A Survey of Computational Geometry

Eds Handbooks in OR MS Vol

Chapter

Joseph S B Mitchel l

Applied Math SUNY Stony Brook NY Email jsbmamssunysbedu

Subhash Suri

Bel lcore South Street Morristown NJ Email suribellcorecom

Introduction

Computational geometry takes an algorithmic approach to the study of geometri

cal problems The principal motivation in this study is a quest for go o d algorithms

for solving geometrical problems Of course several practical and aesthetic factors

determine what one means by a go o d algorithm but the general trend has b een

to asso ciate go o dness with the asymptotic eciency of an algorithm in terms of

the time and space complexity Lately however the implementational ease and ro

bustness also are b ecoming increasingly imp ortant considerations in the algorithm

design

Although many geometrical problems and algorithms were known b efore compu

tational geometry evolved into a cohesive discipline only in the mid to late seventies

An imp ortant event in this development was the publication of the PhD thesis of

M Shamos in During its rst decade the eld of computational geom

etry grew enormously as its fundamental structures were applied to a vast variety

of problems in diverse disciplines and many new to ols and techniques were devel

op ed In the pro cess new insights were gained into interrelationships among some

of these fundamental structures which also led to a unication and consolidation

of several disparate sets of ideas In the last ve or so years the eld has matured

signicantly b oth in mathematical depth as well as in algorithmic ideas

Computational geometry has had strong interactions with other elds in math

ematics as well as in applied computer science A particularly fruitful interplay has

taken place b etween computational geometry and combinatorial geometry The lat

ter is a branch of mathematics concerned primarily with the counting of certain

geometric structures Examples include counting the maximum p ossible number of 1

2 J Mitchel l and S Suri

incidences b etween a set of lines and a set of p oints and counting the number of

lines that bisect a set of p oints Both elds seem to have b eneted from each other

combinatorial b ounds for certain structures have b een obtained by analyzing an

algorithm that enumerates them and conversely the analysis of algorithms often

dep ends crucially on the combinatorial b ound on some geometric ob jects

The eld of computational geometry has also b eneted from its interactions with

other disciplines within computer science such as VLSI database theory rob otics

computer vision computer graphics pattern recognition and learning theory These

areas oer a rich variety of problems that are inherently geometrical

Due to its interconnections with many applications areas the variety of problems

studied in computational geometry is truly enormous Our goal is this pap er is

quite mo dest we survey state of the art in some selected areas of computational

geometry with a strong bias towards problems with an optimization comp onent

In the pro cess we also hop e to acquaint the reader with some of the fundamental

techniques and structures in computational geometry

Our pap er has seven main sections The survey prop er b egins in Section while

Section introduces some foundational material In particular we briey describ e

ve key concepts and fundamental structures that p ermeate much of computational

geometry and therefore are somewhat essential to a prop er understanding of the

material in later sections The structures covered are convex hulls arrangements

geometric duality Voronoi diagram and p oint lo cation data structures The main

b o dy of our survey b egins with Section where we describ e four p opular geometric

graphs minimum and maximum spanning trees relative neighborho o d graphs and

Gabriel graphs Section is devoted to algorithms in path planning The topic of

path planning is a vast one with problems ranging from nding shortest paths in a

discrete graph to deciding the feasible motion of a complex rob ot in an environment

full of complex obstacles We briey mention most of the ma jor developments in

path planning research over the last two decades but to a large extent limit our

selves to issues related to shortest paths in a planar domain In Section we discuss

the matching and the traveling salesman type problems in computational geometry

Section describ es results on a variety of problems related to shap e analysis and

pattern recognition We close with some concluding remarks in Section In each

section we also p ose what in our opinion are the most imp ortant and interesting

op en problems on the topic There are altogether twenty op en problems in this

survey

Fundamental Structures

Convex Hul ls

The convex hul l of a nite set of p oints S is the smallest convex set containing

S In two dimensions for instance the convex hull is the smallest convex p olygon

containing all the p oints of S see Figure for an example In higher dimen sions the convex hull is a p olytop e Before we discuss the algorithms for computing

Ch A Survey of Computational Geometry 3

a convex hull we must address the question of representing it There are several

representations of a convex hull dep ending up on how many features of the cor

resp onding p olytop e are describ ed In the simplest representation we may only

store the vertices of the convex hull The other extreme of the representation is the

face lattice which stores all the faces of the convex hull as well as the incidence

relationships among the faces The intermediate forms of representation may store

faces of only certain dimensions such as the d dimensional faces also called

the facets The dierences among these representations b ecome signicant only in

bdc

dimensions d where the full lattice may have size n while the number

of vertices is obviously at most n Gr ubaumsb o ok is an excellent source for

information on p olytop es

Fig A planar convex hull

In two dimensions several O n log n time algorithms are known Almost ev

ery algorithmic paradigm in computational geometry has b een applied successfully

to the planar convex hull algorithm for instance divideandconquer incremen

tal construction planar sweep and randomization have all b een utilized to obtain

