handle line segments and p oints as
well as lists of p olygons However our underlying p olygon op erators were still regularized
We are currently redesigning these op erators so we will have a complete set of op erators for
closed p olygons
Summary and Conclusions
In this chapter we have made several general observations ab out restrictions for trans
lational containment For example we have observed that replacement p olygons often
have comp onents which intersect the b oundary of the original p olygon and shown when
and why this o ccurs These prop erties represent imp ortant considerations for designers of
translational containment algorithms
We have discussed the valid restriction SSGR in detail We have evaluated the e ec
tiveness of SSGR and of Milenkovic s SSLPR We conclude that SSGR is often much more
e ective on containment examples than SSLPR SSLPR p erforms b est when the con gura
tion spaces are similar to their convex hulls SSGR p erforms b est when a restrictive triple
of items exists Both restrictions p erform p o orly for lo ose ts identical items and other
circumstances that cause the con guration spaces to b e closed curves containing the origin
Both restrictions are e ective in propagating the e ects of selecting a single nearly convex
comp onent of a p olygon for consideration SSGR and SSLPR are complementary There
are cases for which SSGR can restrict more than SSLPR and cases for which SSLPR can
restrict more than SSGR
We have also introduced subset union and characterization based restrictions and
discussed Milenkovic s symmetry breaking restriction
C h a p t e r
Evaluation and Sub division
Introduction
In this chapter we present evaluation and sub division metho ds Recall from Chapter
that given a restricted hypothesis the evaluation step tries to nd a valid con guration
within the hypothesis If no solution is found by the evaluation step then the sub divi
sion step divides the current hypothesis into two sub hypotheses such that if a solution
exists within the current hypothesis then a solution must exist within one of the two
sub hypotheses To divide the current hypothesis we partition one of the con guration
spaces into two parts The algorithm then recurses on each part while keeping the other
con guration spaces constant
Section gives an overview of our work on evaluation Section outlines our
work on sub division Section gives an overview of this chapter
Evaluation
In order to evaluate a hypothesis we can invoke an exact algorithm for k NN at this
p oint or p erhaps a MIP algorithm such as that of DML However these alternatives
are costly For the purp oses of our general containment algorithm given in Section all
we really require is a fast test which generates a con guration and tests to see if it is a valid
con guration t t U i j k If the test succeeds then we have a solution
j i ij
As explained in Section the evaluator decides yes no or maybe It decides
yes if it nds a valid con guration within the current hypothesis It decides no if
it determines that there cannot b e a valid con guration within the current hypothesis
It decides maybe if it fails to nd a valid con guration in the current hypothesis but
it is unable to rule out the p ossibility that one exists In the maybe case additional
sub division is required
Our rst evaluation metho d is a greedy one which op erates directly on the con guration
spaces It shrinks each U to a p oint in turn then propagates the e ects of that shrinking
ij
via SSGR If any U b ecomes empty during this pro cess then that combination of U
ij ij
p oints is not part of a valid con guration and so the evaluator do es not nd a solution
In this case it returns maybe If however no U b ecomes empty then the collection
ij
of single p oint U s determines a valid con guration and the evaluator returns yes The
ij
key to the e ectiveness of this pro cedure is a go o d choice of a shrinking p oint in a U
ij
We present several selection metho ds One of them pro duces valid con gurations without
using sub division in of our test cases This metho d maximizes the intersection of
translations of the con guration spaces
The disadvantage of the greedy evaluation metho d is that it cannot op erate simultane
ously on the con guration spaces As k increases this b ecomes a more serious limitation
An evaluation metho d such as the indirect one of Milenkovic which is based on a con
tainment linear program CLP overcomes this limitation see the discussion of the CLP
in Section and Mil DM It approximates the U s by their convex hulls and uses
ij
linear programming to check if there exists a con guration which is valid with resp ect to
the convex hulls If not then the current U cannot have a valid con guration Thus this
evaluator is able to say no in contrast to the greedy evaluator If a valid con guration
is found with resp ect to the convex hulls and if each p oint found by the linear program is
within its asso ciated U then the con guration is a valid one Otherwise the con gura
ij
tion is an overlapping one We apply the work of Li and Milenkovic on overlap reduction
Li ML LM c LM a to improve this evaluation metho d
Both of these evaluation techniques are p owerful enough so that they can in combination
with restrictions sometimes solve containment problems without any sub division
A containment algorithm based on CLP evaluation can solve a containment problem
using a single CLP evaluation if every U U is convex We present a theorem which
ij
sp eci es a condition under which a single CLP evaluation can solve a containment problem
for a restricted U This result has imp ortant implications It provides a test which can b e
used to decide when to switch from a direct to an indirect approach within a containment
algorithm It also leads in Section to a theorem which guarantees that the number
of worst case sub divisions in certain containment algorithms is not increased by the use of
restrictions
Sub division
In this chapter we also discuss sub division metho ds We de ne a set of p ossible goals for
sub division metho ds and then discuss several sub division metho ds which have one or more
of these goals One goal is to reduce the size of a U where size can b e interpreted in a
ij
variety of ways We present a size based sub division metho d which reduces the maximum
dimension of the b ounding b ox of a U Another goal is to reduce the number of combi
ij
natorial p ossibilities for a valid con guration We discuss two such combinatorially based
sub division metho ds which partition a U by extending one of its edges These metho ds
ij
move away from the current con guration if it is overlapping A di erent way to move
away is o ered by our new distance based sub division metho d
Overview
Section presents our greedy evaluation strategy Section describ es the over
lap reduction evaluation metho d Section gives a condition for which a single CLP
evaluation can solve a containment problem for a restricted U Section presents sub
division techniques Section de nes sub division and discusses goals of sub division
Section discusses a size based sub division technique Section describ es two
combinatorially based sub division techniques Section discusses our distance based
sub division metho d Section summarizes this chapter and states conclusions
Greedy Evaluation
Algorithm
The metho d we present here is a greedy strategy which shrinks each U to a p oint in
ij
turn then propagates the e ects of that shrinking via SSGR If any U b ecomes empty
ij
during this pro cess then the chosen combination of single U p oints cannot b e part of
ij
a valid solution If on the other hand we succeed in shrinking each U to a p oint and
ij
then applying SSGR without any U b ecoming empty then a valid con guration exists see
ij
pro of of Theorem Section gives heuristics for selecting a shrinking p oint for
each U In the pseudo co de b elow the tolerance test varies dep ending on the sub division
ij
metho d used
GREEDY EVALUATE U k
if U fails tolerance test
return BUILD CONFIGURATION U k
while no U U and some U has not b een shrunken to a p oint
ij ij
select smallest U U which is not a p oint
ij
select a p oint u U
ij ij
U fu g
ij ij
U SSGR U k
if no U U
ij
return BUILD CONFIGURATION U k
return MAYBE
BUILD CONFIGURATION U k
t
for j to k
select t U
j j
t t t
k
return YES
Theorem If each U has been shrunken to a point and a subsequent application
ij
of SSGR does not cause any U to be empty then GREEDY EVALUATE yields a valid
ij
con guration for P
Pro of By de nition a valid con guration for P is a set of translations t i k such
i
that t t U i j k Since U j k and consists of a single p oint
j i ij j
let t b e that p oint Since t f g we have U t t Now for U i j k
j j j ij
SSGR U k U U U In particular for h u u u where
ij ih hj ij j i
u U u U and u U Since u t t and u t t u u t t
ij ij j j i i j j i i j i j i
Thus the p oints of U constitute a valid con guration and further restrictions cannot alter
that fact 2
Heuristics for Selecting u
ij
We present several heuristics for choosing a shrinking p oint u U In Section
ij ij
we compare the e ectiveness of these metho ds in practice The most naive heuristic is to
select a p oint at random Our other heuristics are based on the following claim which gives
a necessary but insu cient condition for the existence of a valid con guration
Claim If a con guration ft t g for P is valid then
k
U U t t
hi hj i j
j i
Pro of For xed h for each i U t t contains the origin The claim follows by
hi h i
translating each U t t i j by t t 2
hj h j i h
Claim means that a solution exists only if there are choices for the u which cause
ij
the translated U s to intersect SSGR enforces pairwise intersection of the translated U s
ij ij
However it do es not guarantee the multiple intersection in Claim This is illustrated
by Figure and Figure Figure shows a containment problem for three small
convex items Figure depicts the three U p olygons for this example unrestricted and
j
scaled up U is not translated U is translated by a vector in U and U is translated
by a vector in U In this case a solution is not found b ecause the p olygons do not have
a common intersection
Figure CN example which do es not yield a common intersection of translated U s
j
Avnaim and Boissonnat s NP algorithm works b ecause whenever the container is a
parallelogram the U s are parallelograms with sides parallel to the container Avnaim
j
and Boissonnat prove that for parallelograms with parallel corresp onding sides pairwise
intersection implies multiple intersection This version of Helly s Theorem holds even if
k
Note One might b e tempted to conclude from this that our algorithm provides an exact
p olynomial time solution for k NP However this is highly unlikely to b e true b ecause k NP
is NP complete see Section The key to this paradox is that Claim is necessary
Figure No common intersection of translated U s
j
but not su cient For NP we can select u and know that u and u exist and satisfy
Claim Unfortunately for k we can t necessarily obtain a self consistent set of
u s for which u u u for all h i j
ij ij ih hj
Convexity of Union
Theorem gives su cient conditions for a set of k p olygons convex or nonconvex
to have a common intersection
Theorem Given a set of polygons P fP P P g if P P is convex i j
k i j
i j k then
k
P
i
i
Pro of If P is nonconvex let p b e a p oint which is on the b oundary of P but is not
k k
on the b oundary of the convex hull of P which we denote by H P We claim p P
k k i
i k Since P is a closed p olygon it contains its b oundary thus p P Now let
k k
Q H P n P Because P P is convex H P P P This implies that Q P
k k i k k i k i
Since P is a closed p olygon it contains the b oundary of any subset of itself Therefore
i
P contains the b oundary of Q which contains p This establishes the claim that p P
i i
i k
1
The generalization from to k p olygons was suggested by Dan Roth
Now supp ose P is convex We prove this case by induction The base case for k is
k
trivially true The induction hypothesis is that our claim is true for k Now we consider
P We claim that P P is convex implies that P P for i k This is true
k i k i k
b ecause of the more general fact that if the union of two closed sets is connected then
they have a p oint in common To establish this fact we argue as follows Let P P b e
i k
connected p P and p P If either p P or p P we are done Otherwise since
i i k k i k k i
P P is connected there exists a path t t where p and p
i k i k
and either t P or t P for t Consider t supft j t P g The p oint
i k i
t is a limit p oint of P t as t increases from Let t infft j t P g We
i k
claim t t For if t t then there exists t such that t t t and t do es not
b elong to either P or P Therefore t P This establishes P P for i k
i k k i k
Now we want to apply the induction hypothesis to the set of k nonempty p olygons
T
k
f P P P P g to establish that P We claim we can do this b ecause
k k k i
i
P P P P is convex i j k This is true b ecause
i k j k
P P P P P P P
i k j k k i j
This is the intersection of two convex sets and is therefore itself convex This completes the
induction and establishes the theorem 2
Corollary Suppose P is a set of polygons satisfying the conditions of Theorem
Let P be convex and intersect al l the other polygons P P i k It fol lows
i
that
k
P
i
i
Pro of In the pro of of Theorem for the case where one of the p olygons is convex
we only used the prop erty that P P That prop erty is satis ed here b ecause of the
i k
conditions of the theorem 2
Note that we have an exact solution to NN if V V u and V u satisfy the
conditions of Theorem or Corollary However these results cannot give an exact
solution to k NN see the note in the previous section The results are still useful b ecause
they suggest a heuristic we can use in our k NN algorithm for selecting u Theorem
ij
suggests we choose u which maximizes the convexity of U U u The problem of
ij i j j i
nding the set of translations which maximizes the convexity of the union of a set of p olygons
is an op en one see Section In our implementation we nd an approximate answer
see Section
Maximizing Intersection Area
In GREEDY EVALUATE shrinking a particular U to a single p oint fu g xes the
ij ij
translation of U with resp ect to U Subsequent applications of SSGR can therefore
hj hi
cause the asso ciated p olygons in U to shrink A containment solution can only b e reached
however if each p olygon in U can b e shrunk to a p oint without causing any p olygon in U
to vanish
Supp ose we choose the shrinking p oint for a U to b e the p oint which shrinks U and
ij hi
U the least We might then have a b etter chance of preventing the p olygons in U from
hj
vanishing This suggests the following heuristic for selecting the shrinking p oint choose
u to b e the p oint in U such that the area of U U u is maximized If U and
ij ij hi hj j i hi
U are simple p olygons with a and b vertices resp ectively then the optimal u can b e
j h ij
found exactly in p olynomial time using a brute force con guration space metho d by solving
O a b two dimensional convex optimization problems Mit
Solutions to some closely related problems often provide large intersection area in prac
tice One p ossibility is to align the centroids of the p olygons Another is to maximize the
Hausdor distance b etween the p olygons An inexp ensive approximation to this is given in
ABB Their result states that if you sup erimp ose the lower left corner of the b ounding
p
b oxes of two p olygons the Hausdor distance b etween them is where is
the minimum Hausdor distance p ossible for the two p olygons Of course there is nothing
sp ecial ab out the lower left corner The pro of works for any corresp onding pair of b ounding
b ox corners
In practice we use a metho d which approximately maximizes the area of U U
hi hj
u We overlay a grid on each comp onent of U and at each grid p oint u we measure
j i ij ij
the area of U U u Section describ es this in more detail
hi hj j i
Comparing Heuristics for Selecting u
ij
We exp erimented with three di erent heuristics for selecting a shrinking p oint u U
ij ij
a random p oint a p oint which approximately maximizes the area of U U u
i j j i
and a p oint which approximately maximizes the convexity of U U u
i j j i
Heuristics and use a two dimensional grid to nd an approximate maximum For
we overlay a grid on each comp onent of U and at each grid p oint u measure the
ij ij
area of U U u We choose a grid resolution for each comp onent which gives ten
i j j i
grid p oints along the longest x or y dimension of the comp onent For each comp onent
we nd the average area and the maximum area From the comp onent with the highest
average value we select the p oint yielding the maximum value
For we again overlay a grid on each comp onent of U In this case however we
ij
measure the area of H U U u n U U u For each comp onent we
i j j i i j j i
nd the average area and the minimum area From the comp onent with the lowest average
value we select the p oint yielding the minimum value
The goal of our exp eriments was to see which of the three heuristics was able to solve
the most containment problems using only geometric restrictions and greedy evaluation
We p erformed several exp eriments Our early exp eriments with NN examples using geo
metric restrictions but not a steady state lo op suggested that choices of u leading to valid
ij
con gurations often have regions of large intersection area b etween the translated U s For
ij
that exp eriment we tested CN and NN examples using metho d A solution was
found without sub dividing in of the examples
Once we had the steady state restriction pro cess in place we ran another exp eriment
using test examples for k The results app ear in Table
Table shows that maximizing the area of intersection heuristic p erforms b est
solving of the problems without sub division Maximizing convexity heuristic
solved of the problems and randomization heuristic only
Although the intersection metho d usually outp erforms the convexity metho d there are
cases in which convexity is able to nd a solution without sub division when intersection
cannot For example Figure shows three U p olygons for the example of Figure
j
Type Number of Examples Random Intersection Convexity
CC
CN
NN
CC
CN
NN
NN
Totals
Table Number of solutions found without sub division
The three p olygons almost have a triple intersection when u is chosen by the intersection
ij
metho d When the convexity metho d is used for the same example Figure shows that
a triple intersection results Note in Figure that the algorithm has matched each pair of
U p olygons along one side in an attempt to maximize the convexity of the union of each
ij
pair
The grid analysis metho d for nding an approximate maximum for intersection app ears
to b e e ective However many set op erations are p erformed in order to choose a sin
gle u shrinking p oint We would like a metho d which is as e ective but is faster see
ij
Section Our initial exp eriments with inexp ensive metho ds for encouraging large
intersection area such as aligning centroids and b ounding b ox corners see Section
suggest that these faster metho ds are not as e ective as grid analysis
Overlap Reduction Evaluation
The disadvantage of the greedy evaluation metho d of Section is that it cannot act
simultaneously on the U s Thus our choices of shrinking p oints for the U s do not
ij ij
necessarily pro duce a consistent set of p oints leading to a valid con guration This limitation
can b e overcome by approximating the con guration spaces by their convex hulls
The metho d we discuss here is based on Milenkovic s containment linear program called
the CLP see Section and Mil DM To create the CLP temp orarily replace every
element U of U by its convex hull H U The set of constraints
ij ij
t t H U i j k
j i ij
can b e expressed as a linear program
The use of CLP as an evaluator predates the use of the CLP for restriction If the CLP
is infeasible then no solution exists However if the CLP is feasible then an arbitrary
feasible solution may or may not b e a valid con guration If U U is not convex then it
ij
is p ossible that t t lies in the convex hull H U but not in U itself In that case the
j i ij ij
con guration is feasible with resp ect to the CLP yet P t and P t overlap
i i j j
We susp ected that applying an overlap reduction algorithm to the overlapping con
guration would greatly improve this evaluation metho d We tested this conjecture by
directly invoking an overlap reduction mo dule which was developed for another task within
Figure No common intersection after maximizing intersection
our pro ject Li and Milenkovic have a metho d for overlap reduction based on linear pro
gramming Li ML LM c LM a This was recently generalized and improved by
Milenkovic We decided to apply this algorithm to improve the p erformance of the naive
CLP Thus our rst attempt directly invoked the overlap reduction mo dule for each over
lapping con guration This was a breakthrough for our containment work Previously the
CLP algorithm had only b een fast for three or fewer p olygons We found that reducing
overlaps allowed this algorithm to solve containment for up to ten p olygons often without
sub division
However there was still ro om for improvement The overlap reduction mo dule op erates
directly on the original i e unrestricted U s and so it cannot take into account the
ij
restrictions that SSGR or SSLPR nd for U Our initial success motivated Milenkovic to
develop a new overlap resolution algorithm as describ ed in MD We brie y summarize
the key features of this algorithm here Full details app ear in MD
Finding an overlap minimum even a lo cal minimum is a nonconvex problem Fortu
nately it is p ossible to rapidly nd a lo cal minimum in practice through iterated linear
programs For the problems we encounter in practice minimization app ears to require at
most two or three iterations The linear program used in this case is an extension of the
CLP called the overlap linear program OLP Building the OLP requires constructing a
set of inner convex approximations to the U s in a fashion similar to Li s use of a lo cality
ij
heuristic for compaction Li The OLP minimizes the maximum distance from the U
ij
to those inner approximations which simultaneously satisfy the CLP
2
The implementation was done by Iraklis Kourtidis a Harvard undergraduate
Figure Common intersection after maximizing convexity of the union
This evaluation technique is very p owerful Like the greedy evaluation metho d it can
sometimes solve containment problems without any sub division It is esp ecially adept for
obvious reasons at nding a valid con guration when a lo ose tting one exists We defer
further discussion of its p erformance to Section
Evaluation of a Restricted U
A containment algorithm based on CLP evaluation can solve a containment problem
using a single CLP evaluation if every U U is convex In this section we sp ecify a
ij
condition under which a single CLP evaluation can solve a containment problem for a
restricted U
Following the notation of Chapter in Theorem b elow we denote the replacement
of U in a valid restriction of U by U and the collection of U s by U We also denote
ij
ij ij
orig
the convex hull of U by H U and we use U to denote U b efore any restriction has
ij ij ij
ij
b een applied to U
orig
Theorem If for every U U H U U then a single CLP evaluation
ij ij ij
solves the containment problem for U
Pro of If the CLP is infeasible then the containment problem for U is infeasible If the
orig
CLP is feasible then b ecause for every U U H U U every p oint of the form
ij ij
ij
orig
t t returned by the CLP is within U In this case the con guration t t t is
j i k
ij
valid 2
This result has two imp ortant implications We show in Section that Theorem
can b e used to guarantee that incorp orating restrictions into a CLP based algorithm do es
not increase the worst case number of sub divisions
In addition Theorem provides a test which can b e used to decide when to switch
from a direct to an indirect approach within a restrict evaluate subdivide containment
algorithm Inside any evaluator one can test if the conditions of Theorem hold If
orig
so one can call a CLP evaluator which tests t t against U to solve the containment
j i
ij
problem for the current hypothesis
Sub division
We now turn to a discussion of sub division techniques Section de nes sub division
and discusses goals of sub division Subsequent sections discuss sp eci c sub division metho ds
Section describ es our size based metho d Section discusses combinatorially based
metho ds Section gives a new distance based metho d
De nition and Goals
Subdivision divides the hypothesis into two sub hypotheses Sp eci cally sub division
It creates two sub hypotheses by replacing and U splits one U U into two parts U
ij
ij ij
U with U or U That is it creates U U n fU g fU g and U U n fU g fU g
ij ij ij
ij ij ij ij
Sub division can have a variety of goals
reduce the size of the U s which can b e interpreted as the number of vertices
ij
the number of re ex vertices the number of comp onents the area or the maximum
dimension of the b ounding b ox for example
orig orig
is U b efore re less where U make the convex hull H U stick out of U
ij ij
ij ij
strictions have b een applied to U In particular if we de ne the maximum overlap of
H U to b e
ij
orig
max dist u U
ij
u H U
ij
then sub division should make the maximum overlap of H U and H U smaller
ij ij
move away from the current con guration
reduce the number of combinatorial p ossibilities for a valid con guration
Dep ending on the containment algorithm one or more of the goals applies
The rst goal can b e accomplished by almost any reasonable splitting algorithm The
goal of reducing the maximum dimension of the b ounding b ox is appropriate for a direct
algorithm Reducing the number of re ex vertices is appropriate for an indirect algorithm
which approximates the U s by their convex hulls We call a sub division metho d which
ij
fo cuses on the rst goal a size based sub division metho d
The last three goals make sense if the sub division metho d is used together with an evalu
ation metho d which always pro duces a con guration If the con guration is an overlapping
one then sub division is invoked The direct greedy evaluation metho d of Section do es
not pro duce a con guration when it answers maybe However the indirect overlap re