O n log n time algorithms for planar convex hulls see the textb o ok by Preparata

and Shamos The b est theoretical b ound for the planar convex hull problem

is achieved by an algorithm of Kirkpatrick and Seidel which runs in time

O n log h where h is the number of convex hull vertices

In three dimensions Preparata and Hong prop osed an O n log n time al

gorithm based on the divideandconquer paradigm Theirs was the only known

optimal algorithm in three dimensions until Clarkson and Shor developed

a randomized incremental algorithm that achieved an exp ected running time

O n log n A variant of ClarksonShor algorithm was prop osed by Guibas Knuth

and Sharir which admits a simpler implementation as well as an easier anal

ysis These randomized algorithms are quite practical and considerably easier to

implement than the divideandconquer algorithm Very recently Chazelle and

Matousek have settled a longstanding op en problem by announcing a deter

ministic O n log h time algorithm for the threedimensional convex hull problem

4 J Mitchel l and S Suri

In higher dimensions Chazelle recently prop osed an algorithm whose worst

case running time matches the worstcase b ound on the facet complexity of the

convex hull in any dimension d Chazelles algorithm builds on earlier ideas of

bdc

Kallay and Seidel and runs in worstcase time O n log n n The

algorithm in achieves the b est worstcase p erformance but it do es not dep end

on the actual size of the face lattice An algorithm by Seidel runs in time

prop ortional to the size of the face lattice In particular the algorithm in takes

O n F log n time for facet enumeration and O n L log n time for pro ducing

the face lattice where F is the number of facets in the convex hull and L is the

size of the face lattice Seidels algorithm uses a metho d called giftwrapping and

builds up on the earlier work by Chand and Kapur and Swart

There is a vast literature on convex hulls and the presentation ab ove has hardly

scratched the surface We have left out whole lines of investigation on the convex

hull problem such as the exp ected case analysis of algorithms and the average size

of the convex hull we refer the reader to Dwyer Devroye and Toussaint

and Golin and Sedgewick

Arrangements

Arrangements refer to space partitions induced by lines hyperplanes or other

algebraic varieties A nite set of lines L partitions the plane into convex regions

called cells which are b ounded by straight line edges and vertices The arrange

ment of lines A L refers to this collection of cells along with their incidence re

lations Similarly a set of hyperplanes or other surfaces such as spheres induce

arrangements in higher dimensions

An arrangement enco des the sign pattern for its generating set In other words

an arrangement is a manifestation of equivalence classes induced by a set of lines

or hyperplanes all p oints of a particular cell have the same sign vector with

resp ect to all the hyperplanes in the set This prop erty is a key to solving many

geometric problems in computational geometry It often turns out that solving a

geometric problem requires computing a particular cell or a family of cells in an

arrangement of hyperplanes We will say more ab out this in the section on Voronoi

diagrams

Combinatorial asp ects of arrangements have b een investigated for a long time

the arrangements in two and three dimensions were studied by Steiner The

interested reader may consult Gr ubaumsb o ok for a detailed discussion on

arrangements We will fo cus mainly on the computational asp ects of arrangements

An arrangement of n hyperplanes in dspace can b e computed in time O n by

an algorithm due to Edelsbrunner et al This b ound is asymptotically tight

since a simple arrangement where no more than d hyperplanes meet in a p oint

has n complexity

Although a single cell in an arrangement of hyperplanes can have complexity

bdc d

n not many cells can b e large afterall the combined complexity of n

cells is only O n There has b een a considerable amount of work on b ounding

Ch A Survey of Computational Geometry 5

the complexity of a family of cells in an arrangement For instance in two dimen

sions the total complexity of any m cells in an arrangement of n lines is roughly

O m n m n up to some logarithmic factors Extensions to higher

dimensions and related results can b e found in Arrangements of

b ounded ob jects such as line segments triangles and tetrahedra have also b een

studied see

Geometric Duality

Duality plays an imp ortant role in geometric algorithms and it often provides a

to ol for transforming an unfamiliar problem into a familiar setting In this section

we will give a brief description of two most frequently used transformations in

computational geometry

The rst transform D maps a p oint p to a hyperplane D p and vice versa

p p p p D p x p x p x p x p

d d d d d

Thus in the plane the p oint a b maps to the line y ax b and the line

y mx c maps to the p oint m c This transformation preserves incidence

and order a p oint p is incident to hyperplane h if and only if the dual p oint

D h is incident to dual hyperplane D p and a p oint p lies b elow hyperplane

h if and only if the dual p oint D h lies b elow the dual hyperplane D p

The second transform also called the lifting transform maps a p oints in R

d d

to a p oint in R It maps a p oint p p p p in R to the p oint p

If we treat R as the hyperplane x embedded p p p p p p

d d

in R then the lifting transform maps a p oint p R onto its vertical pro jection

The combination of the on the unit parab oloid U x x x x

lifting transform and the duality map D maps a p oint p R to the hyperplane

p p D p x p x p x p x p

d d d

The hyperplane D p is tangent to the parab oloid U at the p oint p It turns

out that this mapping is esp ecially useful for computing Voronoi diagrams the topic

of our next section

Voronoi Diagram and Delaunay Triangulation

The Voronoi diagram is p erhaps the most versatile data structure in all of com

putational geometry This diagram along with its graphtheoretical dual the De

launay Triangulation nds applications in problems ranging from asso ciative le

6 J Mitchel l and S Suri

searching and motion planning to crystallography and clustering In this section

we give a brief survey of some of the key ideas and results on these structures for

further details consult Edelsbrunners b o ok or a survey by Aurenhammer

A Voronoi diagram enco des the nearestneighbor information for a set of

sites We b egin by explaining the concept in two dimensions Given a set of

n sites or p oints S fs s s g in the twodimensional Euclidean plane

the Voronoi diagram of S partitions the plane into n convex p olygons V V V

such that any p oint in V is closer to s than to any other site

i i

V fx j dx s dx s for all j ig

i i j

where dx y is the Euclidean distance b etween the p oints x and y An interesting

fact ab out Voronoi diagrams in the plane is their linear complexity O n vertices

and edges

Fig The Voronoi diagram left and the Delaunay triangulation right of a set of p oints in

the plane

The Delaunay triangulation of S is the graphtheoretic dual of its Voronoi dia

gram two sites s and s are joined by an edge if the Voronoi p olygons V and V

i j i j share a common edge Under a nondegeneracy assumption that no four p oints of