duction evaluation metho d of Section do es pro duce an overlapping con guration when
its answer is maybe
The second goal is appropriate for an indirect algorithm which approximates the U s
ij
orig
by their convex hulls This goal arises b ecause if H U U then U need not b e split
ij ij
ij
further Recall that Theorem guarantees that if this condition holds then a single
CLP evaluation solves the containment problem for the current hypothesis Recall also that
Theorem can b e used within any evaluator to identify a situation in which a single
CLP evaluation solves the containment problem for the current hypothesis This implies
that the second goal of sub division can b e viewed as an appropriate one for any containment
algorithm within our restrict evaluate subdivide paradigm
Both the third and fourth goals have the e ect of driving the evaluator away from
the current overlapping con guration Without this goal there is no guarantee that a
single sub division will pro duce sub hypotheses which are easier to evaluate Evaluation
might investigate the same overlapping tentative solution in one of the sub hypotheses
Metaphorically evaluation may get stuck in a rut and a large number of sub divisions
may b e required to move it out of the rut
A sub division metho d which emphasizes the third goal by maximizing the distance from
the current con guration is a distance based metho d A metho d based on the fourth goal
is a combinatorially based metho d
Size Based Metho d
As discussed ab ove a size based sub division metho d reduces the size of the U s which
ij
can b e interpreted as the number of vertices the number of re ex vertices the number of
comp onents the area or the maximum x or y dimension of the b ounding b ox for example
We present here one size based metho d which reduces the maximal dimension of the
b ounding b oxes of the U s Our metho d op erates on the V fU j j k g which has
ij j
the maximum b ounding b ox dimension It cuts V with either a vertical or horizontal line
dep ending on the asp ect ratio of V
SIZE SUBDIVIDE U k
V element of fU j j k g with maximum b ounding b ox dimension
j
B b ounding b ox of V
Assume x is the longest dimension of B
U fV left half of B g U n fV g
U fV right half of B g U n fV g
return U U
Combinatorially Based Metho ds
Here we discuss two sub division metho ds which satisfy the third and fourth goals Each
moves away from the current con guration and do es so in a way that reduces the number
of combinatorial p ossibilitie s Both can b e used with an indirect evaluation metho d which
approximates the U s by their convex hulls The rst part of Section discusses a
ij
metho d which extends an edge of a U This metho d was designed by Milenkovic to work
ij
with the CLP evaluator Mil DM The second part of Section describ es an edge
extension technique developed by Milenkovic MD which dep ends on feedback from an
overlap reduction evaluator
Each of these combinatorially based metho ds has an asso ciated worst case number of
sub divisions which dep ends on the number of vertices in the U s If restrictions are applied
ij
to an hypothesis then the number of vertices in U may increase In the third part of
orig
Section we show that if we use the edges of the U s instead of the U s within
ij
ij
the evaluator and sub division metho ds then any valid restriction may b e applied to an
hypothesis without increasing the worst case number of sub divisions
Extending an Edge
The sub division metho d of Mil DM selects a U U and extends an edge e of
ij
U into a line L The line L is used to cut the p olygon This metho d is used with the
ij e e
CLP linear programming evaluator In the absence of restrictions the worst case number
of sub divisions is given by Theorem In Theorem and later in Theorem
LP a b is the time required to solve a linear program with a variables and b constraints
Theorem DM If subdivision is implemented using cutting lines of the form
L described above then a k NN algorithm which uses CLP evaluation runs in
e
k
O mn LP k O k mn k m
time
Milenkovic s pro of of Theorem rests on the fact that there are only O mn cutting
lines of the form L The number of hypotheses which must b e evaluated is shown to b e
e
the pro duct of the number of cutting lines each of which corresp onds to a cutting plane
in the k dimensional con guration space for the hypothesis times the number of di erent
p ossible vertices in the k dimensional con guration space The number of di erent p ossible
k
vertices is in O mn b ecause each vertex is the intersection of k cutting planes
The con gurations asso ciated with a hypothesis are contained within a convex cell of
the k dimensional con guration space The cell is b ounded by the cutting planes for
the hypothesis This sub division metho d pro duces two sub hypotheses asso ciated with
combinatorially di erent con gurations b ecause their con gurations are inside convex cells
b ounded by di erent sets of cutting planes
Extending the Closest Edge
This section summarizes a di erent sub division metho d designed by Milenkovic MD
that drives the evaluator to a con guration which is combinatorially di erent This algo
rithm in combination with SSLPR and a mo di cation of the OLP of Section gives
containment a provable running time b ound but it sacri ces practicality for the sake of
this b ound
The technique receives a p oint u from the OLP evaluator The p oint u corre
max max
sp onds to the pair of p olygons with the most overlap The cutting line is chosen to b e the
extension of the edge of the U which is closest to u
ij max
This type of combinatorially based sub division requires some mo di cations to evalu
ation The mo di ed OLP has only a single distance constraint for each U and this
ij
constraint is determined by the closest edge of U to the current value of t t inside the
ij j i
convex hull H U We imp ose an arbitrary constant b ound on the number of iterations of
ij
overlap minimization to prevent it from solving an exp onential number of OLPs
With this sub division metho d the containment algorithm requires
k
k mn k m
O LP k k k O k mn k m
k
k
time MD Here O k mn k m k is the number of lo cal minima visited by the
algorithm and LP k k k O k mn k m is the time for an evaluation
If we mo dify the containment algorithm we can claim the following running time which
has an exp onent of k instead of k
Theorem MD There exists a k NN algorithm which uses combinatorially based
subdivision and requires
k
k mn k m
O LP k k mn k m
k
time
Milenkovic s pro of is sketched in MD For the pro of the containment algorithm s
notion of a hypothesis is mo di ed Instead of using U as a hypothesis it uses F fF
ij
i j k g where F b egins as the convex hull of the original U All restriction and
ij ij
sub division acts on F without mo difying the U p olygons The restriction is basically
ij
the SSLPR of Section but we do not replace F by H F U and as a result
ij ij ij
restriction automatically terminates after one iteration Finally when we split a hypothesis
we immediately restrict b oth sub hypotheses b efore recursing Evaluation acts on the F
ij
U p olygons and the splitting edge is selected from one of these as usual
ij
This may seem like a small improvement over the simple
k
O mn LP k O k mn k m
LP based algorithm DM the exp onent on mn is smaller but the number of variables
is larger However the new algorithm visits only lo cal overlap minima inside convex cells
b ounded by cutting lines There are reasons to exp ect the number of such minima to b e
indep endent of m and n and dep end only on an exp onential function in k
Incorp orating Restrictions
A containment algorithm which uses either of these combinatorially based metho ds has
an asso ciated worst case number of sub divisions which dep ends on the number of vertices
in the U s If restrictions are applied to a hypothesis then the number of vertices in U
ij
may increase thus increasing the worst case number of sub divisions We show here that
in each case the same small mo di cations to evaluation and sub division allow us to use
restrictions without increasing the worst case number of sub divisions
These mo di cations are based on Theorem The rst mo di cation is within the
orig
linear programming evaluator We test each t t for inclusion in U not U The
j i
ij
ij
second mo di cation is that the cutting lines used for the sub division are extensions of the
orig
edges of the U s not the U s These mo di cations preserve the running times of the
ij ij
algorithms as shown by the following theorem and corollary
Theorem Theorem holds for U if the evaluator tests each t t for inclusion
j i
orig orig
in U and subdivision uses extensions of the edges of the U s
ij ij
orig
Pro of Consider nonconvex p olygon U In the algorithm referred to in Theorem
ij
orig
in the worst case U is sub divided at every re ex vertex The result is a set of convex
ij
orig
p olygons Now use the same partitioning lines to sub divide U U Each subp olygon
ij
ij
orig
of U is inside exactly one convex subp olygon of U hence the convex hull of each
ij
ij
orig
subp olygon of U is in U Now Theorem guarantees that a CLP evaluation which
ij
ij
orig
tests t t for inclusion in U solves the containment problem for U Thus the number
j i
ij
of sub divisions for the worst case is the same 2
Corollary Theorem holds for U if the evaluator tests each t t for inclusion
j i
orig orig
in U and subdivision uses extensions of the edges of the U s
ij ij
Theorem provides a way to incorp orate restrictions into the exact indirect k NN
algorithm of DM Mil without a ecting the worst case running time The sub division
metho d which results from applying Corollary to the metho d asso ciated with Theo
rem app ears in MD
Distance Based Metho d
tj - ti
U U
ij ij
Figure Distance based sub division
Extending the closest edge gives containment a provable running time b ound but it
sacri ces practicality for the sake of this b ound Here we give a new cutting algorithm that
moves the con guration the maximum p ossible distance from the current con guration
Because it fragments the regions much less this algorithm is much more practical although
we do not have a provable running time b ound
Distance based sub division attempts to move the tentative con guration as far as p os
sible from the current con guration As b efore let P t and P t b e a pair of p olygons
i i j j
with maximum overlap under the current con guration The sub division algorithm selects a
line L through t t t and uses L to cut U into U and U It selects the line L that
ij j i ij max
ij ij
maximizes the minimum distance d from t to H U and H U This guarantees that
ij
ij ij
all p oints within distance d of t are removed from further consideration in the algorithm
ij
see Figure In other words it eliminates some of the overlapping con gurations which
are close to the current overlapping one The line L can b e computed in O n log n
max ij ij
time where n jU j using a fairly intricate sweep algorithm We summarize here a
ij ij
simpler and more practical approximate algorithm with essentially the same order running
time
First compute the visibility p olygon V for H U U with resp ect to t If t cannot
ij ij ij ij
see the b oundary of H U in any direction then the maximum distance is zero and
ij
the algorithm chooses the line L through the nearest p oint to t on the b oundary of U
ij ij
Otherwise let ab b e an arc of the b oundary of H U that is visible from t Parameterize
ij ij
ab by piecewise linear for Let the cut line b e L t it partitions U
ij ij
into U and U Let U b e the p ortion of U containing a and U b e the p ortion of U
ij ij
ij ij ij ij
containing b For each value of let d and d b e the distances of t to H U and
ij
ij
H U resp ectively
ij
Theorem For and d d and d d
Pro of Let p b e the rst p oint b eyond t where the ray t intersects V Similarly
ij ij
let q b e the rst p oint b eyond t where the ray t intersects V We show that
ij ij
d d cannot o ccur For let r b e the closest p oint of H U to t We show
ij
ij
for Now all p oints of U on or to the right of L are in H U that r H U
ij
ij ij
for The p oints of U on or to the right of L are on or to the right of t q Let
ij ij
w b e the set of p oints on or to the right of t Because t can see all p oints
ij ij
on the arc ab no p oints of U are in the interior of w Therefore all p oints of U on or to
ij ij
the right of L are also on or to the right of L If r is a p oint of U then it must
ij
for If r is not a p oint of U then it is on a line b etween two therefore b e in H U
ij
ij
p oints of U and thus is also in H U for Therefore d d A similar
ij
ij
argument establishes that d d 2
Because d increases monotonically and d decreases monotonically the maximum
distance can b e found to any degree of accuracy by binary search on The running time
of the practical algorithm is O n log n n log log n where the accuracy is and
ij ij arcs ij
the number n of arcs is the number of comp onents of U which is generally much
arcs ij
less than n
ij
Sub division is only invoked if the OLP fails to resolve all overlaps in this case it is
often true in practice that t t is very close to the b oundary of U It is often true that
j i ij
the overlap is b etween two p olygons as opp osed to b etween a p olygon and the container
Thus the situation often resembles that shown in Figure In this gure the shaded
regions are the comp onents of U
ij
In Section we show that distance based sub division together with the overlap
reduction evaluation technique allow our hybrid algorithm to detect infeasibili ty b etter
than the naive CLP based exact algorithm
3
This can b e found in O n time GA Lee JS or in n log n time if the p olygon has holes
O R Restrictions tend to fragment U so an individu al U may have many comp onents this means
ij
holes may b e encountered
p
Figure Common situation for p t t
j i
Summary and Conclusions
In this chapter we presented several evaluation and sub division metho ds We o er two
evaluation metho ds Our greedy evaluator op erates directly on the con guration spaces and
employs SSGR The key to its e ectiveness is a go o d choice of a shrinking p oint for each
U Selecting a p oint which maximizes the intersection area of translated U s app ears to
ij ij
work well in our exp eriments this strategy solves of the containment problems without
any sub division
The other evaluation metho d is an overlap reduction evaluator This is based on im
provements to an indirect containment linear program CLP evaluator A con guration
which is feasible for the CLP evaluator may b e overlapping If the overlaps can b e elimi
nated a valid con guration can b e found An overlap reduction evaluator is sometimes able
to solve containment problems without any sub division It is esp ecially adept at nding a
valid con guration when a lo ose tting one exists
Our discussion of evaluation metho ds concludes with the observation that if the convex
hull of every U is within its unrestricted U then a single CLP evaluation solves the
ij ij
containment problem for the asso ciated hypothesis This test can b e used to decide when
to stop sub dividin g a hypothesis inside any containment algorithm which is based on our
paradigm
We o er four di erent p ossible sub division goals reduce the size of the U s
ij
make the convex hull of a U protrude less from its original U this is motivated by the
ij ij
observation of the previous paragraph move away from the current con guration and
reduce the number of combinatorial p ossibilities for a valid con guration We present
a size based sub division technique which abides by the rst goal and reduces the maximal
dimension of the b ounding b oxes of the U s We discuss two combinatorially based meth
ij
o ds developed by Milenkovic which satisfy the third and fourth goals We show that a
containment algorithm which uses either of these can use restrictions without increasing
the worst case number of sub divisions We introduce a distance based sub division metho d
which satis es the third goal This practical metho d eliminates some of the overlapping
con gurations which are close to the current overlapping one
C h a p t e r
Algorithms
Introduction
Our techniques can b e combined to form a variety of two dimensional translational
containment algorithms which follow the restrict evaluate subdivide paradigm introduced
in Section As we noted in Section termination of the algorithm dep ends on the
compatibility of the techniques In this chapter we present two ways of combining techniques
into k NN containment algorithms
Our rst algorithm is an approximate algorithm That is it nds an approximate
con guration for k NN refer to De nition in Section The algorithm uses the
geometric restrictions SSGR of Section the greedy evaluator of Section and the
size based sub division metho d of Section All of these metho ds are direct and so this
algorithm is a direct approximate algorithm
The approximate algorithm has additional structure b eyond our paradigm It uses
the containment characterization of Section to create a set of k boundary restricted
containment subproblems Each of the subproblems is solved using restriction evaluation
and sub division Our approximate algorithm app ears in DM DM as well as in this
chapter
The second algorithm is an exact algorithm It uses b oth the geometric restrictions of
Section and the linear programming restrictions SSLPR of Section It uses the
overlap reduction evaluation of Section It employs the distance based sub division of
Section so that when the p olygons of a hypothesis are sub divided into more convex
pieces this is done with the goal of moving the con guration as far as p ossible from the
current con guration We consider this exact algorithm to b e a hybrid algorithm b ecause
it uses an indirect evaluation metho d and b oth direct and indirect metho ds for restriction
Our hybrid algorithm is also describ ed in MD This algorithm currently yields our b est
practical results primarily due to its p owerful evaluation metho d
Ironically the approximate algorithm op erates directly on the actual U s whereas the
ij
exact algorithm uses approximations of the U s
ij
Both of our algorithms have practical running times for small numbers of items The
container and items may b e nonconvex Most of our examples are drawn from apparel
manufacturing For our apparel examples the number of vertices in the container ranges
from to and the number of vertices in each item ranges from to We pro duce
a feasible containment problem from an apparel layout by removing a group of neighboring
items from the layout To create an infeasible problem we move some of the remaining
items a small distance into the container
For ten or fewer items the hybrid algorithm typically requires several minutes or less
on a MHz SPARCstation The current implementation of approximate algorithm is fast
for up to four items For k the current implementation of SSGR which is based on
p olygon set op erations slows the algorithm down considerably Increasing the sp eed of our
SSGR implementation is a sub ject of future work see Section In our exp eriments
SSGR evaluates fewer hypotheses than the faster hybrid algorithm and so we b elieve the
approximate algorithm has the p otential to b e comp etitive with the hybrid algorithm
Implementation note Both of our algorithms require robust geometric techniques LM b
MN Mil a Mil b in order to function well in practice
Overview
This chapter contains one section for each of our two containment algorithms Section
discusses our approximate algorithm and gives exp erimental results Section describ es
our hybrid algorithm and presents exp erimental results Section summarizes this chapter
and states conclusions
An Approximate Containment Algorithm
Introduction
We present a numerically approximate algorithm for nding an approximate con g
uration for k NN refer to De nition in Section The value of is an input to
the algorithm In industrial cutting applications the containment solution acts as a guide
to a machine cutting out p olygonal shap es from a sheet of material If one chooses to
b e a fraction of the cutter s accuracy then the solution to the approximate containment
problem is su cient for industrial purp oses In fact for some cutting applications CAD
vendors round p olygon vertex co ordinates to integers In such cases an overlap of up to
one unit corresp onding to inches is acceptable For the applications we consider the
ratio of the largest value to the smallest nonzero value is usually under this makes
an approximate algorithm a practical choice
Given a containment problem we use our characterization of translational containment
to create a collection of k containment subproblems see Section We solve each sub
problem by rst applying our steady state geometric restrictions see SSGR in Section
and then using the greedy evaluator of Section If necessary we sub divide the con gu
ration space using size based sub division to generate new subproblems The running time
k
of our algorithm is O log k s log s where s is the largest number of vertices in
a p olygon restricted by SSGR see the discussion of size analysis in Section In the
worst case s can b e exp onential in the size of the input but in practice it is usually not
more than quadratic
In a sense our algorithm substitutes sub division for the iteration used in Avnaim s exact
NN algorithm see Avn and the discussion of related work in Section In prac
tice much iteration is required in an exact algorithm b efore hitting the right combination
of edges If the answer is no one must check al l combinations of edges Our SSGR
alone can often detect infeasibility see Section SSGR together with our greedy
evaluation metho d can often nd a valid con guration without requiring the algorithm to
use sub division see Section While restriction and sub division often p erform b et
ter than iteration iteration can have a much b etter theoretical worst case running time
b ound Even when sub division is unnecessary the running times of our algorithm are in
O m n log mn for CN and in O m n log mn for NN These seem much worse than
the iteration based metho ds but in terms of the size s of the largest p olygon generated by
SSGR the running times are b oth in O s log s as opp osed to O s log s for the iteration
based algorithms As mentioned in Section it is not clear that the worst case results
for consecutive p olygon op erations are attainable The tests we have run indicate that the
running times of our algorithm are much faster than the b est times we would estimate for
the iteration based solutions to NN
The high level description of our algorithm in the second part of this subsection draws
its structure from our characterization of containment b ecause the characterization allows
us to decomp ose k NN into a set of subproblems The algorithm solves each subproblem
by rst restricting U using SSGR testing for a solution using the greedy evaluation of
Section and sub dividing the con guration space if a solution is not found Section
establishes the correctness and running time of our algorithm Section discusses our
implementation and also presents containment examples
Applying the Characterization
Our philosophy is to avoid iteration as much as p ossible b ecause iteration contributes
to the worst case cost of the algorithm particularly in terms of s analysis For instance
Avnaim and Boissonnat s algorithm for NP AB Avn has running time O s log s
and Avnaim s algorithm for NN Avn has running time O s log s The algorithm for
NP is faster b ecause it avoids iterating over the edges and vertices of p olygons Like the
algorithm for NP we replace iteration with restriction as much as p ossible
There is another type of iteration one must consider This iteration is based on the
characterization of Theorem To apply the entire characterization one needs to choose
k
contacting pairs in a number of ways which is exp onential in k roughly k If one were to
k
generalize Avnaim s algorithm for NN Avn to an algorithm for k NN the factor of k
would not b e a signi cant part of the cost compared to the factor dep endent on m and n
2
k k k
roughly m mn For our approach however a factor of k is signi cant
It is p ossible to use restriction and sub division and avoid iterating over any contacting
pairs In fact this is what we initially implemented However we feel that considering
contacting pairs is p otentially very useful For instance if we invoke a condition weaker
than the second part of Theorem to assume that a particular P is in contact with P
j
then we can replace U with its b oundary Note that this b oundary restriction covers the
j
cases where P has either a single or a double contact with some P Although we restrict
j
U to its b oundary we are not iterating over vertices or edges of U unlike Avnaim s NN
j j
algorithm
This partial characterization implies a signi cant restriction However the disadvantage
is that we must iterate over all k choices of j by solving k subproblems Our current
1
However as noted in Section these subproblems can b e solved in parallel In our current implemen
conjecture is that iterating over O k choices of partial characterization will allow us to
avoid sub division often enough to o set the extra factor of k in cost We present data in
Section to supp ort this conjecture in the case of k
The b oundary restriction do es not add an extra factor of k in cost for containment
problems in which all the items are identical In this case all k subproblems are identical
and so only one subproblem must b e solved
The Algorithm
In the algorithm b elow P and U are set initially to the values given in Section If
at any p oint a valid con guration is found execution terminates with success We assume
that the tolerance test TOO SMALL is used within GREEDY EVALUATE The used
in TOO SMALL is the same used within SSGR BUILD CONFIGURATION is given in
Section Note that in order to p erform a b oundary restriction on a U we need only
j
orig
j
U represents a copy of U for the j th pro cessor intersect it with the b oundary of U
j
j
is pro cessor j s con guration space for P with resp ect to P and U
j
j
APPROX CONTAIN U k
orig
U U
U SSGR U k
if some U U return NO
ij
if TOO SMALL U k
return BUILD CONFIGURATION U k
Search for a b oundary restricted solution
for j to k do in parallel
j
U U
j orig
U U BOUNDARY OF U
j
j j
j
return SOLVE SUBPROBLEM U k
SOLVE SUBPROBLEM U k
U SSGR U k
if some U U return NO
ij
status GREEDY EVALUATE U k
if status YES return YES