Ch A Survey of Computational Geometry 7

S are cocircular the dual graph is always a triangulation of S Figure shows

an example of a Voronoi diagram and the corresp onding Delaunay triangulation

Just like convex hulls algorithms based on several dierent paradigms are known

for the construction of planar Voronoi diagrams and Delaunay triangulations such

as divideandconquer plane sweep and randomized incremental

metho ds They all run in O n log n time worstcase for deterministic

and exp ected for randomized

The concepts of Voronoi diagram and Delaunay triangulation extend naturally to

higher dimensions as well as to other metrics In d dimensions the Voronoi diagram

of a set of p oints S is a tessellation of E by convex p olyhedra The p olyhedral cell

V consists of all those p oints that are closer to s than to any other site in S The

i i

Delaunay triangulation of S is the geometric dual of the Voronoi diagram there

is a k face for every d k face of the Voronoi diagram In particular there is

an edge b etween s and s if the Voronoi p olyhedra V and V share a common

i j i j

d dimensional face An equivalent way of dening the Delaunay triangulation

is via the emptysphere test a d tuple s s s is a simplex trian

gle of the Delaunay triangulation of S if and only if the sphere determined by

s s s do es not contain any other p oint of S The Voronoi diagram of n

dd e

p oints in d dimensions d can have sup erlinear size n

It turns out that Voronoi diagrams and Delaunay triangulations are intimately

related to convex hulls and arrangements via duality transforms This relationship

was rst discovered by Brown who showed using an inversion map that Voronoi

diagram of a set S R corresp onds to the convex hull of a transformed set in

R Later Edelsbrunner and Seidel extended and simplied this idea using

the parab oloid transforms mentioned in the previous section We now sketch their

idea

Let S s s s g b e a set of n p oints in R We map S to a set of

hyperplanes D S in R using the combination of lifting and duality maps

mentioned in Section In particular a p oint s a a a maps to the

hyperplane D s whose equation is

x a x a x a x a a a

d d d

Let P b e the p olyhedron dened by the intersection of upp er halfspaces b ounded

by these hyperplanes Then the vertical pro jection of P onto the hyperplane x

gives precisely the Voronoi diagram of S in R

A similar and p erhaps easier to visualize relationship exists b etween convex

hulls and Delaunay triangulations using only the lifting transform We map the

p oints S fs s s g to their lifted counterpart S fs s s g

Now compute the convex hull of S The triangles in the Delaunay triangulation

of S corresp ond precisely to the facets of CH S with downward normal

Thus b oth the Voronoi diagram and the Delaunay triangulation of a set of p oints

d d

in R may b e computed using a convex hull algorithm in R This relationship

also explains why the worstcase size of b oth a Voronoi diagram in R and a convex

d bdc

hull in R is n

8 J Mitchel l and S Suri

Point Location

Many problems in computational geometry often require solving a socalled p oint

lo cation problem Given a partition of space into p olyhedral cells and a query p oint

q the problem is to identify the cell containing q For instance if Voronoi diagrams

are used for answering nearest neighbor queries one needs to lo cate the Voronoi

p olyhedron containing the query p oint Typically a large number of queries are

asked with resp ect to the same cell complex thus it makes sense to prepro cess the

cell complex in order to sp eed up queries

The problem has b een studied intensely in two dimensions where several optimal

algorithms and data structures are now known These algorithms can prepro cess

a planar map on n vertices into a linear space data structure and answer a p oint

lo cation query in O log n time see

In higher dimensions the p oint lo cation problem is less wellunderstoo d and no

algorithm simultaneously achieves optimal b ounds for prepro cessing time storage

space and query time We give a brief summary of results and give p ointers to rele

vant literature We denote the p erformance of an algorithm by the triple fP S Qg

which refer to prepro cessing time storage space and query time

Preparata and Tamassia give an fO n log n O n log n O log ng algo

rithm for p oint lo cation in a convex cell complex of n facets in three dimensions

d d

Using randomization Clarkson gives an fO n O n O log ng algo

rithm for p oint lo cation in an arrangement of n hyperplanes in d dimensions the

space and query b ounds are worstcase but the prepro cessing time is exp ected

Chazelle and Friedman were later able to make Clarksons algorithm deter

ministic alb eit at an increased prepro cessing cost resulting in an algorithm with

dd d

resource b ounds fO n O n O log ng

Geometric Graphs

Minimum Spanning Trees

The minimum spanning tree MST problem is one of the b estknown problems of

combinatorial optimization and it has received a considerable attention in compu

tational geometry as well Given a graph G V E with nonnegative realvalued

weights on edges a minimum spanning tree of G is an acyclic subgraph of G that

spans all the no des in V and has minimum total edge weight An MST has obvious

applications in the design of computer and communication networks transp ortation

systems and other kinds of networks But applications of the minimum spanning

tree extend far b eyond network design problems They are used in problems as

diverse as network reliability computer vision automatic sp eech recognition clus

tering and classication matching and traveling salesman problems and surface

homogeneity tests

Ecient algorithms for computing an MST have b een known for a long time

a survey by Graham and Hell traces the history of MST and cites algo

Ch A Survey of Computational Geometry 9

rithms dating back to the b eginning of the century Although the algorithms of

Kruskal and Prim are among the b est known an algorithm by Boruvka

preceded them by almost thirty years Using suitable data structures these

algorithms can b e implemented in O jE j log jV j or O jV j time In the last two

decades several new implementations and variants of these basic algorithms have

b een prop osed and the fastest ones run in almost linear time in the size of the