if status NO return NO
U U SIZE SUBDIVIDE U k
return SOLVE SUBPROBLEM U k or SOLVE SUBPROBLEM U k
TOO SMALL U k
V element of fU j j k g with maximum b ounding b ox dimension
j
B b ounding b ox of V
p p
if heig ht B and w idth B return YES
return NO
tation we solve them sequentially
Analysis
Correctness
Claim If a valid con guration exists then the algorithm wil l nd an approximate
solution
Pro of The algorithm converges b ecause each sub division reduces the height or width of
the b ounding b ox of V by a factor of two stopping when the height and width reach the
p
tolerance
If a solution exists then our characterization guarantees that one of the b oundary
restricted subproblems has a solution We claim that our algorithm do es not throw away
any solutions to these subproblems This is b ecause Theorem guarantees that SSGR
is a valid restriction
Now we argue that any solution which is found is an approximate solution If GREEDY
EVALUATE nds a solution then Theorem guarantees that solution contains no over
lap hence it is a valid con guration A valid con guration is an approximate solution
The only other way the algorithm can nd a solution is through a call to BUILD
CONFIGURATION This o ccurs when the height and width of the b ounding b ox of V
p
In this case the diameter is less than for all p olygons in V In that are b oth
case we argue that no p oint of any p olygon is more than inside of any other p olygon s
b oundary None of the p olygons overlap the container b ecause each t chosen in BUILD
j
CONFIGURATION is in U Thus we need only b ound the overlap b etween a pair of
j
p olygons P and P i j k
i j
Any u U having survived SSGR can b e expressed as t t where t U and
ij ij j i i i
t U If P is placed at t and P is placed at t P and P do not overlap Now let
j j i i j j i j
v and v b e the p oints of U and U chosen in BUILD CONFIGURATION Let D p q
i j i j
denote the distance b etween p oints p and q Since the diameter d of any U is less than
i
we have D v t and D v t Translating P from t to v can therefore cause
i i j j i i i
a p oint of P to b e at most a distance of inside P Translating P from t to v can at
i j j j j
most double this overlap Therefore the maximum overlap b etween two p olygons in the
con guration ft t t g is 2
k
Running Time
Here we calculate the asymptotic running time of our algorithm We assume
s is the largest number of vertices of any p olygon generated by SSGR see the discus
sion in Section
is the maximum overlap b etween two p olygons
is the fractional shrinkage in area required by each iteration of SSGR
k
Theorem The running time of our algorithm is in O log s log s k
Pro of Let S b e the number of sub divisions and R b e the running time of SSGR The
running time is dominated by S k R To nd S we rst assume that the maximum height
or width of the b ounding b ox of V is q The height of the sub division tree is therefore
O k log q l og The k term corresp onds to the total number of p olygons that are
sub divided in the worst case The number of no des in the sub division tree is therefore
k
q q
k log log
2 2
O O
k k
q
O S O
From Section R is in O log k s log s
The total running time is therefore
k
log k s log s S k R O
2
Results
This section describ es our implementation of the algorithm for approximate containment
We set equal to and to SSGR is used for k We have found that for k the
algorithm runs so quickly and so rarely requires sub division that one iteration of geometric
restriction is all that is needed
General Results
Table contains running times for our algorithm for a variety of examples for
k All running times given in the table are to the nearest second on a MHz
SPARCstation The setup time to calculate U p olygons is not included in the running
ij
time Infeasible examples are identi ed by All infeasible examples are ones for which
each pair of p olygons ts into the container and the area of the container is larger than
the total area of the p olygons Our restriction pro cess enables the algorithm to often detect
infeasibility very quickly
Figure through Figure show con gurations achieved for four examples using our
algorithm The tightness of the layouts is typical In each gure the items placed by our
containment algorithm have dark shading Figure and Figure are NN examples
Figure is a NN example Figure is a NN example
Figure NN example time seconds
2
SPARCstation is a trademark of SPARC International Inc licensed exclusively to Sun Microsystems
Inc
Type Time Type Time Type Time
CC CC CC
CC CC CN
CN CN CN
CN CN CN
CN CN CN
CN CN CN
CN CN CN
CN CN CN
CN CN CN
CN CN CN
CN CN CN
CN CN CN
CN CN CN
CN NC NN
NN CN CN
CN CN CN
CN CN NN
NN NN NN
CC NN NN
NN NN NN
CC CC
Table Running times in seconds denotes infeasible example
Figure NN example time seconds
Figure NN example time seconds
Figure NN example time seconds
Table b elow presents running times in seconds for the approximate algorithm on
a di erent small collection of examples for k These tests are for a MHz
SPARCstation For each example we also give the number of hypotheses examined by the
approximate algorithm Judging the approximate algorithm purely by its running time can
b e deceptive Although the algorithm takes anywhere from to minutes in these exam
ples it is only examining or hypotheses Most of the running time is consumed by SSGR
whose current implementation is roughly ten times more slow than that of SSLPR Faster
implementations of SSGR will directly lead to faster running times for the approximate
algorithm In fact for an exp eriment in Section the number of hypotheses examined
by our fastest hybrid algorithm is greater than the number of hypotheses examined by
the approximate algorithm We therefore b elieve that given fast enough p olygon set op
erations the approximate algorithm has the p otential to b e comp etitive with the hybrid
algorithm
Type Running Time Number of Hyp otheses
NN
NN
NN
CN
CN
CC
Table Running time in seconds and number of hypotheses examined by approximate
algorithm
Solving Subproblems
In our comparison of evaluation heuristics for choosing a U shrinking p oint we rep orted
ij
in Table that the metho d of maximizing the intersection area solved of the
examples without sub division Here we maximize the intersection area of translated U s
ij
and we compare the e ect of using k subproblems versus not using subproblems That is
we evaluate the e ectiveness of applying a partial containment characterization
Our examples use k Figure shows an example of a feasible NN problem solved
using the characterization The darkly shaded items numbered and were
placed Note that three of the four items are in contact with the b oundary of the container
We rst compare the sequential cost in terms of the number of hypotheses examined for
infeasible as well as feasible examples Results are shown in Table For the examples
the subproblem approach examines a total of hypotheses whereas the non subproblem
approach examines hypotheses These results suggest that for k applying the
partial characterization to the sequential algorithm is b ene cial
When all the items are identical the k subproblems generated by the characterization
are all identical and so we need only solve one of them The results ab ove contain
problems for which all items are identical The remaining problems are for non identical
items The results for identical item problems are listed separately in Table The total
number of hypotheses examined in the characterization case is whereas the number 25 8 28 7 26 21 27 6 47 5 37 61 63 11
58 10 44 62 45 13 46 32 30 31 55 48 36 12 49 57 33 50 38 15 35 43 14 39
22
16 17 40 41 23 56 42 60
20
Figure NN feasible example solved using partial characterization
Type Subproblems No Subproblems Type Subproblems No Subproblems
CC CC
CC CN
NN NN
NN NN
NN CC
NN CN
CN CN
NN CN
CN CN
CN CN
CN CN
CN CN
CN CN
CN CN
CN NN
NN CN
CN CN
CN CN
CN CN
NN NN
CC NN
NN CC
NN CN
Table Number of hypotheses examined by sequential algorithm denotes infeasible
example
examined in the non characterization case is For the non identical item problems
the total number of hypotheses examined in the characterization case is whereas the
number examined in the non characterization case is Thus for identical as well as
non identical item problems the characterization examines fewer hypotheses for k in
our exp eriments
Type Subproblems No Subproblems Type Subproblems No Subproblems
CN CN
CN CN
CN CN
CN NN
NN CN
CN CN
CN CN
CN CN
NN NN
NN CN
CC
Table Number of hypotheses examined for identical items denotes infeasible ex
ample
Intuitively we exp ect that a single b oundary restricted subproblem will not examine
more hypotheses than the non subproblem approach b ecause we are eliminating so much
of the search space Often this is the case However our exp eriments reveal that it is
p ossible for a b oundary restricted subproblem to evaluate a few more hypotheses than a
problem in which the b oundary restriction is not applied We b elieve that this phenomenon
is related to the evaluation metho d Recall that our evaluation metho d maximizes the
intersection area of U p olygons If a general p olygon contains p olygonal regions plus
ij
isolated edges and p oints the intersection heuristic only considers the p olygonal regions
If no p olygonal regions are present we currently resort to a simple alternative heuristic
Thus if the b oundary restriction causes many of the U s to consist of only isolated edges
ij
and p oints we currently lose the p ower of the evaluation heuristic Section suggests
a p ossible remedy for this
Assessing the cost e ectiveness of applying the partial characterization for k is
a sub ject for future work We b elieve that as k increases the partial characterization
will continue to b e cost e ective for identical item problems b ecause only one subproblem
must b e examined The partial characterization is less likely to have an advantage in
sequential tests for non identical items as k increases due to the increase in the number of
subproblems Once the critical value of k is identi ed for which this advantage is lost a
detailed comparison of the parallel cost will b e necessary in order to justify using the partial
characterization for values of k larger than the critical one Section discussed one way
in which course grained parallelism can b e applied to our algorithm A queue of hypotheses
can b e maintained Each hypothesis represents an indep endent containment subproblem
which is created via sub division Whenever a pro cessor is available it op erates on the
next hypothesis in the queue The subproblems generated by the characterization are also
indep endent from each other and so the hypotheses they generate are indep endent and can
b e added to the queue Note that applying the characterization creates subproblems which
are al l available at the start of the algorithm
We remark that the number of subproblems which must b e examined is smaller in
theory for k CN examples than for k NN This is due to the following theorem and corollary
Theorem If an instance of k CN for k has a valid con guration then that
instance has a valid con guration in which at least three items are in contact with the
boundary of the container
Pro of Consider a valid con guration for an instance of k CN for k Milenkovic
shows Mil that given a set P of non overlapping p olygons at least one of them can move
unobstructed to in nity in the direction This disassembly lemma works for any
direction Thus for every direction d there is some p olygon P P such that P can move
to in nity in direction d Let S b e the subset of p olygons in P which can move to in nity
in at least one direction Let S b e a minimal subset of S one which covers all directions
using the smallest number of p olygons We now show that S contains at least two p olygons
First we observe that either every p olygon in P can move to in nity in some direction in
which case we are done or at least one p olygon Q cannot It is easy to show that one
convex p olygon alone cannot surround another convex p olygon and so it takes at least
two p olygons to prevent Q from moving to in nity Thus S contains at least two p olygons
Since S contains at least two p olygons and all directions are covered by p olygons in S at
least two p olygons P and P in S share a common direction d This means that we can
move b oth P and P in direction d until each hits the b oundary of the container Because
the direction d is indep endent of direction d and there are at least three p olygons we can
apply the disassembly lemma to P n fP g fP g to move a third p olygon P in direction
d until it hits the container 2
We note that three items is tight b ecause there are sets of four convex p olygons for
which only three can b e moved to in nity
Corollary When solving an instance of k CN using the partial characterization ex
amining k subproblems is su cient
Corollary implies that examining one subproblem is su cient for CN examples
This however do es not yield the fastest known algorithm for CN Milenkovic Mil ob
served that for a valid CN con guration each of the three p olygons can b e moved to the
b oundary of the container This forms the basis of his O m n log mn time CN algorithm
Below we compare the p erformance of this very fast CN algorithm with our approximate
algorithm on CN examples
Comparison with Exact Algorithm for Convex Items
Table compares running times for Milenkovic s CN exact algorithm DML Mil
with our approximate algorithm on CN examples Infeasible examples are identi ed by
The exact algorithm tests separating line orientations b etween pairs of convex p oly
gons Because b oth algorithms are so fast timing comparisons are based on calls to
each For feasible examples the exact algorithm p erforms ab out seven times faster than
the approximate algorithm on average For infeasible examples the exact algorithm is only
ab out twice as fast on average This is no accident When the exact algorithm was rst
designed the approximate algorithm s p owerful restriction pro cess gave it a considerable
advantage over the exact algorithm for infeasible examples The exact algorithm was re
designed with the goal of detecting infeasibili ty quickly in mind The ma jor part of the
redesign was a multiple resolution metho d for evaluating sets of separating lines
Example Exact Approximate Example Exact Approximate
Table Exact vs approximate running times for CN examples calls each
Sub division
The vertical and horizontal cut metho d of Section has b een implemented For our
applications CAD vendors round p olygon co ordinates to integers so an of units is
su ciently small It is rare for the algorithm to sub divide down to tolerance Often no
sub division is required at all to nd a valid con guration This is b ecause we have a heuristic
for selecting a shrinking p oint for each U which works well in practice see Section
ij
Sub division is rarely required for infeasible situations b ecause our restriction pro cess often
detects infeasibili ty early in the algorithm
A Hybrid Containment Algorithm
Introduction
Our hybrid algorithm is the result of signi cant improvements to the restriction eval
uation and sub division techniques of an existing indirect algorithm of Milenkovic Mil
Milenkovic s indirect algorithm in its original form was slower than our direct algorithm
for infeasible problems and its practical upp er b ound on k was k The improved
indirect algorithm is currently the fastest containment algorithm for k
Our rst improvement to the indirect algorithm improves its ability to detect infeasibil
ity by proving that restrictions when used in the indirect algorithm do not increase the
algorithm s worst case running time if we p erform an appropriate test within the evaluation
step and sub divide using an appropriate set of cutting lines This improvement is discussed
in detail in Section This improvement allows us to use b oth SSGR Section and
SSLPR Section in the algorithm without increasing the worst case running time We
use some SSGR restriction at the top level and SSLPR at the hypothesis level
The second improvement to the indirect algorithm improves the ability of the evaluator
to lo cate a valid con guration We apply the work of Li and Milenkovic on overlap reso
lution Li ML LM c LM a to improve the evaluation step Their overlap resolution
algorithm is an indirect algorithm It uses inner convex approximations an approach which
complements the outer approximations of the indirect containment algorithm The success
of this exp eriment led Milenkovic to develop the overlap reduction evaluator describ ed in
Section We use that evaluator in the hybrid algorithm
A combinatorially based sub division algorithm yields a running time of
k
k mn k m
O LP k k mn k m
k
where LP a b is the time to solve a linear program with a variables and b constraints
although the actual running time dep ends on the number of lo cal minima which is likely
k
to b e far smaller than mn An alternative heuristically based sub division algorithm
the distance based sub division metho d of Section gives containment an even b etter
running time in practice and is guaranteed to not create new lo cal overlap minima
Our practical distance based sub division technique is our third improvement to the indi
rect algorithm This metho d attempts to maximize the distance from the current evaluation
to the convex approximations of the sub divided con guration spaces This technique is de
scrib ed in Section It moves the evaluator away from the current con guration so that
evaluation do es not get stuck in a rut It do es this by eliminating some of the overlapping
con gurations which are close to the current overlapping one
We call our improved algorithm a hybrid algorithm b ecause it uses indirect metho ds
for evaluation developed by Li and Milenkovic and b oth direct and indirect metho ds for
restriction This algorithm is also describ ed in MD
The hybrid algorithm in addition to b eing a formidable containment algorithm has
the prop erty that a partial containment solution from it is valuable This is b ecause each
evaluation step pro duces either a non overlapping solution or an overlapping arrangement
for which a linear program has reached a lo cal minimum As a result the hybrid algorithm
when terminated early is the most p owerful heuristic known for generating a lo cal overlap
minimum
In our exp eriments the new algorithm clearly outp erforms purely geometric containment
algorithms such as the approximate algorithm of Section However it do es this by com
bining rather than replacing geometric techniques with linear programming This hybrid
algorithm demonstrates the manner in which linear programming can greatly increase the
p ower of geometric algorithms
High Level Algorithm Description
The algorithm starts with a queue Q of hypotheses containing a single hypothesis the
set of original U i j k as calculated using Equation We assume that this
ij
hypothesis has b een restricted using some geometric restrictions
HYBRID CONTAIN Q k
while Q is not empty
U DEQUEUE Q
U SSLPR U k
if some U U continue while lo op
ij
status OVERLAP EVALUATE U k
if status YES return YES
if status NO continue while lo op
U U DISTANCE BASED SUBDIVIDE U k
ENQUEUE U Q
ENQUEUE U Q
return NO
Results
The examples we selected are primarily from the apparel industry We use SSLPR only
at the deep est levels of hypothesis splitting for two reasons The rst is cost Although
SSLPR is several times faster than SSGR it is not necessarily cost e ective to invoke it
prior to evaluating each hypothesis The second is that SSLPR restricts p olygons more as
their convex hulls b ecome b etter approximations to them
Comparison with Other Algorithms
We compared the p erformance of our hybrid algorithm with two algorithms the approx
imate algorithm of Section and the naive exact k NN algorithm of DM Mil which is
based on the CLP Table gives running times for feasible containment problems Run
ning times are expressed in seconds on a MHz SPARCstation Table gives running
times for infeasible containment problems Our new algorithm clearly outp erforms b oth of
the other algorithms For ve or more p olygons the di erence is dramatic Figure and
Figure depict con gurations obtained by our algorithm
The approximate algorithm uses SSGR whose current implementation runs ab out ten
times more slowly than SSLPR A more fair comparison b etween the approximate and
hybrid algorithms counts the number of hypotheses examined instead of simply the running
time Table shows the results of such an exp eriment The approximate algorithm
examines a total of hypotheses whereas the hybrid algorithm examines From
this p ersp ective the approximate algorithm app ears more p owerful For the CC example
b oth algorithms nd a solution without any sub division each evaluates only one hypothesis
However the approximate algorithm requires seconds whereas the hybrid algorithm
nds the solution in only seconds The time di erence here is due to the di erence in the
sp eed of SSGR and SSLPR and to the fact that the greedy evaluator invokes SSGR each
time a U is shrunk to a p oint
ij
SSGR vs SSLPR
Here we discuss the e ectiveness of SSGR and SSLPR within our new algorithm
We tested our algorithm on di erent feasible and infeasible containment examples for
k For each example we varied the type of restriction used no restriction
Type APPROX NAIVE LP HYBRID
NN
NN
CC
CC
CN
NN
NN
NN
NN
CC
NN min min
NN
NN
NN
CC
CN min min
CN
CN
NN min min
NN min min
CC
CN min min
NN min min
CC min min
CC
NN
Table Running times for feasible examples in seconds
SSGR SSLPR and SSGR followed by SSLPR In all cases we b egan the containment
algorithm by applying Equation which b ounds the unbounded U s
ij
Let N b e the number of hypothesis evaluations required if we use no restrictions at the
individual hypothesis level Let L b e the number of evaluations if we use SSLPR for each
hypothesis Let G b e the number of evaluations if we use SSGR for each hypothesis Finally
let GL b e the number of evaluations if we follow SSGR by SSLPR for each hypothesis
Table shows values of N L and G for some of our examples
The table omits GL b ecause it is almost always true that G GL However in one case
SSLPR eliminated something SSGR could not eliminate This o ccurred after many levels
of sub division
In all cases min G L GL N This is true of b oth feasible and infeasible cases
The di erence is often dramatic e g for one NN infeasible example N L and
G It is always the case that b oth G N and L N It is often true that jL Gj is
small In of cases G L The total of G for the examples is only whereas the
total for L is
Sometimes G is slightly larger than L We b elieve that this phenomenon is due to a
sub division metho d we exp erimented with for these tests This metho d isolates an individual
comp onent of a U If restriction pro duces many comp onents just isolating one at a time
ij
can create many hypotheses In this case a cutting algorithm which moves the con guration
the maximum p ossible distance from the current con guration might b e preferable It is
p ossible that the b est sub division strategy might dep end on which type of restriction is
used at the hypothesis level For example since SSLPR is most e ective when a p olygon is
similar to its convex hull sub division in this case should maximize the convexity of U and
ij
For SSGR p erhaps sub division should minimize the amount of closure of the two U
ij
subp olygons We are still evaluating sub division strategies for our hierarchical containment
algorithm
Our results suggest that b oth SSGR and SSLPR can dramatically reduce the number
of hypotheses which must b e evaluated Using SSLPR causes the algorithm to evaluate
more hypotheses than SSGR in our exp eriments However for our current implementations
of SSGR and SSLPR SSLPR is the most cost e ective restriction in this context b ecause
SSLPR is approximately ten times faster than SSGR
Sub division
In order to gauge the success of our sub division strategy we compared the number of
hypotheses evaluated by the naive exact algorithm versus the hybrid algorithm on infeasible
examples For this comparison we disabled the high level geometric restriction as well as
the hypothesis level restrictions The sub division metho d we tested uses distance based
sub division when the U consists of a single comp onent If the U to b e sub divided has
ij ij
multiple comp onents we isolate the comp onent which is nearest to the bad p oint The
results are shown in Table In every case the hybrid algorithm requires fewer evaluations
than the naive exact algorithm This suggests that a sub division strategy which uses the
information from a lo cal overlap minimum can b e e ective
Figure NN example time seconds
Figure NN example time seconds
Conclusion
We now have b oth geometric and linear programming based metho ds for restriction
evaluation and sub division Each app ears to have strengths in di erent areas and they
can b e combined to achieve go o d results In particular geometric restrictions are very
helpful at the top level when the U are very nonconvex and we de nitely apply them
ij
there This work in no way shows the sup eriority of either approach but instead shows how
well they can work together
Summary and Conclusions
This chapter has presented two k NN algorithms based on our restrict sub divide evaluate
paradigm The rst is a numerically approximate algorithm b ecause it pro duces an
approximate con guration It uses a partial containment characterization to pro duce a set
of k subproblems It uses SSGR a greedy evaluator and size based sub division to solve
each subproblem
The second algorithm is a hybrid algorithm It uses b oth geometric and linear program
ming restrictions an overlap reduction evaluator and distance based sub division SSLPR
is used at the hypothesis level Our exp eriments suggest that distance based sub division
helps the hybrid algorithm detect infeasibility This algorithm is sup erior to the original
CLP based algorithm of Milenkovic
We give running times for b oth of these algorithms for a variety of k CC k CN and
k NN containment examples for k The hybrid algorithm is currently faster
than the approximate algorithm for k The hybrid algorithm typically solves our
containment examples in several minutes or less on a MHz SPARCstation for k
m and n We b elieve that the evaluation metho ds of the
two algorithms are of roughly the same e ectiveness and that this di erence in sp eed is
primarily due to the fact that the current implementation of SSLPR is roughly ten times
faster than the current implementation of SSGR This conclusion is supp orted by the fact
that the number of hypotheses evaluated by the approximate algorithm in our exp eriment
is less than the number of hypotheses evaluated by the hybrid algorithm Sp eeding up our
implementation of SSGR is a sub ject of future work see Section We b elieve that
given fast enough p olygon set op erations the approximate algorithm has the p otential to
b e comp etitive with the hybrid algorithm
Type APPROX NAIVE LP HYBRID
NN
NN
NN min min
NN min min
Table Running times for infeasible examples in seconds
Type APPROX HYBRID
CC
CC
CC
CN
CN
CN
CN
NN
NN
NN
NN
CN
NN
NN
NN
CN
CN
CC
Table Number of hypotheses examined denotes infeasible example
Type N L G Type N L G
NN NN
CN NN
CN NN
CN NN
CN NN
CN NN
CN CN
CN CN
CN CN
NN CC
NN
Table Number of hypotheses evaluated for restriction options denotes infeasible
example
Type NAIVE LP HYBRID Type NAIVE LP HYBRID
CN CN
NN NN
NN NN
Table Number of hypotheses evaluated for infeasible problems no hypothesis restric
tion
Part II
Applications of Containment to
Layout
C h a p t e r
Preliminaries
Introduction
This part of the thesis gives examples of layout problems which can b e solved using
two dimensional translational containment for irregular p olygons The problems are of two
types minimal container and maximal item These applications supp ort our claim
that b eing able to quickly solve containment problems for up to ten items allows a variety
of layout problems to b e solved more e ectively
Recall that our containment approach is a b ottom up one in which we maximize the
number of items for which algorithms are practical Clustering techniques for layout also
use a b ottom up approach Some of them even allow the items to rotate However the
metho ds which cluster more than two arbitrary nonconvex p olygons are greedy heuristics
Our containment based solutions to minimal container problems do not allow the items to
rotate but in contrast to the heuristics we establish an actual lower bound on the size of
the container for multiple items
In addition we b elieve that some p otentially promising layout techniques have b een
abandoned by the layout community b ecause they lack strong b ottom up metho ds The
primary example of this is pre packing Recall that a pre packing strategy generates feasible
groups for each container and then assigns groups to containers If the layout world had a
practical metho d for generating groups of items which t into each container this plus an
appropriate assignment strategy would make pre packing viable We demonstrate through
our containment based solutions to maximal item problems that pre packing can work
We also solve a problem which arises in multi stage layout problems and which must b e
solved b efore a strategy such as pre packing can b e applied The problem is to transform a
single highly nonconvex container into a collection of containers
We remark that although the primary use of containment is for automatic solutions
to layout problems containment can also b e part of a collection of computer aided layout
to ols for humans as well as assist in training humans who solve layout tasks manually
Overview of Chapter
Section discusses our work on the container transformation problem Section
gives an overview of our work on minimal container problems Section discusses our
work on maximal item problems Section explains the structure of this part of the thesis
Overview of Container Transformation Work
The motivation for our container transformation work is the trim placement problem in
apparel manufacturing see Section Trim placement is the second stage of a two stage
layout pro cess for pants markers The items to b e placed are either large pattern pieces or
small trim pieces The rst stage places the large pattern pieces The container for the
second stage is the complement of the large placed pieces within the rst stage container
It is typically highly nonconvex Rather than op erate on this unwieldy container directly
we transform it into a collection of containers The trim placement phase places as many
of the small trim pieces as p ossible into the containers
An ideal transformation of the container b oth limits and decomposes it The limiting
pro cess removes unnecessary regions which are not reachable by any of the items to b e
placed We use restrictions see Chapter to remove regions which are clearly unreachable
The decomposition pro cess pro duces a collection of containers which may overlap each other
Overlap among containers makes it p ossible for placed items to overlap each other We
show how to balance the con icting goals of maximizing the number of containers while
minimizing the overlap b etween them Maximizing the number of containers pro duces
manageable containment problems We allow the user to select the probability that the
containers will b e overlapping
For e ciency purp oses we introduce the notion of a breaking polygon This is a p olygon
which can b e translated to t inside any of the items to b e placed In order to nd a breaking
p olygon we nd the maximum area axis parallel rectangle MAAPR in a p olygon Our
theoretical work on the MAAPR problem uses fast matrix searching techniques from the
computer science literature and gives the fastest known asymptotic running time for this
problem O n log n time for an n vertex p olygon
Our work on container decomp osition is applicable in general to multi stage layout
problems b ecause the second stage of a multi stage layout problem may involve a large
highly nonconvex container which needs to b e decomp osed Highly nonconvex containers
also app ear in industries in which defects in material cause the material to b e partitioned
into quality zones see Heistermann and Lengauer s work with leather HL
Overview of Minimal Container Work
From the discussion of related work in layout in Section we see that solutions to
minimal enclosure problems are often required for the rst stage of a multi stage layout
problem Minimal container problems also arise when nding the b est sto ck sheet size for
a given set of items or the b est cloth width in garment manufacturing Current solutions
to minimal enclosure problems for more than two arbitrary nonconvex items often allow
the items to rotate but they use greedy heuristics The heuristics cannot pro duce a lower
b ound on the size of the enclosure
We give four examples of minimal container problems strip packing nding mini
mal square enclosures nding minimal rectangular enclosures and nding tight convex
and nonconvex enclosures Before discussing our work on these problems in Section
we summarize related work on enclosure problems in Section
Related Work
Attempts to treat multiple p olygon enclosure problems in the literature typically use
greedy heuristics Adamowicz and Albano AA Haims and Freeman HF Amaral
et al ABJ Da gli and Tato glu DT and Bailleul et al BTS pack nonconvex
items into rectangles to form comp osite items and then place the rectangles In AA
Adamowicz and Albano give an algorithm for pairwise clustering of irregular items in a
rectangle and describ e limited clustering options for more than two items They allow
p olygons to rotate They p erform pairwise clustering using an ob ject called the No Fit
Polygon NFP They also p erform limited clustering for more than two items by rst
clustering two large items and then placing small items within the b ounding b ox of that
cluster Amaral et al ABJ adopt the approach of AA for clustering items
Da gliand Tota glu DT cluster pairs of items in a greedy fashion and allow the items
to rotate They rst cluster two items make them a unit cluster this unit with another
item and so on Prasad and Somasundaram PS and Babaev Bab form an e cient
cluster of items then build a rep eating pattern in order to nest sheet metal blanks Ismail
and Hon IH use a genetic algorithm to cluster two similar items together the items are
allowed to rotate Dighe and Jakiela DJ use a hierarchical genetic algorithm to cluster
items The items are allowed to rotate Their algorithm uses a binary tree in which each
item together with values for its three parameters x y and constitutes a leaf and a
no de represents a search for the minimal enclosing rectangle of its two children Thus the
genetic algorithm seeks the b est binary tree for a set of items Milenkovic et al MDL
solve the sp ecialized problem of building item stacks of pants panels for layouts in apparel
manufacturing
In computational geometry algorithmic work app ears on various types of enclosure
problems for single items Finding a minimal enclosure for a single p olygon can also b e
viewed as a p olygon approximation problem as noted in CY
Chang and Yap de ne inclusion and enclosure problems CY They de ne an en
closure problem as follows Enc P Q Given P P nd the smallest Q Q that
encloses P where P and Q are families of p olygons and is a real function on p olygons
such that
Q Q Q Q Q Q Q
Chang and Yap CY review work on a variety of such enclosure problems We brie y
mention two examples of enclosure work Melissaratos and Souvaine MS a nd the min
imum concave quadrilateral circumscrib ed ab out a simple p olygon and O Rourke O R
nds the minimal b ox circumscribing a set of three dimensional p oints
1
The NFP of p olygon A with resp ect to p olygon B is really just B A see Section
2
Chang and Yap s de nition of P could b e extended to p oint sets for this case
Our Minimal Container Work
Our containment based solutions to minimal container problems do not allow the items
to rotate but in contrast to heuristics we establish an actual lower bound on the size of the
container for multiple items Our general approach to nding minimal enclosures is to nd
an optimal value for a parameter via binary search We remark that the binary search can b e
accelerated by using compaction techniques such as those developed by Li and Milenkovic
Li ML LM c LM a to improve a feasible solution and thereby help up date the
upp er b ound Our approach to these problems follows the common op erations research
approach of tightening upp er and lower b ounds to obtain an optimal value Containment
provides the lower b ounds in our case Our hybrid containment algorithm is appropriate
for minimal enclosure problems It is extremely fast for lo ose ts and so the initial stages
of the search are fast
For the problem of nding tight convex and nonconvex enclosures we show that re
moving unreachable p ortions of a tight rectangular enclosure can yield tight convex and
nonconvex enclosures which preserve all the containment solutions for the rectangle
Overview of Maximal Item Work
We give four examples of maximal item problems For this discussion let k
pr actical
b e the largest number of items for which current containment algorithms are feasible Our
problems are given a container and a collection of items nding maximally scaled copies
of items which t in the container given a container and a collection of k items
pr actical
for which the sum of the item areas do es not exceed the area of the container nding a
large sub collection which ts in the container given a container and a collection of
items for which either the sum of the item areas exceeds the area of the container or the
number of items is k nding a large sub collection of k items which
pr actical pr actical
ts in the container and problem for a collection of containers
Section contains an extensive survey of the relevant literature on layout and packing
problems It also reviews the limited layout literature on the assignment problem which
arises as part of problem
We discuss our work on maximal item problems b elow in Section
Our Maximal Item Work
For problem we use binary search combined with containment In problem the
given items cannot b e scaled they can only b e included in or excluded from the set The
key to solving this problem is to carefully choose which item to eliminate from an infeasible
subset We reject the most troublesome item which is identi ed using the results of the
overlap reduction evaluation metho d of the hybrid containment algorithm For problem
we randomly construct a set of items and solve problem This yields a rich collection of
n
sets which t into the container For each set of size n which ts we automatically have
subsets which t and which need not b e individual ly represented This work provides the
rst practical metho d for constructing sets of up to ten p olygons which t in a container
For problem we solve problem for each container and then apply a lo ok ahead
assignment strategy This is a pre packing strategy Containment thus allows us to trans
form a geometric optimization problem into a combinatorial optimization problem Our
results challenge the existing b elief that pre packing metho ds are impractical for layout
Arb
Overview of Part Two
Chapter discusses our work on container decomp osition Our container decomp osition
work also app ears in DM Our theoretical work on the MAAPR problem is summarized
in Chapter and also app ears in DMR DMR Chapter solves four minimal container
problems Chapter solves four maximal item problems including pre packing
C h a p t e r
Limited Gaps
Introduction
In this chapter we consider the following problem given a complex container and a set
of items to b e placed in the container transform the container by limiting and decomposing
it The limiting pro cess removes unnecessary regions which are not reachable by any of
the items to b e placed The decomposition pro cess pro duces a collection of containers
Maximizing the number of containers pro duces manageable containment problems
The motivation for our container transformation work is the trim placement problem in
apparel manufacturing see Section Trim placement is the second stage of a two stage
layout pro cess for pants markers The items to b e placed are either large pattern pieces or
small trim pieces The rst stage places the large pattern pieces The container for the
second stage is the complement of the large placed pieces within the rst stage container
This container is highly nonconvex and may have many comp onents An example is shown
in Figure In that gure the darkly shaded regions together form the second stage
container They represent gaps formed by neighboring pants panels The items to b e
placed during the second stage app ear stacked at the b ottom of the gure These items are
trim pieces Rather than op erate on this unwieldy container directly we transform it into a
collection of containers The trim placement phase places as many of the small trim pieces
as p ossible into the containers
Our work on container decomp osition is applicable to multi stage layout problems in
general b ecause the second stage may involve a large highly nonconvex container Highly
nonconvex containers also app ear in industries in which defects in material cause the ma
terial to b e partitioned into quality zones see Heistermann and Lengauer s work with
leather HL
An ideal transformation of the container b oth limits and decomp oses it We show how
the container can b e limited with the help of restrictions see Chapter The limiting
pro cess removes regions which are clearly not reachable by any item Identifying al l un
necessary regions is an NP hard problem The need for eliminating unreachable regions has
b een suggested by Amaral et al ABJ It is necessary to implement algorithms that
identify those gaps which cannot contain any shap e due to its dimensions or geometry
Figure Trim placement task
Karasic Kar conjectures that humans p erform this function intuitively when they place
items in a container
The decomp osition pro cess pro duces a collection of containers which may overlap each
other Overlap among containers makes it p ossible for placed items to overlap each other
We show how to balance the con icting goals of maximizing the number of containers while
minimizing the overlap b etween them As mentioned earlier maximizing the number of
containers pro duces manageable containment problems Our metho d allows the user to
select the probability that the containers will b e overlapping
For e ciency purp oses we introduce the notion of a breaking polygon This is a p olygon
which can b e translated to t inside any of the items to b e placed In order to nd a breaking
p olygon we nd the maximum area axis parallel rectangle MAAPR in a p olygon Our
theoretical work on the MAAPR problem uses fast matrix searching techniques from the
computer science literature and gives the fastest known asymptotic running time for this
problem O n log n time for an n vertex p olygon
This chapter describ es our e orts to provide a sound mathematical basis and a set of
algorithms for container limiting and decomp osition We call each comp onent of the limited
and decomp osed container a limited gap
Decomp osing the Container into Gaps
Given a collection of k p olygonal items P and a container C we seek a gap set G
fg g g g such that g C with the containment property if p P is placed inside
l i i
the container then it is contained entirely inside some g G Smaller gaps and more gaps
are desirable overlap among gaps is undesirable
The set of connected comp onents of the container has the containment prop erty but
it usually has to o few elements to improve tractability much in practice We will de ne a
subset R of the container called the reachable region of P The connected comp onents of
P
R form a much b etter gap set than the container itself but R still is often insu ciently
P P
decomp osed However it is the largest cardinality gap set we know of with non overlapping
gaps
If we p ermit gaps to overlap then it is p ossible for p olygons placed in di erent gaps
to overlap However if we construct the gaps prop erly then the probability of overlap
1
After creating the term limited gap we noticed the double connection of the name to the retail side of
the apparel business
will b e small We will give a rigorous notion of this probability Furthermore for any
given probability we will show how to construct a gap set called limited gaps such
that the probability that items in di erent gaps overlap is less than assuming a uniform
distribution
There is a tradeo b etween the number and size of gaps and the probability Many small
gaps imply small k for a containment problem for an individual gap but the probability of
overlap b etween items in di erent gaps is higher Since is a parameter to the construction
of the limited gaps we allow the p ossibility of nding the b est value of in the tradeo
Overview
Section describ es the container decomp osition metho ds we exp erimented with b efore
arriving at the notion of the limited gaps Section de nes the limited gaps in terms of
the probability and gives an algorithm for constructing them Section presents two
metho ds for improving the e ciency of limited gap construction These metho ds introduce
the new concepts of breaking polygons and inking polygons Section summarizes our
work on nding the maximum area axis parallel rectangle in a p olygon This is used to
construct a breaking p olygon Section summarizes this chapter and states conclusions
The Challenge of Constructing Gap Sets
Before de ning the limited gaps we review the other container decomp osition metho ds
we exp erimented with Our metho ds assume that the container is the pro duct of the rst
stage of a two stage layout problem The rst stage places a collection of items which
b ecome obstacles in the second stage That is the container for the second stage is the
region within the original container which is the complement of the obstacles We also
assume that a transformation of the obstacles which improves the rst stage layout is legal
Compaction
Li and Milenkovic have developed a compaction to ol MDL LM d Li ML LM c
LM a which plans a motion for a set of placed items under a set of applied forces We
attempted to eliminate holes and narrow necks in the container by applying a leftward
force to each obstacle p olygon In theory this pro cess is guaranteed to eliminate holes
b ecause each obstacle will b e in contact with at least one other obstacle However we
found that this strategy has three shortcomings it fails to close all the narrow necks
although the container comp onents b ecome simply connected unnaturally large regions
still exist and contact detection su ers from numerical instability Through the use of
a con guration space approach the compaction algorithm avoids the question of contact
detection in order to run quickly and numerically robustly However in order to generate
a go o d decomp osition it is necessary to explicitly detect contacts and this proves to b e
numerically unstable
Partitioning Squares
Our second attempt involved choosing a narrowness tolerance We formed a partition
ing square with side of length and used this to partition the container The partitioning
Figure A partitioning square which fragments the container
pro cess is conceptually analogous to centering a partitioning square at each vertex of each
obstacle and then removing those squares for which the intersection with the set of ob
stacles is connected see Figure this illustrates a square which is not removed The
di culties with this approach are it introduces artifacts pieces of squares into the
layout it fragments the container in a way that leaves many small regions which are
not usable by the set of items to b e placed and choosing a universal is inherently
imp ossible The fragmentation issue can b e resolved by testing each region but this is
computationally exp ensive Cho osing is not p ossible b ecause P do es not naturally give
rise to a narrowness value Adopting a value corresp onding to regions which no item can
t through is insu cient b ecause an item might b e able to reach into an arbitrarily narrow
neck from b oth sides thus making the neck usable see Figure The only natural
solution is to let the left and right regions overlap
Figure A neck which is usable although items cannot pass through it
Other Metho ds
We considered other metho ds such as computing a Voronoi diagram of the obstacles us
ing edges of the obstacles as sites For However such metho ds also rely on a narrowness
tolerance which is a fundamentally awed approach
Limited Gaps
We rst de ne the set of limited gaps for P when P P for i j k Then
i j
we give a straightforward extension of the theory to noncongruent shap es For ease of
notation we denote P by p We also denote U by V recall from Section that U
j p
j j
is the replacement of U in a valid restriction of U
j
Reachable Regions
De nition The set R V p is the reachable region of p with respect to P
p p
2
One could use a breaking p olygon see Section to reduce the exp ense
orig
Note that if we use U the unrestricted U in place of V the reachable region
j p
j
b ecomes what is known in mathematical morphology and image analysis as the opening of
a p oint set Ser Ser At the other extreme if we use U in place of V we obtain V
p
j p
The set V p is a tight upp er b ound on the reachable region under set inclusion
p
A gap set G consisting of the connected comp onents of R has the containment prop
R p
erty and the comp onents have the added advantage of b eing non overlapping Unfortu
nately comp onents of R can contain arbitrarily narrow necks Ideally each comp onent
p
of R is generated by a single comp onent of V Unfortunately dilation by p can blend
p p
together two comp onents of V for cases in which p cannot t through a narrow neck of the
p
container but two copies of p can touch if they reach into the neck from di erent sides as
in Figure
By dilating each individual comp onent of V we can create a gap set G with higher
p V
cardinality than G and with no narrow necks Unfortunately these gaps can b e highly
R
overlapping The limited gap set will b e somewhere in b etween G and G
V R
One advantage of R is that it provides a nontrivial upp er b ound on the e ciency of a
p
container as follows The e ciency of packing copies of p in a container C cannot exceed
the ratio of area R to area C Most packing density b ounds in the literature apply only
p
when the p olygons are convex but this one works for any type of p olygon
Karasic Kar conjectures that human layout exp erts employ the op ening as a heuristic
criterion He notes that the op ening can b e used not only to evaluate the quality of a
p olygon s placement but also for selecting the next p olygon to place The area of R can
p
also b e used within a greedy layout algorithm as a criterion for judging the placement of
p olygons Larger R is b etter in this case and the algorithm chooses a placement p osition
p
for each item which results in the most usable space for the remaining unplaced items
De ning Limited Gaps using Probability of Interference
Let v v V b e two connected comp onents If v p and v p do not intersect then
p
we can indep endently place copies of p into the container using translations from v and
v without interference If v p v p then we want to compute a probability
Prob v v that p t p t We assume a uniform distribution on the connected
comp onents of V so t v and t v are chosen uniformly Assuming we can compute
p
Prob v v we can nally de ne limited gaps Given a probability comp onents v and v
interfere with each other if Prob v v Partition the comp onents of V according to
p
the transitive closure of the interference relation into interference sets Finally dilate each
interference set by p to generate a gap The resulting set G of gaps is the limited gap
L
set for probability There are variations on this de nition but the p oint is that one can
select the probability of interference for the gap set
We calculate Prob v v as follows For each t v the set of t v such that p t p
t is p t p v and the probability that this choice of t results in interference
is Prob ftg v area p t p v area v The integral of Prob ftg v over all
t v gives us Prob v v
We do not have an exact algorithm for this integral but we can approximate it by
sampling t v We can improve the e ciency by sampling only the region v v p p
that can result in a non zero value of Prob ftg v After computing the average value of
3
E ciency is the ratio of used area to total area
this probability we normalize by multiplying it by area v v p p area v
Finally if P contains noncongruent shap es then we can similarly de ne and compute