graph

The interest of computational geometry researchers in MST stems from the ob

servation that in many applications the underlying graph is Euclidean we want to

compute a minimum spanning tree for a set of n p oints in R for d The set

of n p oints in this case completely sp ecies the graph without an explicit enumer

ation of the edges Since the edgeweights in this geometric graph are not entirely

arbitrary a natural question is if one can compute an MST in asymptotically less

than n steps that is without insp ecting every edge

Surprisingly it turns out that for a set of n p oints in the plane an MST can

b e computed in O n log n time A key observation is the following lemma which

states that the edges of an MST are contained in the Delaunay triangulation graph

we omit the pro of but an interested reader may consult the b o ok of Preparata

Shamos

Lemma Let S be a set of n points in the plane and let DT S denote the

Delaunay triangulation of S Then MST S DT S

We recall from Section that the Delaunay triangulation in two dimensions

is a planar graph Thus by running an ecient graph MST algorithm on DT S

we can nd a minimum spanning tree of S in O n log n time In fact given the

Delaunay triangulation a minimum spanning tree of S can b e extracted in linear

time by using an algorithm of Cheriton and Tarjan which computes an MST

in linear time for planar graphs

Lemma holds in any dimension however it no longer serves a useful purp ose

for computing minimum spanning trees in higher dimensions since the Delaunay

graph can have size n in dimensions d Nevertheless the under

lying geometry can b e exploited to compute an MST in sub quadratic worstcase

time Yao has prop osed a general metho d for computing geometric minimum

d d

where is a dimensiondep endent log n spanning trees in time O n

constant Yaos algorithm is based on the following idea if we partition the space

around a p oint p into p olyhedral cones of suciently small angular diameter then

there is at most one MST edge incident to p p er cone and this edge joins p to its

nearest neighbor in that cone In order to nd these nearest neighbors eciently

Yao utilizes a data structure that after a p olynomialtime prepro cessing can de

termine a nearest neighbor in logarithmic time In the original pap er of Yao

the constant had value of thus making his algorithm only slightly sub

quadratic however the interesting conclusion is that an MST can b e computed

without checking all the edges

The exp onent in the general algorithm of Yao has steadily improved as b etter data structures have b een developed for solving the nearestneighbor problem Re

10 J Mitchel l and S Suri

cently Agarwal et al have shown also that computationally the twin problems of

computing a minimum spanning tree and computing a bichromatic nearest neigh

b or are roughly equivalent The constant in the running time their algorithm

is roughly In three dimensions the algorithm of computes an MST

dd e

in O n log n time An alternative and somewhat simpler alb eit randomized

algorithm of the same complexity is given by Agarwal Matousekand Suri

There also has b een work on computing an approximation of the MST

Vaidya constructs in O n log n time a spanning tree with length at most

times the length of an MST If the n p oints are indep endently and uniformly

distributed in the unit dcub e then the exp ected time complexity of Vaidyas algo

rithm is O ncn n where is the inverse Ackermann function

The b est lower b ound known for the MST problem is n log n in any xed

dimension d The lower b ound holds in the algebraic tree mo del of computation

for any input consisting of an unordered set of n p oints on log n time algorithms

are p ossible for sp ecial congurations of p oints such as the vertices of a convex

p olygon if the p oints are given in order along the b oundary of the p olygon It is

an outstanding op en problem in computational geometry to settle the asymptotic

time complexity of computing a geometric minimum spanning tree in dspace

Op en Problem Given a set S of n unordered points in E compute its Eu

clidean minimum spanning tree in O c n log n time where c is a constant de

d d

pending only on the dimension d Alternatively prove a lower bound that is better

than n log n

There is an obvious connection b etween MST and nearest neighbors the MST

neighbors of a p oints s include a nearest neighbor of s Thus the al lnearest

neighbors problems which asks for a nearest neighbor for each of the p oints of

S is no harder than computing the MST S A few years ago Vaidya gave

an O c n log n time algorithm for the allnearestneighbors problems for any xed

dimension the constant c in Vaidyas algorithm is of the order of

Unfortunately no reduction in the converse direction given all nearest neighbors

compute MST is known However the result of Agarwal et al p oints out an

equivalence b etween the MST and the bichromatic closest pair problem The bi

chromatic closest pair problem is dened for two ddimensional sets of p oints R

and B and it asks for a pair r R and b B that minimizes the distance over

all such pairs It is shown in that the asymptotic time complexities of the two

problems are the same if they have the form n for any otherwise

they are within a p olylogarithmic factor of each other This leads to the following

op en problem

Op en Problem Given two unordered sets of points B R E compute a bi

chromatic closest pair of B and R in time O c n log n where n jB j jRj and

c is a constant depending only on the dimension d Alternatively prove a lower

bound better than n log n

Ch A Survey of Computational Geometry 11

Maximum Spanning Trees

A maximum spanning tree is the other extreme of the minimum spanning tree

it maximizes the total edge weight In graphs a maximum spanning tree can b e

computed using any minimum spanning tree algorithm by simply negating all the

edge weights But what ab out a geometric maximum spanning tree Is it p ossible

to compute the maximum spanning tree of a set of p oints in less then quadratic

time

As a rst attempt we could try to generalize Lemma Instead of a Delaunay

triangulation we would consider the socalled furthestpoint Delaunay triangula

tion which is the graphtheoretic dual of the furthestpoint Voronoi diagram In

a furthestp oint Voronoi diagram of a set of p oints S the region asso ciated with a

site s S consists of all the p oints x that satisfy dx s dx s for all s s

i i j i j

see PreparataShamos for details Unfortunately the maximum spanning tree

edges do not necessarily lie in the furthestp oint Delaunay triangulation One of the

reasons why this relationship do es not hold is trivial the furthestp oint Delaunay

triangulation only triangulates the convex hull vertices of S the interior p oints of

S have an empty furthestp oint Voronoi p olygon The trouble in fact go es deep er

even if all p oints of S were to lie on its convex hull the maximum spanning tree

do es not always lie in the Delaunay triangulation Consider the example in Fig

ure which is due to Bhattacharya and Toussaint In this gure AC D

is an equilateral triangle and B lies on the line joining D with the center O of the

circle AC D such that dD O dD B dD A It is easy to check that the

furthestp oint Delaunay triangulation consists of triangles AB C and AC D and

do es not include the diagonal BD On the other hand the maximum spanning tree

of fA B C D g necessarily contains the edge BD the other two edges can b e any

two of the three edges of the equilateral triangle AC D

An optimal O n log n time algorithm for computing a maximum spanning tree of

n p oints in the plane was prop osed a few years ago by Monma Paterson Suri and

Yao Their algorithm starts by computing the furthest neighbor graph connect

each p oint to its furthest neighbor This results in a forest whose comp onents are

called clusters in Monma et al show that these clusters can b e cyclically

ordered around their convex hull and that the nal tree can b e computed by

merging adjacent clusters where merging two clusters means adding a longest edge

b etween them The algorithm in runs in O n time if all the p oints lie on their

convex hull

Sub quadratic algorithms for higher dimensional maximum spanning trees were

obtained a little later by Agarwal Matousekand Suri who prop osed random

ized algorithms of exp ected time complexity O n log n for dimension d

Agarwal Ma for dimension d where is roughly and O n

dde

tousek Suri also present a simple approximation algorithm that computes in

O n log n time a spanning tree with total length at least times the optimal

12 J Mitchel l and S Suri

B . D O

Fig MXST is not a subset of farthestp oint Voronoi diagram

Applications of Minimum and Maximum Spanning Trees

We said earlier that minimum spanning trees have several applications some

are obvious such as network design problems and some are less obvious such as

pattern recognition traveling salesman and clustering problems In this section we

mention some constrained clustering problems that can b e solved eciently using

minimum and maximum spanning trees

Given a set of p oints S in the plane dene a k partition of S as a decomp osition

of S into k disjoint subsets fC C C g We want to nd a k partition that

maximizes the minimum intercluster distance min minfds t j s C t C g

ij i j

Asano et al show that an optimal k partition is found by deleting the k

longest edges from the minimum spanning tree of S

Next consider the problem of partitioning a p oint set S into two clusters sub ject

to the condition that the larger of the two diameters is minimized recall that the

diameter of a nite set of p oints is that maximum distance b etween any two p oints

in the set An O n log n time solution of this problem was prop osed by Asano

et al and also by Monma and Suri based on the maximum spanning

tree The metho d of Monma and Suri is particularly simple compute a maximum

spanning tree of S and color its no des p oints The partition induced by the

coloring is an optimal minimum diameter partition

A related bipartition problem is to minimize the sum of measures of the two

subsets Monma and Suri gave an O n time algorithm for computing a bi

partition of n p oints minimizing the sum of diameters This result was subsequently

improved to O n log n time by Hershberger An interesting problem in this

class is to nd a subquadratic algorithm for the twodisk covering of a p oint set with a minimum radius The relevant results on this problem app ear in Hershberger

Ch A Survey of Computational Geometry 13

and Suri and Agarwal and Sharir the former gives an O n log n time

algorithm to check the feasibility of a covering by two disks of given radii and the

latter gives an O n log n time algorithm for nding the minimum radius It is

an op en problem whether a minimum radius twodisk covering of n p oints can b e

computed in subquadratic time

Op en Problem Given n points in the plane give an on time algorithm for

computing the minimum radius r such that al l the points can be covered with two

disks of radius r also nd the corresponding disks

Gabriel and Relative Neighborhood Graphs

Nearest neighbor relationships play an imp ortant role in pattern recognition prob

lems One of the simplest graphs enco ding these relationships is the nearest neighbor

graph which has a directed edge from p oint a to p oint b if b is a nearest neighbor

of a The minimum spanning tree is a next step which rep eatedly applies the near

est neighbor rule until we obtain a connected graph Building on this theme several

other classes of graphs have b een introduced We discuss two such graphs in this

section the Gabriel graph and the relative neighborho o d graph The Gabriel graph

was introduced by Gabriel and Sokal in the context of geographical variation

analysis while the relative neighborho o d graph was introduced by Toussaint

in a graphtheoretical context Matula and Sokal studied several prop erties of

the Gabriel graphs with applications to zo ology and geography A recent survey

pap er by Jaromczyk and Toussaint is a go o d source for additional information

on these graphs Let us rst describ e the Gabriel graph

Let S fs s s g b e a set of p oints in the plane and dene the circle of

inuence of a pair s s S as

i j

C s s fx R j d x s d x s d s s g

i j i j i j

We observe that C s s is the circle with diameter s s The Gabriel graph

i j i j

GGS has the set of p oints S as its vertex set and two vertices s and s have

i j

an edge b etween them if the circle C s s do es not include any other p oint of S

i j

In other words s s is an edge of GGS if and only if d s s d s s

i j i k j k

d s s for all s This denition immediately implies that the Gabriel graph

i j k

of S is a subgraph of the Delaunay triangulation DT S recall the empty circle

denition of Delaunay triangulations cf Sec

Matula and Sokal give an alternative denition of the Gabriel graph an

edge s s of DT S is in GGS if and only if s s intersects its dual edge

i j i j

in the Voronoi diagram of S This latter characterization leads immediately to

an O n log n time algorithm for computing the Gabriel graph rst compute the

Delaunay triangulation DT S and then delete all those edges that do not intersect