Let V b e the set of all comp onents of the V regions Prob v v for v V and v V
P p p
p
We partition V according to interference In this case v and v can only interfere if they
P
For each partition set V we dilate each comp onent come from di erent V and V
P p
p
v by the appropriate p P and then take the union v p to generate the gap
v
corresp onding to
E cient Computation of Limited Gaps
In order to e ciently construct the limited gaps we want to quickly partition V into in
P
terference sets We present two helpful partitioning techniques The rst technique involves
nding a breaking polygon a p olygon B that can b e translated to t inside any p P
The second technique uses the notion of inking polygons q inks p if the reachable region of
q inside p is all of p Section discusses the breaking p olygon technique Section
introduces the more p owerful concept of inking p olygons
Breaking Polygons
Let us denote a breaking p olygon of P by B To simplify the presentation we assume
that each p P has its origin p ositioned so that B p without translation Since B p
it follows that every p olygon in V is a subset of some p olygon in V
P B
We can use B to quickly group together comp onents of V which might intersect each
P
other Clearly two comp onents of V cannot intersect each other unless they are in the same
P
comp onent of V Furthermore since each connected comp onent of V is entirely within
B P
a connected comp onent of V the inclusion test consists only of a single p oint in polygon
B
test
Since we want to maximize the number of comp onents of V we want to maximize the
B
area of B Finding the p olygon of maximum area which can b e translated to lie inside of
each p P is an interesting op en problem which we do not solve here In practice we nd
the maximum area axis parallel rectangle MAAPR of each p P Finding the MAAPR
quickly is a challenging problem in its own right and is the sub ject of DMR DMR We
brie y summarize our MAAPR work in Section
We rep osition each p so that its origin is at the center of its MAAPR We choose B
to b e the intersection of all p P This guarantees that B contains a rectangle with the
minimum width and minimum height of all the MAAPRs This in turn provides a lower
b ound on the area of the b est breaking p olygon
Unfortunately the reachable region R of the breaking p olygon is not necessarily a
B
sup erset of R for each p P Consequently we are not guaranteed that if two comp onents
p
v and v V are in di erent connected comp onents of V then they do not interfere In
P B
fact there can b e a signi cant amount of interference
Inking Polygons
We say p olygon q inks p olygon p if and only if p q q p A set of p olygons
Q is an inking set of a set of p olygons P if for every p P p q q p
q Q
4
We use the term breaking polygon b ecause V breaks or partitions V
B P
An inking set is valuable b ecause R R is a sup erset of each reachable region R
Q q Q q p
are connected comp onents then v p and Therefore for p p P if v V and v V
p
p
v p can intersect and hence Prob v v only if they lie in the same comp onent of
R Furthermore since v p either lies entirely inside or entirely outside each comp onent
Q
of R we can simply p erform a p oint in polygon test of any p oint of v p It is easy to
Q
generate this p oint without computing all of v p
For an inking set to b e useful R should not b e to o much larger than R On the
Q P
other hand R should b e easier to compute The rst condition implies that we desire
Q
the inking set whose smallest member has maximum p ossible area The second condition
implies that we desire the inking set of minimum complexity total number of vertices
The inking set of maximal area is the set P itself The inking set of minimum complexity
is a single p oint but if we sp ecify non zero area then this is an op en question We desire
the answer to a doubly op en question of the inking sets of minimum complexity nd the
one whose smallest member has maximum area The next section gives a lower b ound on
the complexity of an inking set based on the notion of a hitting set
Hitting Sets
Let a b c b e three consecutive vertices of some p olygon p P such that angle abc op ens
into the interior of p If an inking set Q inks p then there must b e a set of consecutive
vertices a b c on some q Q such that angle a b c is a subset of abc if q is translated to
place b on top of b then a and c are on or in the interior of angle abc We say that angle
a b c hits angle abc
We can map angle abc to an interval arc on the unit circle by translating b to the
center of the circle and taking the arc in the interior of abc Clearly a b c hits abc if and
only if its interval is a subset We refer to the set of angles or intervals of P as that set of
arcs generated by all triples of consecutive vertices abc of p olygons p P
Given a set of intervals on the unit circle a hitting point set S is a set of p oints such
that each interval contains at least one p oint of S in its interior Clearly the complexity of
Q is at least as great as the size of the minimum hitting p oint set of the angles of P
In the following discussion let M b e the total number of vertices in P
Theorem A minimal hitting point set for M intervals on the unit circle can be found
in O M time
Pro of We rst give an O M log M time algorithm for nding the minimal hitting set
for a set of M intervals of the real line Then we show how to use this algorithm as part of
an O M time algorithm for intervals on a circle
The rst step in the line interval algorithm is to initialize the hitting set H to We
then sort the intervals by increasing x co ordinate of their right endp oints and pro cess them
in this order For each interval i if H contains a p oint that hits i then we do not change
H Otherwise we add the right endp oint of i to H
This algorithm runs in O M log M time b ecause we sort the intervals It can b e shown
that each step of the algorithm yields the minimal hitting set for the current set of intervals
whose rightmost p oint is furthest to the right
5
Actually to avoid degeneracy we add a p oint just slightly to the left of the right endp oint of i
Our algorithm for intervals on a circle rst sorts the interval endp oints creating a set
of M event p oints and M sub intervals b etween event p oints The algorithm has an
outer lo op which considers each of these M sub intervals in turn We add a single hitting
p oint to a sub interval and thus we can eliminate from consideration any interval that
contains this sub interval We can now cut the circle anywhere in this sub interval and then
apply the line algorithm This gives the minimal hitting set for this choice of sub interval
After considering all M sub intervals we know which choice yields the minimum hitting
set We only have to sort once so the running time is in O M 2
We can generalize the notion of hitting p oint set to hitting interval set recall an interval
of Q hits an interval of P if it is a subset Clearly we can replace each p oint in the
hitting p oint set by the sub interval in which that p oint lies However this might not
yield the hitting interval set whose minimum length interval is maximized Since we do not
want skinny inking p olygons to avoid inking narrow necks of the container we want to
maximize this minimum angle This will also tend to maximize the minimum area of an
inking p olygon
For a given arc length L we can nd the minimum hitting interval set whose members
are of length L We can do this in O M time using a mo di cation of the p oint set
algorithm The algorithm on the line is exactly the same but unraveling the circle into
a line b ecomes more intricate For any given hitting set size h we can do binary search on
the value of L to nd the smallest value of L whose minimum hitting interval set is h or
greater Since L can take on at most M M values we only have to run the O M
hitting interval set algorithm O lg M times
Forming Inking Sets using Hitting Sets
Given the minimal sized hitting interval set Q of h intervals for P with maximum
minimum length L we show here how to construct an inking set consisting of a set of h
isosceles triangles and a single square The isosceles triangles ink regions near the vertices
of P and the rectangle inks the remaining area If h is the minimum complexity of the
min
inking set then this algorithm yields an inking set of complexity h
min
We rst discuss the isosceles triangles We asso ciate an isosceles triangle with each
hitting set interval i as follows Consider the set S of angles abc of p olygons p P which
i
are hit by i For each angle abc nd the maximum height of a corner isosceles triangle
whose top is b and whose sides are aligned with ba and bc such that the triangle is in the
interior of p Let H b e the minimum height over all vertices in S Now form an inking
i i
set p olygon T which is the isosceles triangle of height H and top angle determined by i
i i
T can ink a corner isosceles triangle for abc of height H
i i
The size of the square s is constrained by two factors the narrowest neck in P and
the narrowest and most diagonal angle in P To treat the rst constraint we build a
Voronoi diagram of each p olygon in P under the L metric Lee If we surround each
site with the largest square which ts then s must b e smaller than the smallest of these
squares Let w b e the resulting width The second constraint requires that we calculate
p
w min H sin over all abc in P with angle and hitting interval i The width
i
of s is min w w
MAAPRs
This section considers the geometric optimization problem of nding the maximum area
axis parallel rectangle MAAPR in an n vertex general p olygon This rectangle problem
arises naturally in applications where a quick internal approximation to a p olygon is useful
One of its applications is nding breaking polygons see Section
Section summarizes our MAAPR work from DMR DMR This work gives an
O n log n time algorithm for an n vertex general p olygon Section describ es a naive
algorithm which we use in practice to nd a breaking p olygon
Summary of MAAPR Results
Here we summarize our work on MAAPRs which is describ ed in detail in DMR
DMR We present the fastest algorithmic results for general p olygons with holes a
O n log n time algorithm We also prove a lower b ound for this type of p olygon of time in
n log n In order to develop this algorithm we rst characterize the MAAPR for general
p olygons by considering di erent cases based on the types of contacts b etween the rectangle
and the p olygon
We present a general framework for solving the two contact case of the MAAPR problem
which dominates the running time for a variety of p olygon types The framework involves
transforming the p olygon via vertex pro jection and inner rectilinear approximation into
another p olygon for which we can solve a related problem In the related problem we are
given a p oint set S and two subsets L and R of S The task is to nd the largest rectangle
containing no p oint of S which has lower left corner in L and upp er right corner in R This
problem can b e solved by transforming it into a monotone matrix searching problem Fast metho ds exist in the literature for nding the maximum element of such matrices
xy-monotone orthogonally convex horizontally convex vertically separated horizontally convex general
(n) O (n a(n)) O (n a(n) log n) O (n log n) O (n log 2 n)
Figure Algorithmic results for a variety of p olygon types
Using these metho ds leads to the following MAAPR results n time for an xy
monotone p olygon O n n time for an orthogonally convex p olygon where n
is the slowly growing inverse of Ackermann s function O n n log n for horizontally
vertically convex p olygons O n log n for a sp ecial type of horizontally convex p olygon
6
We thank Jacqueline Huang for implementing this algorithm while she was an undergraduate at Harvard
University
7
A matrix is monotone if the column index of the row maximum moves to the right as the row index
increases
8
A simple p olygon consisting of two xy monotone chains is an xy monotone p olygon A chain is xy
monotone if it is monotone with resp ect to b oth the x and y axes A chain is monotone with resp ect to a
line l if a line orthogonal to l intersects the chain in exactly one p oint PS
9
An orthogonally convex p olygon is b oth horizontally and vertically convex This class contains the class
of convex p olygons
consisting of two y monotone chains on opp osite sides of a vertical line and O n log n
for general p olygons
MAAPR Characterization and Naive Algorithm
Here we characterize the MAAPR contained in a general p olygon P by considering
di erent cases based on the types of contacts b etween the MAAPR and the b oundary of P
and outline a naive algorithm for nding the MAAPR based on this characterization Others
have used contact classi cation for algorithmic development see for example MS a
MOS DKO The rst part of this subsection classi es the types of contacts b etween
the MAAPR and the b oundary of P The second part introduces the notion of a determining
set of contacts The third part shows how to solve maximization problems asso ciated with
certain determining sets of contacts The fourth part presents our characterization theorem
for MAAPRs The fth part gives a naive algorithm based on the characterization theorem
Types of Contacts
Intuitively if an axis parallel rectangle is inside P it has four degrees of freedom param
eters and can grow until each of its four sides is stopp ed by contact with the b oundary
of P Contacts b etween the MAAPR and P are of two types a side of the MAAPR
with a vertex of P and a corner of the MAAPR with an edge of P In order to discuss
the rst type we require the notion of a re ex extreme vertex introduced in SRW
De nition A vertex v of P is a vertical re ex extreme vertex if there exists
such that for an open ball of radius about v and S exterior P either no point of
S has y coordinate above v or no point of S has y coordinate below v A horizontal re ex
extreme vertex is de ned similarly
For type contacts a re ex extreme vertex of P touches a side of the MAAPR and
stops growth in one direction we call this a re ex contact Each re ex contact can remove
one degree of freedom Two re ex contacts with adjacent sides of the MAAPR x a corner
of the MAAPR For type contacts a corner of the MAAPR touches an edge of P forming
an edge contact
Determining Sets of Contacts
De nition A set of contacts C is a determining set of contacts if the MAAPR R
satisfying C has nite area and if the MAAPR R satisfying any proper subset C C has
greater or in nite area
For example a set of four re ex contacts one on each side of the rectangle is a deter
mining set
Note A determining set determines the area of the MAAPR but it do es not necessarily
determine a unique rectangle or MAAPR
Within a determining set we distinguish b etween two di erent subtypes of edge contacts
An edge contact is xed if the set of constraints uniquely determines the p oint of contact
with the rectangle not necessarily the MAAPR Otherwise it is a sliding contact
A xed contact can arise when there is no freedom to slide along an edge b ecause a
re ex contact xes a co ordinate For example in Figure a the re ex contact of the (a) Fixed contact (b) Sliding contact (c) Two dependent
sliding contacts
Figure Edge contact types for a determining set
determining set xes the x co ordinate of the edge contact which completely determines the
lo cation of the edge contact An edge contact with an adjacent side that has either a re ex
or xed contact must also b e a xed contact
Two sliding edge contacts are dependent if the p osition of one determines the p osition
of the other otherwise they are independent An indep endent sliding contact requires that
the two adjacent sides of the MAAPR do not have any contact with P see Figure b
A sliding contact adjacent to another sliding contact is dep endent b ecause the two contacts
must share a co ordinate see Figure c
Maximization Problems
Here we examine maximization problems asso ciated with certain determining sets of
contacts Finding the MAAPR asso ciated with a determining set of contacts requires solving
a maximization problem if the set contains a sliding contact For a given set of contacts the
number of degrees of freedom is the number of undetermined parameters of the rectangle
Degrees of freedom within a determining set can only arise from sliding contacts b ecause
any other degree of freedom would result in a rectangle of in nite area and therefore the
contacts would not form a determining set It follows that if a determining set consists of
only re ex or xed edge contacts no maximization is required For each indep endent sliding
contact in the set we can parameterize the asso ciated edge The maximization problems
can then b e classi ed based on the number of parameters
Parameter Problems
The set of parameter maximization problems can b e further sub divided according to
the number of dep endent sliding contacts
The Basic Parameter Problem
The simplest parameter problem involves no dep endent sliding contacts just a single
indep endent one This is the basic parameter problem and it arises when one corner of
the MAAPR has a sliding contact and the opp osite corner is xed The basic parameter
problem can b e solved by parameterizing the edge asso ciated with the sliding contact and
maximizing a quadratic in one variable This can b e solved in O time
An alternate constant time solution to the parameter problem is based on the following
lemma which demonstrates that the slop e of the MAAPR diagonal dep ends only on the
slop e of the p olygon edge We assume here that the edge is neither vertical nor horizontal
Lemma Slop e Lemma Given a point p and a line L with slope s the MAAPR
with one corner at p and opposite corner at point q on L has diagonal pq where the slope
of pq s
In other words if one corner of a MAAPR is incident on L the slop e of the MAAPR s
diagonal is the negative of L s slop e
L: ax + by = d
s = -a/b
increasing area
(x c , y c ) = q
area of MAAPR: xy = c
p
Figure Slop e lemma
Pro of Assume p is at the origin and that the line L is given by ax by d see
Figure W l o g assume L intersects the x and y axes Note we ignore the
degenerate cases of horizontal and vertical lines for which the MAAPR is unde ned The
family of hyperb olas given by xy r represents curves corresp onding to rectangles of
constant area Moving away from the origin into the rst quadrant the hyperb ola branch
which is tangent to L provides the area c of the MAAPR asso ciated with L Let x y b e
c c
the co ordinates of the p oint at which L is tangent to the hyperb ola xy c
Now the tangent to xy c at x y is equal to L so the normal to xy c at x y
c c c c
is equal to the normal to L at x y The normal to xy c is F F y x evaluated
c c x y
at x y it is y x The normal to L at x y is a b Therefore
c c c c c c
a y
c
slop e slop e L pq
x b
c
2
Note that as a consequence of this lemma if p moves downward q moves downward
along its edge
Two Dep endent Sliding Contacts
Consider a pair of dep endent sliding contacts which share a co ordinate w l o g y and
have an opp osite re ex contact which determines one co ordinate of a side w l o g y see
Figure a
To nd the MAAPR we parameterize edge p p by t yielding the following quadratic
in t to maximize
F t x x y y
A B t C D t p p 4 4 p p 2 t 2 t
(x,y) (x',y) (x,y) (x',y)
p p 1 1 p p 3 p 3 5 y' (x, y' ) p 6
(a) Pair of Constrained Sliding Contacts (b) Triple of Constrained Sliding Contacts
Figure Parameter problems with two and three dep endent sliding contacts
where
y b y y
A x B x x
m m
C y y D y y
y y
b y mx m
x x
p p Note that x is obtained by solving the line equation asso ciated with edge
Three Dep endent Sliding Contacts
For the case of three dep endent sliding contacts see Figure b we again parameterize
edge p p and express x in terms of y for edge p p Similarly we express y in terms of x
p p We again obtain a quadratic in t to maximize The di erence b etween this for edge
and the previous case is in the values of C and D
C y m x b D y y m x x
y y
m b y m x
x x
The Parameter Problem
There is only one type of parameter problem It has two indep endent sliding contacts
The following lemma allows us to reduce a parameter problem to a set of parameter
problems
Lemma Let e and e be non intersecting line segments Consider the set E of
empty axis parallel rectangles which have diagonal ly opposite corners on e and e There
is a maximum area rectangle in E with at least one corner at an endpoint of e or e
Pro of One way to prove the claim is to parameterize each edge to obtain a quadratic in
two variables It is easily shown that the resulting surface is a saddle surface so that the
maximum is achieved along the b oundary of the unit patch implying that the MAAPR has
a corner at a vertex of one of the edges
A geometric argument is more intuitive however Let c on e and c on e b e opp osite
corners of a MAAPR let r b e the diagonal connecting c and c and by way of contra
diction assume that neither c nor c is at a vertex of e or e Now replace e and e by
the lines l and l containing them Consider the set S of all line segments that connect
l and l and are parallel to r The length of these line segments as a function of their
distance from r is monotonic If l and l are parallel the length is constant Otherwise
the length increases moving away from r in one direction and decreases in the opp osite
direction Move away from r in the direction of increasing length either direction if l is
parallel to l until either c or c is at a vertex of e or e and let r b e the new diagonal
Since jr j jr j the area of the rectangle whose diagonal is r is at least as large as the
area of the rectangle whose diagonal is r contradicting the assumption that the latter is a
MAAPR
2
Having established that the MAAPR has a corner at a vertex in this case we can
nd the MAAPR by considering in turn each of the four endp oints of e and e solving
the asso ciated parameter problems and then comparing the four resulting parameter
MAAPR areas
Characterization Theorem
To characterize the MAAPR we examine the p ossible determining sets of contacts By
enumerating the re ex contacts b etween the MAAPR and P we derive the set of ve cases shown in Figure
# RC
4
3 1 FC
RC = Reflex Contact FC = Fixed Contact 2 2 FC 2 FC 1 ISC 2 FC 2 FC ISC = Independent Sliding Contact DSC = Dependent Sliding Contact
1 FC 1 3 FC 3 FC 2 DSC 1 ISC
0 2 ISC 3 DSC 4 FC
Figure Determining sets of contacts for the MAAPR
Theorem The determining set of the MAAPR of a general polygon P conforms up
to symmetry to one of the ve cases in Figure
Pro of A determining set has by de nition at most one re ex contact with each side of
the MAAPR For each of the p ossible numbers of re ex contacts we show that the deter
mining set of a MAAPR of that type conforms up to symmetry to one of the con gurations
shown in Figure In each case we must eliminate degrees of freedom b eyond those of
sliding contacts We observe that a determining set cannot contain b oth two adjacent re ex
contacts and the xed contact b etween them this would b e redundant
Case In this case there is one re ex contact with each side of the MAAPR Since each
re ex contact removes one degree of freedom from the rectangle this set of contacts
is su cient to determine the MAAPR Removing any one re ex contact allows the
rectangle to grow so all four contacts are necessary
Case Three re ex contacts are not su cient to determine the MAAPR since the fourth
side can move outward We must add an edge contact which is xed b ecause it is
adjacent to a re ex contact There is only one choice of p osition for this contact up
to symmetry It xes the remaining side of the rectangle
Case Two re ex contacts can b e either adjacent or diagonally opp osite In b oth cases edge
contacts are needed to determine the MAAPR Since two degrees of freedom remain
the determining set contains at most two edge contacts
In the adjacent case we examine the p ossibility of edge contacts at the corners of the
rectangle excluding the corner b etween the two re ex contacts There are two ways
that two edge contacts can app ear In b oth cases b oth contacts must b e xed so they
x the remaining sides of the rectangle If only one edge contact app ears it must
b e a sliding contact otherwise a degree of freedom remains The sliding contact is
diagonally opp osite to the corner xed by the re ex contacts This contact is su cient
b ecause it represents a basic parameter problem
In the opp osite case any edge contact is xed hence there must b e two of them
There are two p ossible ways they can b e con gured In b oth cases all four sides of
the rectangle are xed
Case One re ex contact is not su cient to determine the MAAPR We can choose to
place edge contacts at any three of the four corners of the rectangle four would b e
redundant If there are three edge contacts they are all xed and they completely
determine the MAAPR There are two ways to con gure three such contacts If
there are two edge contacts they can b e adjacent or diagonally opp osite If they are
adjacent they must b oth b e opp osite the re ex contact otherwise a degree of freedom
remains These adjacent contacts are dep endent sliding contacts and they determine
the MAAPR b ecause they represent a parameter problem If they are diagonally
opp osite one is xed and the other is a sliding contact this is also a parameter
problem One edge contact is not su cient together with the one re ex contact to
determine the MAAPR
Case In this case there are no re ex contacts One sliding contact is not su cient to
determine the MAAPR If there are two sliding contacts they must b e opp osite if
they are to determine the MAAPR This is a parameter problem If there are
three edge contacts they are all dep endent sliding contacts They are su cient to
determine the MAAPR b ecause they represent a parameter problem If there are
four edge contacts they yield four equations in four unknowns for which the MAAPR
is completely determined This therefore forms a set of xed contacts
2
Corollary Given a determining set C for a MAAPR of a general polygon it fol lows
that jC j
A Naive Algorithm
Based on the ab ove characterization we can nd the MAAPR in a general p olygon by
nding the MAAPR under the constraints of each of the ve cases and selecting the largest
one
Theorem For each determining set of contacts in Figure the MAAPR can be
found in constant time
Pro of A re ex contact yields in constant time one of the four parameters of the rectan
gle as do es a xed contact In all cases we can pro cess these contacts rst The remaining
situations are all either parameter problems or parameter problems which can b e solved
in constant time b ecause they only require maximizing a constant number of quadratic
forms 2
A naive MAAPR algorithm can use Theorem and supply it in each case all
p ossible determining sets for P These can b e identi ed using an algorithm with up to four
nested lo ops one for each element of the determining set For each MAAPR candidate we
can check if it is empty i e contains no edges or vertices of P in O n time We conclude
Theorem The MAAPR of an n vertex general polygon can be found in O n time
Summary and Conclusions
This chapter has discussed our container limiting and decomp osition work Limiting is
based on the reachable regions of the items A reachable region is found by using restrictions
combined with the opening morphological op erator Our decomp osition pro cess allows the
user to control the probability that gaps will overlap each other For e ciency reasons
we introduce the notion of breaking polygons and inking polygons We have implemented
the breaking p olygon approach and found it to b e practical for forming limited gaps Our
implementation uses a naive algorithm for nding the maximum area axis parallel rectangle
MAAPR in a p olygon Our theoretical work on the MAAPR problem yields the fastest
known asymptotic running time for this problem O n log n time for an n vertex p oly
gon Figure is a screen dump which shows limited gaps for an example from apparel
manufacturing
We note that in applications which op erate on U s and do not directly op erate on the
ij
containers themselves the metho ds we present for clustering together U s can b e used
ij
without carrying out the nal construction of the limited gaps themselves
Figure Limited gaps example
C h a p t e r
Minimal Container Problems
Introduction
In this chapter we address four minimal container problems strip packing nding
minimal square enclosures nding minimal rectangular enclosures and nding tight
convex and nonconvex enclosures
In Chapter we cited the following de nition of an enclosure problem given by Chang
and Yap CY Enc P Q Given P P nd the smallest Q Q that encloses P
where P and Q are families of p olygons and is a real function on p olygons such that
Q Q Q Q Q Q Q
This de nition is not su cient for our purp oses We need a more general de nition of
an enclosure problem Our de nition replaces P with P where P is a tuple of elements of
P We also let M b e a set of p olygon transformations We de ne
E nc P M Q Given P P P P where P P nd the smallest Q
k i
Q that encloses P where P and Q are families of p olygons M is a set of p olygon
M
transformations P is the transformation of each item in P by a p ossibly di erent
M
transformation in M and is a real function on collections of p olygons such that
Q Q Q Q Q Q Q
Work on minimal enclosures in computational geometry uses the identity transformation
for M and do es not allow a collection of p olygons The particular two dimensional enclosure
problems we address here let P b e a collection of p olygons M b e translations b e the
area function and Q vary dep ending on the problem For example in strip packing Q is
the set of axis parallel rectangles of xed width
From the discussion of related work in layout we see that solutions to minimal enclosure
problems are often required for the rst stage of a multi stage layout problem Minimal
container problems also arise when nding the b est sto ck sheet size for a given set of items
or the b est cloth width in garment manufacturing Section discusses computational
geometry work on various types of enclosure problems for single items Current solutions to
minimal enclosure problems for more than two arbitrary nonconvex items often allow the
items to rotate However b ecause they use greedy heuristics they cannot provide a lower
b ound on the size of the enclosure Popular approaches either place small items in gaps
inside the b ounding b ox of two large items AA or solve the problem incrementally by
adding one item at a time DT DJ
Our containment based solutions to minimal container problems do not allow the items
to rotate but in contrast to heuristics we establish an actual lower b ound on the size of
the container for multiple items Our approach to nding minimal enclosures is to nd an
optimal value for a parameter via binary search combined with containment The idea of
solving a geometric optimization problem using binary search on a parameter is found for
example in Chazelle s seminal pap er on p olygon containment Cha We observe that the
binary search can b e accelerated by using compaction techniques such as those developed
by Li and Milenkovic Li ML LM c LM a to improve a feasible solution and hence
up date the upp er b ound Our hybrid containment algorithm is appropriate for minimal
enclosure problems It is extremely fast for lo ose ts and so the initial stages of the search
are fast
We prop ose the following general approach to minimal enclosure problems involving a
single parameter Our approach is a binary search with compaction used to up date the
upp er b ound obtained by containment Note that the binary search approach works only
if the function b eing minimized decreases monotonically as the parameter decreases In
this binary search the function can b e thought of as a binary one which returns if the
containment problem is feasible and otherwise
MINIMAL ENCLOSURE P k
do
large initial value
max
min
tolerance value
while
max min
cur r ent max min
Q container of size
cur r ent
if CONTAINMENT P k Q
place p olygons according to con guration
COMPACTION P k Q
max
else
min cur r ent
return
max
Overview
In Section we use binary search on length together with containment to solve a two
dimensional strip packing problem In Section binary search on length is combined with
containment to nd tight axis parallel square enclosures for collections of items We extend
this approach in Section to nding tight axis parallel rectangular enclosures which is
a common prepro cessing step in many layout algorithms which pack non rectangular items
It can also b e applied to the problem of nding the b est sto ck sheet size for a given set
1
Our current implementations do not use compaction adding it is a sub ject of future work
of items or the b est cloth width in garment manufacturing In higher dimensions this
technique could help determine the b est packaging for a pro duct In Section we show
that removing unreachable p ortions of a tight rectangular enclosure can yield tight convex
and nonconvex enclosures which preserve all the containment solutions for the rectangle
Section summarizes this chapter and states conclusions
Strip Packing
Here we use binary search on length together with containment to solve a two dimensional
strip packing problem We illustrate how containment can b e used for strip packing by solv
ing an incremental layout task from the apparel industry Figure shows an incremental
layout task A garment containing items has already b een laid out and the task is to
add a second garment to this layout while minimizing the length of the strip This type
of problem is b ecoming more imp ortant as the apparel industry moves in the direction of smaller lot sizes and customized garments
5 3
0 2 14
6
7
15 10 11
14 12
8
13
9
Figure Incremental layout task
Let the length b e Figure shows a very lo ose initial layout corresp onding to
inches Note that only the six large items are placed The two small ones can easily
b e inserted into the nal layout in a number of places so our search on length ignores them
Figure shows the tightest layout obtained for this problem inches
One alternative to this approach is to use an improvement strategy Improvement meth
o ds for layouts of nonconvex items are surveyed by Dowsland and Dowsland DD Meta
heuristic techniques such as simulated annealing genetic algorithms and tabu search can
b e applied However they are unable to establish a lower b ound on Another improvement
strategy uses the compaction techniques of Li and Milenkovic Li ML LM c LM a
MDL LM d Compaction is also unable to establish a lower b ound Compaction alone
can b e applied in a given direction to simulate a gravitational force However this technique
can only pro duce a continuous motion of the items It is unlikely to pro duce the tightest
p ossible strip packing esp ecially if the items are nonconvex In order to pro duce a tight Name: multnew1 Width: 23.50 in Length: 101.24 in Pieces: 16 Efficiency: 56.35%
5 13 3
0 10 2 8 14 9
6 15
7 14
11
12
Figure Lo ose layout
Name: multnew4 Width: 23.50 in Length: 77.09 in Pieces: 16 Efficiency: 74.93%
5
3 15
0 13 2
14 8 9 10 11 12
6
7 14
Figure Final layout
packing we must allow the items to move discontinuously as decreases This jumping
b ehavior can b e obtained using forms of compaction based on mixed integer programming
but current implementations are currently to o slow to b e practical
Name: multiple Width: 23.50 in Length: 85.48 in Pieces: 16 Efficiency: 67.58%
5 13
0 8 2 10 1 9
3 11 4 12
6 14
7 15
Figure Compaction solution
We compare our solution based on binary search and containment with the solution
obtained by rst compacting the layout of Figure and then placing the two small items
Compaction gives see Figure whereas for the containment ap
proach
Although the binary search technique obtains a tighter layout than compaction alone
compaction can still play an imp ortant role in this problem and in all our enclosure prob
lems It can certainly help to up date an upp er b ound quickly thereby accelerating the
binary search as noted in Section
Minimal Square Enclosures
In our second problem binary search on a scaling factor can b e combined with contain
ment to nd the minimal enclosing axis parallel square for a set of p olygonal items where
the orientation of each p olygon is xed One parameter su ces to characterize the square
We denote this by it is equal to the length of a side of the square We use binary
search on combined with containment
We b egin with a square which is guaranteed to b e large enough to hold the items Our
rst guess consists of a square whose side is the maximum of the height and width of the
b ounding b ox of the following shap e
P P P P P
k k
The k items can certainly t into this square
Because the area function decreases monotonically as decreases minimizing
yields the minimum area square enclosing the items The same is true if we apply a pure
scaling to any p olygon Thus the same metho d can b e used for example to nd the
minimal regular hexagon of xed orientation enclosing a collection of items
Figure Minimal square enclosure for nonconvex items
Figure shows the minimal square enclosure for three items An improvement metho d
which only allows continuous motion of the items is incapable of nding this square enclosure
b ecause there is no continuous motion of the items which can nest the rectangle inside
the C shap ed item
Figure shows a minimal square enclosure for four nonconvex apparel pattern pieces
This runs in seconds on a MHz SPARCstation Each of the four p olygons has
vertices The pieces are identical except that two are rotated by The con guration pro
duced by the containment algorithm is a rep eating pattern Indeed one p ossible application
of our minimal enclosure work is to identify go o d rep eating patterns for identical nonconvex
ob jects Although much work has b een done on lattice packings for convex ob jects there
is still ro om for work on nding go o d rep eating patterns for identical nonconvex ob jects
It would b e interesting to exp eriment with di erent types of regular p olygonal enclosures
e g square hexagon for a given set of identical items to see which gives the b est result
Figure shows the minimal square for ve items this takes seconds on a MHz
SPARCstation
Figure Minimal square enclosure for items
2
We thank Matt Buonomano of Gerb er Garment Technologies for suggesting this application
Figure Minimal square enclosure for nonconvex items
Implementation note an appropriate form of compaction for the square enclosure prob
lem is one which gravitates items toward the center of the square
Minimal Rectangular Enclosures
We extend this approach to nding tight rectangular enclosures which is a common
prepro cessing step in many current algorithms which pack non rectangular items It can
also b e applied to the problem of nding the b est sto ck sheet size for a given set of items
or the b est cloth width in garment manufacturing For layout algorithms which build strips
AA or columns MDL of items this approach can help determine the b est cluster In
higher dimensions this technique could help determine the b est packaging for a pro duct
Characterizing a rectangle requires two parameters One could parameterize by asp ect
ratio a and a scaling factor Unfortunately although is monotonic with resp ect to
it is not monotonic with resp ect to a Thus we cannot p erform binary search on a Instead
we can sample a
We choose instead to parameterize the rectangle by height h and width w In this case
as well the area function is only monotonic with resp ect to one of the parameters We
choose to sample w and p erform binary search on h for each sampled value of w It is
straightforward to obtain upp er and lower b ounds on w Let h b e the maximum height
max
of any item and w b e the maximum width Then one b ound on w can b e obtained
max
by letting the width of the container b e w and using binary search on height combined
max
with containment to nd the minimum asso ciated height The other b ound is achieved by
letting the height of the container b e h and using binary search on width combined
max
with containment to nd the minimum width w The minimal rectangular enclosure
min
algorithm contains an outer lo op which samples w in b etween the two b ounds For each w
it invokes a pro cedure which p erforms binary search on h for a particular value of w
Implementation note there are two ways to use compaction to accelerate the minimal
rectangle algorithm First of all downward compaction can b e used within the binary search
on h for xed w The second application of compaction relies on the fact that the minimal
h as a function of w increases monotonically as w decreases Steps in the function can
b e identi ed using compaction as follows After the minimal h is obtained for a given w
place the items within a w by h rectangle and apply leftward compaction to the right hand
b oundary of the rectangle Hop efully compaction will move w b eyond the next sample
p oint so that a binary search is avoided For this reason we suggest that w b e sampled in
decreasing fashion
We present three examples of nding minimal rectangles The rst example is three
rectangles in Figure This was found in seven seconds on a MHz SPARCstation
The second is three identical nearly nonconvex apparel pattern pieces in Figure This
required less than ve minutes The third is three nonconvex p olygons in Figure
Milenkovic has recently suggested MD an alternative metho d for nding minimal
rectangular enclosures of multiple items He p oses the following problem given shap es
P P P and area is there a rectangle of area which contains translated copies of
k
the shap es He solves the problem by mo difying the hybrid containment algorithm With
the mo di cation one can nd the minimum numerically by binary search
Figure Tight rectangular enclosure for convex items
Figure Tight rectangular enclosure for nearly identical nonconvex items
Figure Tight rectangular enclosure for nonconvex items
Tight Convex and Nonconvex Enclosures
Figure Tight nonconvex enclosure for items
Section observed that one parameter binary search together with containment can
not only nd the minimal enclosing square but it can also nd the minimal scaling factor
applied to any p olygonal enclosure
In this section we introduce a di erent way to obtain tight convex and nonconvex enclo
sures One can easily generate tight enclosures from a con guration of items within a tight
square or rectangular enclosure C For example one can use the convex hull of the items
to obtain a convex enclosure
However if we want the enclosure to preserve all the valid con gurations of C we need
a di erent approach We use the reachable region as de ned in Chapter to form a
container The pro cedure is simple First nd U for the items with resp ect to C Next
S
P This C is a container which U apply restrictions to U Then form C
j
j k
j
has all the valid con gurations asso ciated with C It reveals p ortions of C which cannot b e
reached by any of the items when they are placed simultaneously in C
Figure shows a tight nonconvex container obtained from the rectangular enclosure
of Figure In Figure the container is the black lled region inside the rectangle
and the outlines of the original items are shown to the right for reference This container
was formed using reachable regions which were in turn obtained from U s All of the
j
white space inside the rectangle and outside the black region is unreachable and can b e
removed with no consequences In the gure we can easily see that three of the four corners
of the rectangle are sup er uous for the three given items The convex hull of this container
is a convex container which has the same valid con gurations as the rectangle
This technique can b e applied for container design In some cases it can b e used to help
humans solve containment problems Observe that the container and items of Figure
immediately suggest a legal con guration to the human eye
Summary and Conclusions
This chapter has addressed four minimal container problems strip packing nding
minimal square enclosures nding minimal rectangular enclosures and nding tight
convex and nonconvex enclosures Our general approach to solving these problems is to
use binary search combined with containment The binary search can b e accelerated using
compaction techniques
Previous work on minimal enclosures in computational geometry considers only one
item Work in the layout literature allows multiple items but is heuristic and typically
greedy for more than two items Our work represents the rst non heuristic multiple
item minimal enclosure work for more than two items Our containment based solutions to
minimal container problems do not allow the items to rotate but in contrast to heuristics
we establish an actual lower bound on the size of the container for multiple items
C h a p t e r
Maximal Item Problems
Introduction
In this chapter we give four examples of maximal item problems given a container
and a collection of items nding maximally scaled copies of items which t in the container
given a container and a collection of k items for which the sum of the item areas
pr actical
do es not exceed the area of the container nding a large sub collection which ts in the
container given a container and a collection of items for which either the sum of the
item areas exceeds the area of the container or the number of items is k nding
pr actical
a large sub collection of k items which ts in the container and problem
pr actical
for a collection of containers
For problem we use binary search combined with containment In problem the
given items cannot b e scaled they can only b e included in or excluded from the set The
key to solving problem is to carefully choose which item to eliminate from an infeasible
subset We reject the most troublesome item which is identi ed using the results of the
overlap elimination evaluation metho d of the hybrid containment algorithm For problem
we randomly construct a set of items and solve problem This yields a rich collection
of sets which t into the container For each set of size n which ts we automatically have
n
subsets which t and which need not b e individuall y represented This work provides the
rst practical metho d for constructing sets of up to ten p olygons which t in a container
For problem we solve problem for each container and then apply a lo ok ahead
assignment strategy This is a pre packing strategy Containment thus allows us transform
a geometric optimization problem into a combinatorial optimization problem Our results
challenge the existing b elief that pre packing metho ds are impractical for layout Arb
Overview
Section solves problem Section solves problem Section solves prob
lem Section solves problem Section summarizes this chapter and states
conclusions
Maximally Scaled Items
In this section we are concerned with positive homothets of items A p ositive homothet
of an item P is a scaled copy of P with scaling factor We do not allow rotation
i i i
of the homothets Given a container a set of k items which has a valid con guration in
the container and a real number we p ose the following general problem nd a valid
con guration of p ositive homothets of the items in the container for which the sum of the
areas of the homothets is at least We call this the maximal positive homothet problem
MAX HOMOTHET for a container Maximizing the value of yields a tight packing
of the container
Claim The maximal positive homothet problem is NP hard
Pro of We reduce an instance of the NP complete PARTITION problem to an instance
of MAX HOMOTHET An instance of PARTITION has a set A of items Each a A has a
P
p ositive integer size s a Let s a Q We build a two component container Each
a A
comp onent is a rectangle of height and width Q For each a A construct a rectangle
of height and width s a Now if jAj scale each of these rectangles by
where and
X
s a Q
a An
where is the item in A of minimum width
This guarantees that the scaled rectangles t into the container Let Q Now solve
MAX HOMOTHET We claim MAX HOMOTHET has a solution if and only if PARTI
TION has a solution For if PARTITION has a solution then there exists a valid con gu
ration of rectangles of height and width s a for a A This con guration has area Q
and it represents a scaling of at least for each item we constructed for MAX HOMOTHET
so it is a solution to our instance of MAX HOMOTHET
Now we argue that if MAX HOMOTHET nds a solution then that solution is a solution
to PARTITION Because we scaled by no two items can b e stacked vertically within
a comp onent of the container This means that in order to completely ll the container
each item must b e restored to its full height This yields a solution to PARTITION 2
We examine heuristics for obtaining a go o d but not necessarily optimal solution to the
problem of maximizing the value of for MAX HOMOTHET One p ossible approach is to
force all items to have the same scaling factor This is easy to solve using binary search on
the scaling factor It has the advantage that a solution can b e found either by scaling the
container or the pieces If the number of vertices in the container is much smaller than the
sum of the number of vertices in the items then it is appropriate to scale the container
Another p ossibility is a greedy approach A greedy scaling algorithm would x the
scaling factor of one item during each iteration The xed item would b e the one whose
p ositive homothet can have maximum area
Each of these strategies can b e appropriate under di erent circumstances To illustrate
this we consider a maximal homothet problem for a collection of identical circular items
and a circular container Supp ose the container has a radius of and there are items It
is easy to show that if each of the given circles has radius less than then the area of the
Figure Scaling circles
p ositive homothets is greatest under a greedy strategy If however each circle has radius
greater than the greedy approach is inferior to applying the same scaling factor to each
circle The left p ortion of Figure shows a common scaling factor applied to the circles
and the right p ortion shows a greedy scaling
Yet another approach is an enhancement to the common scaling factor approach sug
gested by Milenkovic Mil a In this approach one maintains a set Q of items whose
scaling factors are xed and a set S of items whose scaling factors are not yet xed While
S the algorithm scales items in S up as much as p ossible by the same scaling factor
The scaled items of S must t into the container along with the items in Q Then for
each item of S if it cannot b e scaled more than by we remove it from S and add it to Q
We present an approach to the maximal homothet problem which has most of the
advantages of b oth the greedy and the enhanced common scaling factor approaches and
is sup erior to b oth on some inputs As in the enhancement to the common scaling factor
algorithm we maintain a set Q of items whose scaling factors are xed and a set S of items
whose scaling factors are not yet xed We use the area of SSGR U as a rough measure
i
of the freedom of an item P We asso ciate a parameter with the area of these U s
i i
The value of is a real number whose minimum and maximum values are the minimum and
maximum areas resp ectively over all SSGR U for i k For a given value of we
i
de ne a subset S of S S fP S jar ea S S GR U g As stated this causes
i i
b oth P and P to b e in S if ar ea S S GR U ar ea S S GR U Thus whenever
i j i j
an item is in S all identical copies of it are also in S To avoid this di culty we
mo dify the area function slightly so that for i j ar ea S S GR U ar ea S S GR U
i j
As ranges from to S changes k times For each of these k events we
max min
scale S up as much as p ossible using a common scaling factor for the items in S
The maximum common scaling factor is found using binary search on a parameter For
the maximum common scaling factor we record the total area of the scaled and unscaled
items Let b e the scaling factor asso ciated with the event of maximum total area and
max
S b e the asso ciated subset We scale items in S by If an item in S
max max max max
cannot b e scaled by more than we remove it from S and add it to Q We then up date
max
the set of events and rep eat the pro cess
Pseudo co de for the complete algorithm is given b elow The tolerance for the search on
is We denote the scaling of each element of a set S by a scaling factor as follows
S
MAX HOMOTHET P
Q
S P n P
max ar ea S S GR U
max P S i
i
min ar ea S S GR U
min P S i
i
Form event list
while S
best best max
for each event in event list
S fP S jar ea S S GR U g
i i
SCALABLE S P Q S n S