their dual edges in the corresp onding Voronoi diagram

In dimensions d the complexity of the Gabriel graph dep ends on whether

or not many p oints are cospherical Note that this is not the case for Delaunay

14 J Mitchel l and S Suri

triangulation If no more than a constant number of p oints lie on a common d

sphere then GG has only a linear number of edges Computing this graph in less

than quadratic time is still quite nontrivial Slightly subquadratic algorithms are

presented in Without the nondegeneracy assumption the Gabriel graph can

have n edges even in three dimensions The following example gives a lower

b ound construction for d Take two orthogonal interlocking circles of radius

each passing through the center of the other In particular let the rst circle lie in

the xy plane with as the center while the second circle lies in the y z plane

with as the center Place n p oints on the rst circle very close to the p oint

and n p oints on the second circle close to the p oint Then it is

easy to see that the Gabriel graph of these n p oints contains the complete bipartite

graph b etween the two sets of n p oints

Op en Problem Given n points in E such that only O d points lie on a com

mon sphere give an O c n log n time algorithm to construct their Gabriel graph

where c is dimensiondependent constant Alternatively prove a lower bound better

than n log n

The basic construct in the denition of a relative neighborho o d graph is the lune

of inuence Given two p oints s s S their lune of inuence Ls s is dened

i j i j

as follows

Ls s fx R j maxfdx s dx s g ds s g

i j i j i j

Thus the lune Ls s is the common intersection of two disks of radius ds s

i j i j

centered on s and s The relative neighborhood graph RN GS has an edge b e

i j

tween s and s if and only if the lune Ls s do es not contain any other p oint

i j i j

of S Again it easily follows that RN GS DT S in fact the relative neigh

b orho o d graph is also a subgraph of the Gabriel graph since the circle of inuence

is a subset of the lune of inuence Thus we have the following ordered relations

among the four graphs we have discussed in this section

MST RN G GG D T

Characterization of the DT edges not in RN G however is not so easy as it

was for the Gabriel graph In two dimensions Sup owit presents an O n log n

time algorithm for extracting the RN G from the Delaunay triangulation If the

p oints form the vertices of a convex p olygon then the minimum spanning tree

relative neighbor graph Gabriel graph and Delaunay triangulation can each b e

computed in linear time The b ound for MST GG and DT is implied by a linear

time algorithm for computing the Voronoi diagram of a convex p olygon and the

result on RN G is due to Sup owit

In dimensions d the size of the relative neighborho o d graphs dep ends on

whether or not the p oints are cospherical If only a constant number of p oints lie on

a common d sphere then the RN G has O n edges but without this restric

tion it is easy to construct examples where RN G has n edges in any dimension

Ch A Survey of Computational Geometry 15

d In R the b est upp er b ound on the size of the relative neighborho o d graph

currently known is O n

Op en Problem Given a set S of n points in E such that only a constant

number of points lie on a common sphere show that the relative neighborhood graph

of S has only O n edges Alternatively prove a superlinear lower bound on the

size of the relative neighborhood graph

The size of the relative neighborho o d graph is related to the number of bi

chromatic closest pairs

Op en Problem Given two unordered sets of points B R E what is the

maximum number of pairs b r such that r R is a closest neighbor of b B

Path Planning

Introduction

The shortest path problem is a familiar problem in algorithmic graph theory

Given a graph G V E whose edges have nonnegative realvalued weights as

so ciated with them a shortest path b etween two no des s and t is a path in G from

s to t having the minimum p ossible total edge weight The shortest path problem is

to nd such a path Generalizations of this basic shortest path problem include the

single source and the al lpairs shortest paths problems the former asks for shortest

paths to all the vertices of G from a sp ecied source vertex s and the latter asks for

shortest paths b etween all pairs of vertices The b estknown algorithm for comput

ing shortest paths is due to Dijkstra If prop erly implemented his algorithm

can nd a shortest path b etween two vertices in time O minn m log n here n

and m denote the number of vertices and edges of G A considerable amount of re

search has b een invested in improving this time complexity for sparse graphs that

is graphs with m n Only a few years ago Fredman and Tarjan succeeded

in devising an implementation of Dijkstras algorithm using their Fib onnaci heap

data structure that achieved a worstcase running time of O m n log n this time

b ound is optimal in a comparisonbased mo del of computation

The shortest path problem acquires a new richness when transp orted to a geo

metric domain Unlike a graph an instance of the geometric shortest path problem

is typically sp ecied through the description of some geometric ob jects that im

plicitly enco de the graph This raises the following rather interesting question is

it p ossible to compute a shortest path without explicitly constructing the entire

graph There are some intriguing p ossibilities asso ciated with this question For

instance a set of geometric ob jects can enco de some very large sup erp olynomial

or even exp onential size graphs implying that an ecient shortest path algorithm

must necessarily avoid building the entire graph Even if the graph is p olynomial

size considerable eciency gain is p ossible if the shortest path problem can b e

solved by constructing only a small relevant subset of the edges We will address

16 J Mitchel l and S Suri

these issues in more detail later but for now let us just say that there is a diverse

variety of shortest path problems dep ending up on the type of geometric ob jects

considered the metric used and the dimension of the underlying geometric space

We start with a discussion of some common basic concepts

Basic Concepts

The most commonly studied shortest path problems in computational geometry

typically involve a set of p olyhedral ob jects called obstacles in Euclidean dspace