if total area for total area for
best
best best
S fP S jar ea S S GR U g
best i i best
S S
best best best
S fP S jSCALABLE fP g P Q S n fP g g
best i best i i
Q Q S
best
S S n S
best
SCALABLE S C S
max init
min
while
max min
cur r ent max min
if S S ts in C
cur r ent
min cur r ent
else
max cur r ent
return
min
P
Here is a simple way to establish Set ar ea C ar ea P
init init i
P S S
i
We stated ab ove that this algorithm p ossesses most of the advantages of the greedy
and the enhanced balanced strategies Here we elab orate on this p oint We rst show how
our algorithm is similar to the greedy algorithm Our event list contains an event which
corresp onds to scaling up a single item If we included an event for scaling up each P
i
individuall y we would b e testing all the p ossibilities considered by the greedy algorithm
In contrast our algorithm has only one single item scaling event However the scaled item
is the P whose U has the largest area This is a rough measure of the freedom of an
i
i
item When all items are identical our algorithm clearly makes the same choice as the
greedy algorithm We exp ect that in many cases in practice it will make the same choice
as the greedy algorithm Figure gives an example of MAX HOMOTHET b ehaving in
a greedy fashion for three items In this example only one of the items is scaled up its
scaling factor is
Figure Scaling up items in a greedy fashion
We now show how our algorithm is similar to the enhanced balanced strategy Our event
list contains an event which corresp onds to scaling up all items simultaneously by the same
scaling factor This is also done by the enhanced balanced strategy In addition we only
x the scaling factor for an item if it cannot b e scaled up further This is also a feature
of the enhanced balanced algorithm Figure gives an example of MAX HOMOTHET
b ehaving like the enhanced balanced strategy In this example all of the items are scaled
up by a factor of
Figure Scaling up items in an enhanced balanced fashion
There exist inputs for which our algorithm obtains a b etter solution than the greedy
algorithm A simple example is the previously mentioned circle problem in which the
container is a circle of radius and each of items is a circle of radius greater than In
general if the solution found by the enhanced balanced strategy is b etter than the solution
found by the greedy strategy for a given input then our algorithm s solution will b e b etter
than the greedy algorithm s solution for that input
There also exist inputs for which our algorithm obtains a b etter solution than the en
hanced balanced algorithm One such input consists of a circular container of radius and
circular items each of radius less than Because the items are identical our algorithm
makes the same choice as the greedy algorithm in this case
Our algorithm not only combines the p ower of the greedy and enhanced balanced strate
gies It also considers p ossibilities not considered by either of these two strategies These
p ossibilities corresp ond to events which scale up j items simultaneously for j k
Figure is an example in which MAX HOMOTHET scales up two items by the same
factor of the third item is not scaled
Figure Scaling up two items
Implementation note In applications where a scaling factor is involved one can use the
subset restriction of Section to attempt to sp eed up the containment algorithm
An Elimination Technique
In this section we consider the following problem given a container and a collection of
k k items for which the sum of the item areas do es not exceed the area of the
pr actical
container nd a large subset of the items which ts in the container An example of such
a problem is illustrated by Figure In this gure we are given nine items and must nd
a subset of them which ts into the container The container is depicted as a black lled
region The given items cannot b e scaled they can only b e included in or excluded from
the set
We present here a heuristic approach to this problem which is based on an elimination
technique and is therefore a top down metho d The ow of the pro cedure is shown b elow
for a container C and an initial collection of items S The advantage of this top down
approach is that a subset is found using at most k calls to containment Of course once
n
a subset of size n is found we obtain subsets for free and need not represent them
separately
GENERATE SUBSET S C
while S do es not t into C
S S n CHOOSE S
return S
However nothing is ever truly for free The price we pay here is that we must resolve the
di cult issue of how to choose an item to remove from the subset If this is not done well
the following scenario can easily o ccur Supp ose we always eliminate the item of smallest
area If a small number of the largest items in the subset cannot t together into the
container the subset pro duced by our pro cedure might only contain one or two items This
might not pack the container tightly
Figure Find a large subset which ts
Intuitively it makes sense to ask why a given infeasible subset do es not t If the
b ottleneck can b e found we can eliminate one of the items involved in the b ottleneck We
know how to obtain a go o d answer to this question b ecause when the hybrid containment
algorithm fails it can give us information ab out the lo cal overlap minimum it has obtained
Recall that our distance based sub division metho d of Section selects a U to sub divide
ij
based on feedback from the evaluation step In that case we sub divide the U which is
ij
1
One measure of size is the number of items another is the total area of the items
Figure A large subset which ts
asso ciated with the greatest overlap b etween two items These two items are in some sense
the most troublesome In the current application we can eliminate one of these two items
from the subset We choose to eliminate the smaller of the two This approach works
well Figure shows a solution to the subset problem of Figure Our approach nds
a collection of seven items which t into the container This packing is fairly tight
Implementation note To control running time it is p ossible to limit the number of
hypotheses examined by the hybrid algorithm This causes it to give up early in some
cases However b ecause every hypothesis evaluation pro duces a lo cal overlap minimum we
still obtain useful overlap information when the algorithm terminates early
Randomized Subset Construction
Here we address the following problem given a container and a collection of items for
which either the sum of the item areas exceeds the area of the container or the number
of items is k nd a large sub collection of k items which ts in the
pr actical pr actical
container
Whenever the total area of the items exceeds the area of the container the containment
problem is infeasible and there is no need to run a containment algorithm Whenever
the number of items exceeds k it is also not appropriate to run a containment
pr actical
algorithm In b oth of these cases our goal is to nd a subset S of the given set of items
which satis es the conditions of the problem addressed in Section The elimination
technique of Section can then b e applied to S
One way to pro duce S is to use randomness Using randomness has the advantage that
we can create a diverse collection of subsets by rep eating the pro cess We then apply the
elimination technique to each subset We select the resulting subset which has the largest
area and therefore provides the tightest packing of the container One can randomly exclude
items from the collection until the subset satis es the requirements of S Alternatively one
can select items randomly in a b ottom up fashion When the number of items is large is
large i e k as in our applications the latter approach is more e cient
pr actical
We o er two examples of the success of this randomized approach In b oth examples
some of the items can b e re ected in the x axis or rotated by a multiple of Our con
struction pro cess takes these p ossibilitie s into account by randomly selecting orientations
Figure p ortrays a collection of items and a container lled in black Figure shows
a collection of three items which pack the container tightly We generated the following ten
Figure Items and container for randomized construction task
16 17 52 18 36 19 53 21 20 54 57 37 55 22 38 56 23 24 39 58 25 40 59 26 60 27 41 42 61 28 43 62 29 44 63 30 45 64 65 31 46 66 67 47 68 32 69 70 48 71 33 49 50 34 51
35
Figure Randomly constructed item group
Figure Randomly constructed item group
groups for this container
Figure gives another example In this case a collection of ve items is
placed in the container We generated groups for this container
Pseudo co de for a group generation pro cedure for a container C and a collection of items
P is given b elow We maintain a t list G and a mis t list M which are b oth initially
empty To generate a group g which ts we rst initialize the current group g Then
while the total area of items in g do es not exceed the container area and the number of
items in g is k we randomly add items to g Let random P return a randomly
pr actical
selected item P i from P Now we attempt to disqualify g by checking if any group
i
in the mis t list is a subset of g If g is not disquali ed we obtain a subset of g which ts
by invoking GENERATE SUBSET from Section We add the subset to G Let b e a
stopping condition for generating groups for a container
GENERATE GROUPS P C
G
while not satis ed
g
P
while ar ea P ar ea C and jg j k
i pr actical
P g
i
g g random P
if there exists m M such that m g
M M g
else
g GENERATE SUBSET g C
Add g to G
return G
The question of what constitutes an e ective stopping criterion is an op en one A
running time packing density tradeo clearly exists Our exp erience suggests that a go o d
balance b etween these two factors can b e achieved if the number of groups is inversely pro
p ortional to the capacity of the container We have tested our group generation pro cedure
on hundreds of examples and veri ed that our stopping criterion pro duces tight packings
Ultimately the b est stopping condition can dep end on what the groups will b e used for
We therefore defer further discussion of this to the next section
Pre packing
Given a packing problem for a collection of containers and a collection of items a
pre packing approach has two steps generate groups of items for each container and
assign groups to containers The goal is to place a maximal subset of the items
Section o ers a group generation pro cedure which uses containment This solves problem
Problem is a combinatorial optimization problem There is a rich literature on
combinatorial optimization
The optimal solution to the assignment problem can b e obtained using a matching mo del
based on mixed integer programming We developed one such mo del A Bo olean variable
2
We did not implement the mis t list so it is not clear whether or not it is cost e ective in general
3
Area is one p ossible measure
g in our mo del represents the assignment of group k to container j A Bo olean variable p
k j ij
represents the assignment of item i to container j The mo del has three types of constraints
The rst type of constraint is of the form
X
g
k j
k
There is one constraint of this type for each container Each constraint prevents a container
from having more than one group assigned to it The second type of constraint is
X
p
ij
j
There is one constraint of this type for each item Each constraint prevents an item from
b eing assigned to more than one group The third type of constraint forces all items in a
group to b e assigned together This is of the form
p g
ij k j
There is one constraint of this type for each item in each group One p ossible ob jective
function is to maximize the following
X
g
k j
k j
We originally intended to use this mo del under the assumption that each group contains
at most three items If we allow up to ten items in a group it is not clear that this approach
is practical given the current state of the art in mixed integer programming Consider for
example a typical pre packing application in apparel marker making A typical pants
marker has approximately containers If containment pro duces a item group for each
container this yields subgroups for each container The number of Bo olean variables
could b e as large as in this case
At the other extreme a greedy algorithm can b e used for assignment There are many
p ossible approaches in b etween mixed integer programming and a greedy strategy Finding
the ultimate assignment technique for pre packing is b eyond the scop e of this containment
oriented thesis However we o er an intermediate approach which runs quickly provides
lo ok ahead and is quite exible Our results challenge the b elief that pre packing strategies
are impractical Arb
Assignment Strategy
Our assignment strategy requires three functions The rst is a comparison function
f g g which accepts two groups g and g returns g if g g and returns g otherwise
Let C b e a collection of containers The second function is an ordering function f C C
for containers The third is an evaluation function h C g which accepts a container C C
and a group g and returns a real number
The rst step of our assignment pro cess is to sort the groups for each container according
to f Then while some item remains unassigned p erform the following For each container
select the rst group in its ordering for which there are su cient unassigned items Apply
h C g to obtain a value for this group Select the C g pair with the b est value and
p erform this assignment
The evaluation function h uses a greedy lo ok ahead strategy which requires the ordering
function f for containers For a given container and its rst group tentatively p erform this
assignment Now examine each remaining container according to the container ordering
For each container tentatively assign the highest group according to f whose items are
currently tentatively unassigned When tentative assignments have b een p erformed for
all containers return which is the total area of all tentatively assigned items
In the pseudo co de b elow C is a collection of containers and G is the output of GENERATE
GROUPS for all the containers C C is an individual container and G G is the
i i
collection of groups pro duced by GENERATE GROUPS for C An available group g G
i i
is one whose items are all available i e unassigned
ASSIGN P C G
for each container C C
i
Sort G G according to f
i
while some P P i is available
i
for each container C C
i
g rst available group of G
i
h C g
i i
B estI index of container with b est
Assign items of g for C to C
B estI B estI
Results
To justify our claim that pre packing can b e practical we present the results of our
exp eriments with applying pre packing to a trim placement problem for pants markers in
apparel manufacturing see Section for a description of the trim placement problem
Table presents results of our group generation and assignment pro cedures for ex
amples For each example we give the name of the marker layout the number of trim
items which must b e placed and results for three trim placement strategies The three
strategies are First Fit our assignment strategy called Match in the table and
our assignment strategy followed by First Fit on the unplaced items For each strategy we
give the number of items left unplaced and the total area of the unplaced items
Our First Fit strategy is a greedy heuristic which places each item at the leftmost
available p osition The items are sorted by decreasing area and are placed in that order
The heuristic has two additional features which allow it to pro duce layouts whose waste
p ercentage is on the average only ab out larger than what exp erienced humans achieve
The rst additional feature tries all legal ips and discrete rotations by multiples of
for each item and selects the b est one The second feature uses a variation on compaction
to bump each placed item in a particular direction in an e ort to reduce fragmentation
of the available space
There is a number of ways to interpret this data Comparing the total number of items
left unplaced assignment plus First Fit places more items than First Fit alone Averag
ing the p ercentage of the items which are left unplaced shows that assignment plus First Fit
leaves only an average of of the trim items unplaced whereas First Fit leaves of
First Fit Match Match First Fit
marker name ntrim nleft area left nleft area left nleft area left
c e e
b e e e
b e e e
b e e e
d e e e
c e e e
k t e e e
k t e e e
k t a e
k t e e e
a e e
a e e e
a e e e
a e e e
i t e e e
i t a e e e
i t e e e
e e
Table Pre packing results
them unplaced assignment plus First Fit is b etter by this measure It is also valuable
to compare the area of the unplaced items A raw comparison of total area shows that
assignment plus First Fit placed more area than First Fit To remove any di erences
in scale averaging the p ercentage of the area which is left unplaced shows that assignment
plus First Fit leaves only an average of of the area unplaced whereas First Fit leaves
of the area unplaced assignment plus First Fit is b etter by this measure All of
these comparisons show the quality of assignment plus First Fit to b e sup erior to First Fit
alone
We illustrate our results by showing layouts for two examples For the rst example Fig
ure shows the trim placement task Figure shows the layout obtained by First Fit
Figure shows the layout obtained by our assignment pro cedure and Figure shows
the layout after assignment followed by First Fit Figure Figure Figure and Figure show the same progression for our second example
Name: 37188c-trim Width: 59.75 in Length: 130.28 in Pieces: 54 Efficiency: 76.45%
5 3 7
11 6
1
8 9 0
10 2 4
49 12 23 38 50 13 29 51 39 52 24 14 30 53 15 40 31 16 25 41 32 17 42 18 33 26 43 34 19 44 35 20 27 45 36 21 46 37 22 28 47
48
Figure Trim placement task for c mrk
Additional quality comes at the cost of higher running time Our First Fit strategy
requires under ve minutes whereas our new strategy currently requires one to two hours
for a item example on a MHz SPARCstation In apparel manufacturing a human
takes ab out hour for an example of this size One to two hours is within the realm
of practicality For our current implementations of containment and assignment ab out
of the running time is consumed by group generation and of this almost all the
time is sp ent in the containment algorithm This means that a dramatic increase in the
sp eed of our hybrid containment algorithm will also have a dramatic impact on the sp eed Name: 37188c-ffit Width: 59.75 in Length: 130.28 in Pieces: 54 Efficiency: 86.78%
51 50 48 14 18 24 45 22 31 16 5 3 7 35 15 34 33 23
11 20 30 13 6 29 28 47 1 26 25 46 17 8 9 0
32 12 10 2 4
21 27 53 52 49
19 36 37 38 39 40 41 42 43
44
Figure First Fit for c mrk
Name: 37188c-matchk Width: 59.75 in Length: 130.28 in Pieces: 54 Efficiency: 87.27%
52 51 30 21 25 26 37 5 3 7 34 49 32 20
15 29 47 11 17 23 6 35 39 36 27 48 31 1 46 24 45 38 8 9 0
33 12 14 10 2 4 40 16 13 28 41 22 50 53
18 19 42 43
44
Figure Assignment for c mrk Name: 37188c-matchk2 Width: 59.75 in Length: 130.28 in Pieces: 54 Efficiency: 87.82%
52 51 30 21 25 26 424344 37 5 3 7 34 49 32 20
15 29 47 11 17 23 6 35 39 36 27 48 31 1 46 24 45 38 8 9 0
33 12 14 10 2 4 40 16 13 28 41 22 50 53
18
19
Figure Assignment plus First Fit for c mrk
Name: 41132-trim Width: 59.75 in Length: 182.13 in Pieces: 72 Efficiency: 78.65%
6 13 2 9
7 3 12 8
14 10 5 1
15 11 4 0
16 49 17 34 18 50 19 35 51 20 52 21 36 53 22 54 23 37 24 55 25 38 56 26 57 27 39 58 28 59 40 29 60 41 61 30 42 31 62 43 63 44 64 32 65 45 66 67 46 68 33 69 70 47 71
48
Figure Trim placement task for b mrk Name: 41132-ffit Width: 59.75 in Length: 182.13 in Pieces: 72 Efficiency: 89.04%
28 26 47 16 24 29 40 23 22 17 69
6 13 2 9 36 37 62 55 33 54 32 21 53 42 7 3 12 8 20 45 46 14 10 5 44 1 38 34 39 63 19 43 71 35 68 15 11 4 0 70 18 25 31 30 41 27
48 49 50 51 52 56 57 58 59 60 61 64 65 66
67
Figure First Fit for b mrk
Name: 41132-matchk Width: 59.75 in Length: 182.13 in Pieces: 72 Efficiency: 89.67%
67 69 26 53 24 40 31 56 66 68
6 13 2 9 43 18 70 57 21 39 37 34 20 59 36 54 55 7 3 12 8 47 44
23 45 14 10 5 42 1 41 35 38 63 33 17 58 32 46 22 62 15 11 4 0 61 71 65 30 52 28 19 60 27 29
16 25 48 49 50 51
64
Figure Assignment for b mrk Name: 41132-matchk2 Width: 59.75 in Length: 182.13 in Pieces: 72 Efficiency: 90.41%
67 69 26 53 24 40 495051 31 56 66 64 16 68
6 13 2 9 43 18 70 57 21 39 37 34 20 59 36 54 55 7 3 12 8 47 44
23 45 14 10 5 42 1 41 35 38 63 33 17 58 32 46 22 62 15 11 4 0 61 71 65 30 52 28 19 60 27 29
25
48
Figure Assignment plus First Fit for b mrk
of pre packing
Our b est trim placement strategy for pants markers in apparel manufacturing is neither
First Fit nor pre packing It places items in approximately ten minutes and its waste is in
the pro duction quality range Neither First Fit nor our current pre packing implementation
currently generate results which are consistently in the pro duction quality range Our b est
strategy employs sp ecial heuristics to place the largest items rst uses containment to
construct small groups of at most three items a naive greedy assignment strategy and
compaction based techniques combined with First Fit to place any remaining items
Although it achieves excellent results for pants markers its applicability to other do
mains is limited In contrast our pre packing approach is completely general It can b e
applied to any domain The same is true of First Fit
Our b est pants strategy also do es not have much growth p otential That is we do not
exp ect that adjustments to parameters of this algorithm will signi cantly reduce the waste
of the asso ciated layouts The same is true of First Fit However our pre packing approach
has tremendous growth p otential Its initial results are go o d and improvements in the
assignment strategy or the group generation pro cess will improve the quality of its results
Future research will b ene t from determining whether the current group generation
pro cess or the assignment strategy is the limiting factor We have attempted to p erform
assignment manually using the groups we generate and our initial results show that sur
prisingly the assignment strategy is less limiting than group generation
There are many parts of the group generation pro cedure which can b e altered the ran
dom pro cedure for building a group the elimination technique the containment algorithm
and the number of groups Our initial exp eriments have imp osed a number of arti cial
limitations Our current implementation limits the size of a subset to ten our waste gures
would probably improve with a higher limit We also limit the number of hypotheses eval
uated by the hybrid algorithm to less than and we build no more than groups for a
container
This returns us to the stopping condition issue raised in Section It is impractical to
generate al l p ossible groups for each container How do we determine if a given collection of
groups is satisfactory for the assignment phase without running the assignment algorithm
This is b eyond the scop e of the thesis Investigating ways to obtain a lower b ound on
waste for a given collection of groups is a sub ject of future work Given a lower b ounding
technique one could iteratively determine a b ound and selectively generate more groups
until the desired waste level was achieved
Summary and Conclusions
This chapter has addressed the following maximal item problems given a container
and a collection of items nding maximally scaled copies of items which t in the container
given a container and a collection of k items for which the sum of the item areas
pr actical
do es not exceed the area of the container nding a large sub collection which ts in the
container given a container and a collection of items for which either the sum of the
item areas exceeds the area of the container or the number of items is k nding
pr actical
a large sub collection of k items which ts in the container and problem
pr actical
for a collection of containers
For problem we o er a heuristic which uses binary search combined with contain
ment The heuristic has most of the advantages of the greedy and the enhanced balanced
strategies It b eats b oth of these approaches on some inputs
The remaining three problems are all related to our pre packing strategy We use an
elimination strategy for problem which uses feedback from the hybrid containment
algorithm to identify troublesome items For problem we randomly construct a subset
of items and then solve problem This pro cess considers multiple discrete orientations
for items For problem we nd groups of items which can t into each container and
then apply a lo ok ahead assignment strategy We demonstrate what we b elieve is the rst
practical application of pre packing We are able to generate a su ciently rich collection of
groups for each container so that our assignment strategy followed by a First Fit strategy
can outp erform a First Fit strategy This pro cess takes one or two hours for a item
example using the current implementation of our hybrid containment algorithm There are
many ways to improve our group generation and assignment pro cesses We b elieve that in
addition to demonstrating the feasibility of pre packing this approach has the p otential to
eventually match the pro duction quality results of our b est algorithm for trim placement
in apparel manufacturing
Part III
Conclusion and App endix
C h a p t e r
Conclusion and Future Work
Conclusion
This thesis demonstrates that although containment is NP hard it is fruitful to
develop algorithms for containment as opp osed to heuristics design containment algo
rithms so that they say no almost as fast as they say yes use geometric techniques
not just mathematical programming techniques and maximize the number of items for
which the algorithms are practical
Our approach to containment has a number of imp ortant features First we intro
duce a restrict evaluate subdivide paradigm Second we develop new theory and practical
techniques for the op erations within the paradigm The techniques are appropriate for two
dimensional containment problems in which the items and container may b e irregular i e
nonconvex and have multiple comp onents and in which the items may b e translated but
not rotated Our techniques can b e combined to form a variety of two dimensional transla
tional containment algorithms which follow the restrict evaluate subdivide paradigm The