d and the goal is to nd an obstacleavoiding path of minimum length b etween

two p oints Much of our discussion will b e limited to shortest paths in the plane

d A connected subset of the plane whose b oundary consists of a union

of a nite number of straight line segments will b e called a polygonal domain

The b oundary segments are called edges their endp oints are called vertices A

p olygonal domain P is called a simple polygon if it is simplyconnected that is it

is homeomorphic to a disk A multiplyconnected p olygonal domain P is also called

a polygon with holes

Triangulation

A triangulation of a p olygonal domain P is a decomp osition of P into triangles

with disjoint interiors with each triangle having its corners among the vertices

of P If we allow triangles whose corners are not among the vertices of P the

triangulation is called a Steiner triangulation we do not use Steiner triangulations

in this section It is a wellknown fact that a p olygonal domain can always b e

triangulated without using Steiner p oints Since a triangulation is a planar graph

the number of triangles is linearly related to the number of vertices of P

A p olygonal domain with n vertices can b e triangulated in O n log n time

This time complexity is worstcase optimal in the algebraic tree mo del of compu

tation The lower b ound however do es not apply if P is a simple p olygon raising

the p ossibility than a b etter algorithm might b e p ossible for triangulating a simple

p olygon Indeed the problem of triangulating a simple p olygon b ecame one of the

most notorious problems in computational geometry in the eighties Despite the

discovery of numerous algorithms the O n log n time b ound remained unbeaten

in the worstcase p erformance Then in a breakthrough result by Tarjan and

van Wyk pro duced an O n log log n time triangulation algorithm Finally

Chazelle recently managed to devise a lineartime algorithm for triangulating

a simple p olygon thus settling the theoretical complexity of the problem

For a p olygon with holes it is p ossible to p erform a triangulation in running time

dep endent on the number of holes or the number of reex ie nonconvex vertices

In particular a p olygonal domain P can b e triangulated in O n log r time where

r is the number of reex vertices of P or in time O n h log n where h is the

number of holes in P and is an arbitrarily small p ositive constant

Ch A Survey of Computational Geometry 17

Visibility

Visibility is a key concept in geometric shortest paths We say that p oints s and

t are mutually visible if the line segment joining them lies within the p olygonal

domain P The relevance to shortest path planning is clear If p oints s and t are

visible to one another then the shortest obstacleavoiding path b etween them is

simply the line segment joining them

The visibility polygon V s with resp ect to a p oint s P is dened to b e the set

of p oints that are visible to s A visibility p olygon can b e found in time O n log n

by applying the sweepline paradigm of computational geometry We simulate the

sweeping of a ray r angularly ab out s keeping track of the ordered crossing list

of edges of P intersecting r When the ray r encounters a vertex of P we

insert andor delete an edge from the crossing list and we make any necessary

up dates to the visibility prole the cost p er up date is O log n We can always

know which vertex is encountered next by the sweeping ray if we sort the vertices

of P angularly ab out s in O n log n time If P has h holes then a recent result

of Heernan and Mitchell shows that one can compute a visibility p olygon in

optimal time O n h log h

The visibility graph VG of P is dened to b e the graph whose no des are the

set of vertices of P and whose edges join pairs of vertices that are mutually visible

Refer to Figure We let E denote the number of edges in VG note that

E for an nvertex domain P Visibility graphs were rst introduced in the

work of Nilsson who used them for computing shortest paths for a mobile

rob ot

Fig A visibility graph

The most naive algorithm for computing the VG runs in time O n checking

each pair of vertices u v for visibility by testing against all edges of P A sub stantially improved algorithm is p ossible based on the idea of computing visibility

18 J Mitchel l and S Suri

p olygon of each vertex The visibility graph edges incident to a vertex v can b e

found in O n log n time by rst constructing the visibility p olygon of v and hence

the entire visibility graph can b e computed in O n log n using only O n working

space

The state of the art in VG construction remained at the O n log n level until

when Welzl and indep endently Asano et al obtained algorithms

whose worstcase running times were O n These new algorithms rely on the

trick of mapping the vertices of P to their dual lines building the arrangement

of these lines in time O n and then using the information present in the

arrangement to read o the sorted order of vertices ab out each vertex v in total

time O n Thus the O n angular sorts are not indep endent of each other as

they can b e done collectively in total time O n Once the angular order is known

for vertices ab out every other vertex a further trick is necessary to pro duce the

VG without the logarithmic overhead p er pair for example Welzl uses a

top ological sort available from the arrangement to guide the construction of the

visibility proles ab out every vertex Edelsbrunner and Guibas have shown how

to use a metho d of top ological sweep to compute the VG in time O n using only

O n working storage ie avoiding the need to keep the entire line arrangement

in memory during VG construction

In the worst case we know that it takes quadratic time O n to compute a

visibility graph since visibility graphs exist with this size In some cases however

the visibility graph is very sparse linear in size Thus ideally we would like an

algorithm whose running time is outputsensitive taking time prop ortional to the

size E of the output

Ghosh and Mount have developed such an outputsensitive algorithm

achieving a time b ound of O n log n E with a working storage requirement of

O E Their algorithm b egins with a triangulation of P and constructs VG edges

by a careful analysis of the prop erties of funnel sequences Indep endently Kap o or