paradigm is designed so that unlike existing iteration based algorithms containment algo
rithms based on the paradigm are adept at saying no even for slightly infeasible problems
Infeasible problems o ccur frequently in practice
We develop geometric restrictions which are p owerful con guration space pruning tech
niques They often detect infeasibility without any evaluation or sub division We give two
algorithms which use geometric restrictions and are based on the restrict evaluate subdivide
paradigm We obtain the rst practical running times for NP complete two dimensional
translational containment problems for up to ten nonconvex items in a nonconvex container
Our examples are drawn primarily from the apparel industry Each item is represented as
a p olygon which typically has from to vertices The container is also represented
as a p olygon it typically has from to vertices The items and container may b e
nonconvex
We demonstrate that viewing containment as a feasibility problem has many b ene ts
for layout optimization problems We present an e ective metho d for nding minimal
enclosures which uses containment to p erform binary search on a parameter We note that
compaction can b e used to accelerate the search This work represents the rst practical
approach to nding minimal enclosures for multiple nonconvex items Clustering is used
often in layout but current clustering techniques for more than two items are heuristic
Our minimal enclosure work should lead to more p owerful clustering metho ds and is an
example of how strong b ottom up techniques from computational geometry can help the
layout community
We b elieve that some p otentially promising layout techniques have b een abandoned by
the layout community b ecause they lack strong b ottom up metho ds The primary example
of this is pre packing We challenge the view that pre packing is impractical for multiple
container packing problems by generating a rich collection of groups for each container
using containment and then applying an assignment strategy
Our advances in containment will help automate layout for industrial pro cesses that rely
on two dimensional packing of ob jects Our results are particularly relevant to industries
which deal with irregularly shap ed items This includes industries such as aerospace ship
building apparel and sho e manufacturing furniture pro duction and steel construction
Future Work
Section invites other researchers to exchange containment data with us We hop e
this will result in a broad set of containment b enchmarks which can b e used to evaluate
the e ectiveness of di erent containment algorithms In addition to soliciting containment
problems from a variety of sources we plan to automate the pro cess of creating containment
problems from existing layouts
Here we o er directions for future research on containment problems and applications of
containment to layout Section describ es problems related to containment algorithms
and Section discusses applications
Containment
There are several directions for future work in containment One direction is to increase
the dimensionality of the containment problems we address For instance many imp ortant
three dimensional packing and layout problems could b e solved using three dimensional
containment algorithms Another way to increase the dimensionality of the containment
problem is to allow rotation Rotational containment adds an extra degree of freedom to
each item by allowing it to rotate continuously within a range of angles Many layout
problems allow continuous rotation of items particularly problems related to sheet metal
work Section reviewed work in this area There is clearly ro om for much work on
rotational containment involving more than one item
There are ways rotation could b e dealt with in a heuristic fashion The rst way in
volves our minimal enclosure work Recall that compaction can b e used to up date the
upp er b ound Li and Milenkovic have rotational as well as translational compaction al
gorithms Rotational compaction could b e applied to the translational solution as a nal
p ost pro cessing step In our hybrid translational containment algorithm one could de
velop a rotational overlap reduction evaluator Using this evaluator the hybrid algorithm
would try to pro duce a valid rotational con guration from each overlapping translational
con guration found by the linear program
A second direction for future work involves improving our algorithms for translational
containment so that larger numbers of items more than ten may b e placed in a practical
amount of time One way to obtain a sp eed up is to apply course grained parallelism
Throughout the thesis we have identi ed parts of our work which are natural candidates
for course grained parallelism
Some layout applications represent ob jects using circular arcs in addition to p olygons
and so enriching our underlying ob ject representation is a worthwhile goal
There is a considerable literature on rectangle packing and integer programming that
remains untapped for general k NN containment Integer programming requires a great
deal of nesse to apply correctly and we have b een disapp ointed so far with results we
have obtained using state of the art mixed integer programming packages Our work on
combining computational geometry and linear programming is a large step towards the
ultimate goal of usefully combining computational geometry and integer programming
Exact covering problems such as solving jigsaw puzzles can b e solved using contain
ment algorithms For the non exact case we hop e to develop algorithms which decide if a
collection of items can cover another item
We plan to develop more valid and semi valid restrictions for translational containment
We will continue to explore the question of which restriction op erations allow s the max
imum number of vertices in a restricted p olygon to remain nite even when an in nite
number of op erations is applied
The following problems are asso ciated with our greedy evaluation metho d
Given p olygons A and B is there a translation t for which H A t B is convex
Find t which maximizes the convexity of A t B
Solve these problems for convex A and B
Given a collection of p olygons P how fast can we nd the translation t for each
p
p P such that the area of p t is maximized This is also useful for nding
p P p
the maximum area breaking p olygon
The shrinking p oint heuristic which maximizes the area of intersection of translated U s
ij
works well unless the U s consist only of isolated edges and p oints This case can o ccur
ij
when we form subproblems using a b oundary restriction In this case we prop ose using a
distance based heuristic instead of an area based one
These problems arise in relation to sub division
Sub divide in a that encourages translations of U s to intersect
ij
Given a set of p olygons P how fast can we nd the line L which partitions P into P
and P such that the area of H P H P H P is minimized
Potential improvements to the approximate algorithm include
Use multiple resolution techniques
Invoke a linear programming based algorithm when the convex hulls of the restricted
U s are all inside their resp ective unrestricted U s
ij ij
Extend the size based sub division used in this algorithm so it sub divides U s for
ij
which i
We plan to evaluate the cost e ectiveness of using a partial characterization to pro duce k
subproblems for k Our current exp eriments suggest that it is b ene cial for k
even when the subproblems are solved sequentially We exp ect that for non identical item
problems a critical value of k exists b eyond which the sequential advantage is lost A de
tailed comparison of the parallel cost will b e necessary in order to justify using the partial
characterization for values of k larger than the critical one Course grained parallelism can
b e applied to our algorithm by maintaining a queue of hypotheses Each hypothesis repre
sents an indep endent containment subproblem which is created via sub division Whenever
a pro cessor is available it op erates on the next hypothesis in the queue The subproblems
generated by the characterization are also indep endent from each other and so the hypothe
ses they generate are indep endent and can b e added to the queue We plan to construct a
program which uses a queue and simulates the parallel execution of our algorithm
Applications of Containment to Layout
In our work on container transformation we currently approximate the area integral of
Prob ftg v for t v via sampling We seek e cient algorithms for computing this integral
exactly
Pursuing more e cient algorithms for nding the exact MAAPR is certainly one di
rection for future work We are interested in faster algorithms or lower b ounds for these
p olygon types
Orthogonally convex
Horizontally convex
Vertically separated horizontally convex
Another direction of practical imp ortance is to nd a fast approximate MAAPR algorithm
Such an algorithm would b e very helpful in our applications
There are many op en problems asso ciated with the concept of inking sets The problem
of minimizing the number of vertices while maximizing the minimum area of inking set
pieces is op en even for convex p olygons Given a collection of p olygons the problem of
nding a p olygon representing the maximal common subset of translations of these p olygons
is useful not only for inking sets but also for the subset substitution restriction
Future work on our minimal enclosure applications includes implementing our idea of
using compaction to accelerate binary search We also plan to exp eriment with using our
minimal enclosure algorithms for nding rep eating patterns of irregular ob jects
Future work on maximal item problems includes improving our algorithm for solving
the maximal homothet problem Our pre packing work can b e improved by obtaining lower
b ounds for assignment and feeding these results back into the group generation pro cess
C h a p t e r A
Containment File Format
This app endix describ es the simple le format we prop ose for two dimensional p olygonal
containment data The format rst describ es the p olygonal container and then each of the
p olygonal items to b e placed For each p olygon the number of vertices precedes the list of
x and y vertex co ordinates Data for each vertex app ears on a separate line The vertices
are listed in counterclockwise order so that the inside of the p olygon is to the left of each
directed edge
The small example on the next page is for a CC problem for which our approximate
algorithm nds a valid con guration in one second on a MHz SPARCStation In this
example the container is a triangle and each of the three items is also a triangle
container vertices
items
item vertices
item vertices
item vertices
Bibliography
AA M Adamowicz and A Albano Nesting two dimensional shap es in rectangular
mo dules Computer Aided Design
AB F Avnaim and J Boissonnat Simultaneous Containment of Several Polygons
In Proceedings of the rd ACM Symposium on Computational Geometry pages
AB F Avnaim and J Boissonnat Polygon placement under translation and rota
tion In Proceedings of the th Annual Symposium on Theoretical Aspects of
Computer Science Lecture Notes in Computer Science pages
Springer Verlag
ABB H Alt B Behrends and J Bl omer Approximate matching of p olygonal shap es
In Proceedings of the th Annual ACM Symposium on Computational Geometry
pages
ABJ C Amaral J Bernardo and J Jorge Marker Making Using Automatic Place
ment of Irregular Shap es for the Garment Industry Computers and Graphics
Ada M Adamowicz The optimum two dimensional al location of irregular multiply
connected shapes with linear logical and geometric constraints PhD thesis
Electrical Engineering Department New York University
Arb A Arb el Large scale optimization metho ds applied to the cutting sto ck problem
of irregular shap es International Journal of Production Research
AS A Albano and G Sapupp o Optimal Allo cation of Two Dimensional Irregular
Shap es Using Heuristic Search Metho ds IEEE Transactions on Systems Man
and Cybernetics SMC
AS B Aronov and M Sharir On Translational Motion Planning in Space In
Proceedings of the th Annual ACM Symposium on Computational Geometry
pages
Avn F Avnaim Placement et d eplacement de formes rigides ou articul ees PhD
thesis Universit ede Franche Comt e France
Bab P V Babaev The optimal cutout of materials with help of a computer
Mashinostro enje
BFM B Baker S Fortune and S Mahaney Polygon Containment under Translation
Journal of Algorithms
BG D B ohme and A Graham Practical Exp eriences with Semiautomatic and
Automatic Partnesting Metho ds In C Kuo K MacCallum and T Williams
editors Computer Applications in the Automation of Shipyard Operation and
Ship Design III pages North Holland Publishing Company
BHW J Blazewicz P Hawryluk and R Walkowiak Using a tabu search approach
for solving the two dimensional irregular cutting problem Annals of Operations
Research
BK A Bezdek and G Kert esz Counter examples to a packing problem of L Fejes
T oth In K B or oczky and G Fejes T oth editors Intuitive Geometry North
Holland New York
BTS J Bailleul K Tiaibia and R So enen Nesting Two Dimensional Irregular
Shap es In Anisotropic Material In T M R Ellis and O I Semenkov editors
Advances in CAD CAM pages North Holland
CG R Cuninghame Green Geometry Sho emaking and the Milk Tray Problem
New Scientist August
Cha B Chazelle The Polygon Containment Problem In Preparata editor Advances
in Computing Research Volume Computational Geometry pages JAI
Press Inc Greenwich Connecticut
CK L P Chew and K Kedem Placing the largest similar copy of a convex p olygon
among p olygonal obstacles In Proceedings of the th Annual ACM Symposium
on Computational Geometry pages
CMTS N Christo des A Mingozzi P Toth and C Sandi Combinatorial Optimiza
tion John Wiley and Sons Chichester
CY J S Chang and C K Yap A Polynomial Solution for the Potato p eeling Prob
lem Discrete and Computational Geometry
DBB D Dori and M Ben Bassat E cient Nesting of Congruent Convex Figures
Communications of the ACM
DD Kathryn A Dowsland and William B Dowsland Solution Approaches to Ir
regular Nesting Problems Invited review to appear in European Journal of
Operational Research
DD K A Dowsland and W B Dowsland Packing Problems European Journal of
Operational Research
Dev O Devillers Simultaneous Containment of Several Polygons Analysis of the
Contact Con gurations Technical Rep ort INRIA
DF H Dyckho and U Finke Cutting and Packing in Production and Distribution
A Typology and Bibliography Physica Verlag Heidelb erg
DJ R Dighe and M Jakiela Solving Pattern Nesting Problems With Genetic Al
gorithms Employing Task Decomp osition And Contact Detection In Evolu
tionary Computation MIT Press Cambridge MA
DKO N DePano Y Ke and J O Rourke Finding Largest Inscrib ed Equilateral
Triangles and Squares In Proceedings of the th Al lerton Conference on Com
munications Control and Computing pages
DM K Daniels and V J Milenkovic Multiple Translational Containment Part
I An Approximate Algorithm Algorithmica special issue on Computational
Geometry in Manufacturing accepted subject to revisions
DM K Daniels and V J Milenkovic Limited Gaps In Proceedings of the th
Canadian Conference on Computational Geometry pages
DM K Daniels and V J Milenkovic Multiple Translational Containment Approx
imate and Exact Algorithms In Proceedings of the th Annual ACM SIAM
Symposium on Discrete Algorithms
DML K Daniels V J Milenkovic and Z Li Multiple Containment Metho ds Tech
nical Rep ort Center for Research in Computing Technology Division of
Applied Sciences Harvard University
DMR K Daniels V J Milenkovic and D Roth Finding the Maximum Area Axis
Parallel Rectangle in a Simple Polygon Computational Geometry Theory and
Applications accepted subject to revisions
DMR K Daniels V J Milenkovic and D Roth Finding the Maximum Area Axis
Parallel Rectangle in a Polygon In Proceedings of the th Canadian Conference
on Computational Geometry pages
DT C Da gliand M Tato glu An approach to two dimensional cutting sto ck prob
lems International Journal of Production Research
Due G Dueck New Optimization Heuristics The Great Deluge Algorithm and the
Record to Record Travel Journal of Computational Physics
Dyc H Dyckho A typology of cutting and packing problems European Journal of
Operations Research
Ede H Edelsbrunner Algorithms in Combinatorial Geometry Springer Verlag New
York
For S Fortune A Fast Algorithm for Polygon Containment by Translation In
Proceedings of the th Col loquium on Automata Languages and Programming
pages Springer Verlag
For S Fortune A Sweepline Algorithm for Voronoi Diagrams Algorithmica
GA H El Gindy and D Avis A Linear Algorithm for Computing the Visibili ty
Polygon from a Point Journal of Algorithms
GC R Grinde and T Cavalier Containment of a Single Polygon Using Mathemat
ical Programming Technical Rep ort IMSE Working Paper The Penn
sylvania State University Department of Industrial and Management Systems
Engineering
GG P Gilmore and R Gomory Multistage Cutting Sto ck Problems of Two and
More Dimensions Operations Research
GJ M Garey and D Johnson Computers and Intractability A Guide to the Theory
of NP Completeness W H Freeman and Company San Francisco
GRS L Guibas L Ramshaw and J Stol A Kinetic Framework for Computa
tional Geometry In Proceedings of the th IEEE Symposium on Foundations
of Computer Science pages
Gur a O Gurel Additional considerations on marker layout problem by graph theory
Technical Rep ort IBM Scienti c Centre
Gur b O Gurel Marker layout via graph theory An attempt for optimal layout of
irregular patterns Technical Rep ort IBM Scienti c Centre
Gur O Gurel Circular graph of marker layout Technical Rep ort IBM
Scienti c Centre
HF M Haims and H Freeman A multistage solution of the template layout prob
lem I E E E Transactions on Systems Science and Cybernetics
Hin A Hinxman The trim loss and assortment problems A survey European
Journal of Operations Research
HL J Heistermann and T Lengauer E cient Automatic Part Nesting on Irregular
and Inhomogeneous Surfaces In Proceedings of the rd Annual ACM SIAM
Symposium on Discrete Algorithms
HL R Heckmann and T Lengauer A Simulated Annealing Approach to the Nesting
Problem in the Textile Manufacturing Industry In R Burkhard P Hammer
T Ibaraki and M Queyranne editors Mathematics of Industrial Systems Vol
ume Annals of Op erations Research J C Baltzer AG Science Publishers
Amsterdam
HT D Hinkle and C Toomey Clavier Applying Cased Based Reasoning to Com
p osite Part Fabriciation In AAAI Sixth Innovative Applications of Arti cial
Intel ligence Conference pages
HW N Hyer and U Wemmerl ov Group technology in the U S manufacturing indus
try a survey of current practices International Journal of Production Research
IH H Ismail and K Hon New approaches for the nesting of two dimensional shap es
for press to ol design International Journal of Production Research
IS S Israni and J Sanders Two dimensional cutting sto ck problem research A
review and a new rectangular layout algorithm Journal of Manufacturing Sys
tems
IS S Israni and J Sanders Performance testing of rectanular parts nesting heuris
tics International Journal of Production Research
JGJ E G Co man Jr M R Garey and D S Johnson Approximation Algorithms for
Bin Packing An Up dated Survey In G Ausiello M Lucertini and P Sera ni
editors Algorithm Design for Computer System Design pages Springer
JL E G Co man Jr and G S Lueker Probabilistic Analysis of Packing and Par
titioning Algorithms Wiley and Sons Inc New York
JS B Jo e and R B Simpson Corrections to Lee s Visibility Polygon Algorithm
BIT
Kar J B Karasic Formalization of Heuristic Tactics Used by Human Beings in
Searching for the Closest Packing of Arbitrary Ob jects in a Given Domain
Cybernetics and Systems An International Journal
Kha L G Khachian A p olynomial algorithm in linear programming Soviet Math
ematics Doklady
Kir D G Kirkpatrick Optimal Search in Planar Sub divisi ons SIAM Journal on
Computing
KK G Kup erb erg and W Kup erb erg Double Lattice Packings of Convex Bo dies
in the Plane Discrete and Computational Geometry
KL F Karoupi and M Loftus Accommo dating diverse shap es within hexagonal
pavers International Journal of Production Research
Law E Lawler Combinatorial Optimization Networks and Matroids Holt Rinehart
and Winston New York
Lee D T Lee Two dimensional Voronoi diagrams in the L Metric Journal of the
p
ACM
Lee D T Lee Visibili ty of a Simple Polygon Computer Vision Graphics and
Image Processing
Li Z Li Compaction Algorithms for Non Convex Polygons and Their Applications
PhD thesis Harvard University Division of Applied Sciences
LLKS E Lawler J Lenstra A Rinno oy Kan and D Shmoys editors The Traveling
Salesman A Guided Tour of Combinatorial Optimization John Wiley and
Sons Chichester
LM a Z Li and V J Milenkovic A Compaction Algorithm for Non Convex Polygons
and Its Application In Proceedings of the th Annual ACM Symposium on
Computational Geometry pages May
LM b Z Li and V J Milenkovic Compaction and Separation Algorithms for Non
Convex Polygons and Their Applications Technical Rep ort TR Center
for Research in Computing Technology
LM c Z Li and V J Milenkovic The Complexity of the Compaction Problem In
Proceedings of the th Canadian Conference on Computational Geometry A
Lubiw and J Urrutia Editors pages August
LM d Z Li and V J Milenkovic The Complexity of the Compaction Problem In
Proceedings of the th Canadian Conference on Computational Geometry pages
Waterloo Canada
MD V J Milenkovic and K Daniels Translational Polygon Containment and Min
imal Enclosure using Geometric Algorithms and Mathematical Programming
In Submitted to the th Annual IEEE Conference on Foundations of Computer
Science
MDL V J Milenkovic K Daniels and Z Li Automatic Marker Making In Proceed
ings of the rd Canadian Conference on Computational Geometry
MDL V J Milenkovic K Daniels and Z Li Placement and Compaction of Non
convex Polygons for Clothing Manufacture In Proceedings of the th Canadian
Conference on Computational Geometry pages
Meg N Megiddo Applying Parallel Computation Algorithm in the Design of Serial
Algorithms Journal of the ACM
MH G Moreau and P Hardavin Marker layout problem An exp erimental attempt
Technical Rep ort IBM New York Scienti c Center
Mil V J Milenkovic Exact Algorithms for Multiple Containment Algorithmica
special issue on Computational Geometry in Manufacturing accepted subject to
revisions
Mil a V J Milenkovic Applications of Geometric Rounding to Polygon and Polyhe
dron Mo deling In ARO MSI Computational Geometry Workshop
Mil b V J Milenkovic Robust p olygon mo delling Computer Aided Design
September
Mil a V J Milenkovic Enhanced Balanced Scaling Strategy Personal communication
April
Mil b V J Milenkovic A Symmetry Breaking Restriction Personal communication
March
Min H Minkowski Volumen und Ob er ache Mathematische Annalen
Mit J Mitchell Comment at MSI Workshop Octob er
MK Y Mo on and Y Kao Automatic Generation of Group Technology Families
during the Part Classi cation Pro cess The International Journal of Advanced
Manufacturing Technology
ML V J Milenkovic and Z Li A Compaction Algorithm for Nonconvex Polygons
and Its Application European Journal of Operations Research to appear
ML A Mangen and N Lasudry Search for the Intersection Polygon of any Two
Polygons Application to the Garment Industry Computer Graphics Forum
MN V J Milenkovic and L R Nackman Finding Compact Co ordinate represen
tations for p olygons and p olyhedra In Proceedings of the th Annual ACM
Symposium on Computational Geometry pages
MOS M McKenna J O Rourke and S Suri Finding the largest rectangle in an
orthogonal p olygon In Proceedings of the rd Al lerton Conference on Com
munication Control and Computing pages
Mou D M Mount The Densest Double Lattice Packing of a Convex Polygon Tech
nical Rep ort UMIACS TR CS TR University of Maryland January
MS a E Melissaratos and D Souvaine On Solving Geometric Optimization Prob
lems Using Shortest Paths In Proceedings of th Annual ACM Symposium on
Computational Geometry pages
MS b D Mount and R Silverman Packing and covering the plane with translates of
a convex p olygon Journal of Algorithms
NH M Nakajima and K Hayashi Automated Parts Layout Planning in Garment
Industry By Using Group Technology In Computers in the World of Textiles
pages Textile Institute Annual World Conference
OF J F Oliveira and J A Ferreira Algorithms for Nesting Problems In R V V
Vidal editor Lecture Notes in Economics and Mathematical Systems Applied
Simulated Annealing Springer Verlag
O R J O Rourke Finding Minimal Enclosing Boxes International Journal of Com
put Inform Sci June
O R Joseph O Rourke Art Gal lery Theorems and Algorithms Oxford University
Press
PS C Papadimitriou and K Steiglitz Combinatorial Optimization Algorithms and
Complexity Prentice Hall Englewoo d Cli s New Jersey
PS F Preparata and M Shamos Computational Geometry An Introduction
Springer Verlag New York
PS Y Prasad and S Somasundaram CASNS a heuristic algorithm for the nesting
of irregular shap ed sheet metal blanks Computer Aided Engineering Journal
pages April
QS W Qu and J Sanders A nesting algorithm for irregular parts and factors
a ecting trim losses International Journal of Production Research
Ree Colin Reeves editor Modern Techniques for Combinatorial Problems Halsted
Press John Wiley and Sons Inc New York
Rog C A Rogers The closest packing of two dimensional convex sets Acta Math
Ser J Serra Image Analysis and Mathematical Morphology volume Academic
Press New York
Ser J Serra editor Image Analysis and Mathematical Morphology volume The
oretical Advances Academic Press New York
Sha U Shankar Restricting Search Spaces for Translational Containment Using
Linear Programming Technical rep ort MIT Summer Research Institute
SP P E Sweeney and E R Paternoster Cutting and Packing Problems A Catego
rized Application Oriented Research Bibliography Journal of the Operational
Research Society
SRW S Schuierer G J E Rawlins and D Woo d A Generalization of Staircase
Visibility In Proceedings of the rd Canadian Conference on Computational
Geometry pages
TW M Tanaka and T Wachi Computerized Marker Making Journal of The Textile
Machinery Society of Japan
YLW C Yuzu L Lujun and Wangwei An exp ert system for automatic allo cation of
d irregular shap es In Expert Systems in Computer Aided Design Conference
IFIP pages