and Maheshwari obtained a similar b ound and also show how one can compute

the subgraph of VG relevant for shortest path planning in time O n log n E and

space O E where E is the size of the resulting subgraph In other words

SP SP

only those edges of the VG that app ear along some nontrivial shortest path are

actually discovered and output

Overmars and Welzl give two very simple easily implementable algorithms

for computing the visibility graph that use only O n space The rst algorithm

runs in time O n and is based on rotation trees the second is outputsensitive

requiring time O E log n See also Alt and Welzl The main op en problem

in visibility graph construction is summarized b elow

Op en Problem Given a polygonal domain with n vertices and h holes compute

the visibility graph in time O h log h E where E is the number of edges of

VG VG

the resulting graph Ideally do this computation using only O n working storage

Mitchell and Welzl have developed an online algorithm to construct a VG

by showing that one can up date a VG when a new obstacle is inserted in time

Ch A Survey of Computational Geometry 19

O n k where k is the number of VG edges that must b e removed when the

new obstacle is inserted Note that k may b e as large as n Vegter

shows that a VG can b e maintained under b oth insertions and deletions in time

O log n K log n p er up date where K is the size of the change in the VG We

are left with an interesting op en question

Op en Problem Devise a dynamic algorithm for maintaining a visibility graph

in O log n K time per insertion or deletion of an obstacle where K denotes the

number of changes in the visibility graph

Shortest ObstacleAvoiding Paths

The most basic kind of geometric shortest path problem is that of nding a

shortest path from s to t for a p oint rob ot that is conned to the interior of a

p olygonal domain P We assume that P has h holes which can b e thought of as

the obstacles and a total of n vertices In this subsection we measure path length

according to the usual Euclidean metric in the following subsections we discuss

variants on the ob jective function

Paths in a Simple Polygon

Assume that there are no holes in P ie h Then there is a unique

homotopy class of any path from s to t and the shortest path from s to t will b e

the unique taut string path If we triangulate p olygon P then there is a unique

path in the triangulation dual graph which is a tree in this case from the triangle

containing s to the triangle containing t This gives us a sleeve within P that is

known to contain the shortest st path Chazelle and Lee and Preparata

show that in time linear in the number of triangles dening the sleeve one can

pull taut the sleeve pro ducing the unique shortest st path

The algorithm pro ceeds incrementally considering the eect of adding the tri

angles in order along the sleeve At a general step of the algorithm when we are

ab out to add triangle abc we know the shortest path from s to a vertex r of the

sleeve and the concave shortest subpaths from r to a and from r to b which

dene a region called the funnel with base ab and ap ex r Refer to Figure

In order to add abc we must split the funnel according to the tautstring path

from r to c which will in general include a segment uc joining c to some vertex

of tangency u along one of the concave chains of the funnel We need to keep only

one of the two funnels based on ac and ab since only one can lead through the

sleeve to t which allows us to charge o the work of searching for u to vertices that

can b e discarded

Since a simple p olygon can b e triangulated in linear time Chazelle the

result of establishes that shortest paths in a simple p olygon can b e found

in O n time which is worstcase optimal This result has b een generalized in several directions

20 J Mitchel l and S Suri

u c r

Fig Splitting a funnel

Guibas et al show that one can construct the shortest path tree and

its extension into a shortest path map ro oted at a p oint s in O n time af

ter which the length of a shortest path to any query p oint t can b e rep orted in

O log n time and the shortest path can b e output in time prop ortional to its

size Their result relies on using nger search trees to do funnel splitting which

now must b e done without discarding either of the two new funnels Hershberger

and Sno eyink have given a considerably simpler algorithm to compute shortest

path trees without any sp ecial data structures

Guibas and Hershberger show that a simple p olygon can b e prepro

cessed in time O n into a data structure of size O n such that one can answer

shortest path length queries b etween a pair of p oints in O log n time In fact

within the O log n query time one can construct an implicit representation of the

shortest path so that one can output the path explicitly in time prop ortional to

its length number of vertices

ElGindy and Go o drich give a parallel algorithm to compute short

est paths in time O log n using O n pro cessors in the CREW PRAM mo del

Go o drich Shauck and Guha show how with O n log n pro cessors and

O log n time one can compute a data structure that supp orts O log n sequential

time shortest path queries b etween pairs of p oints in a simple p olygon They also

give an O log n time algorithm using O n pro cessors to compute a shortest path

tree Hershberger builds on the results of and gives an algorithm for short

est path trees requiring only O log n time and O n log n pro cessors CREW he

also obtains optimal parallel algorithms for related visibility and geo desic problems

Many other problems have b een studied with resp ect to shortest path

geodesic distances within a simple p olygon Aronov shows how to compute

in time O n log n the Voronoi diagram of a set of p oint sites in a simple p oly

gon if the metric is the geo desic distance The geodesic diameter is the length of

Ch A Survey of Computational Geometry 21

the longest shortest path b etween a pair of vertices it can b e computed in time

O n log n The geodesic center is the p oint within P that minimizes

the maximum of the shortest path lengths to any other p oint in P Pollack Sharir

and Rote give an O n log n algorithm Suri studies problems of com

puting geo desic furthest neighbors The furthestsite Voronoi diagram for geo desic

distance is computed in time O n log n by Aronov Fortune and Wilfong

All of the ab ove lineartime algorithms rely on a triangulation of a simple p olygon

Its an interesting op en problem whether a shortest path inside a simple p olygon

can b e computed optimally without a triangulation

Op en Problem Given a simple polygon with n vertices devise an O n time

algorithm for computing the shortest path between two points without triangulating

the polygon

Paths in General Polygonal Spaces

In the general case in which P contains holes obstacles shortest paths can b e

computed using the visibility graph as justied in the following straightforward

lemma proved in

Lemma Any shortest path from s P to t P in a polygonal domain P

must lie on the visibility graph VG of P where VG includes s and t in addition

to vertices of P as nodes

This lemma implies that after constructing the VG we can search for shortest

paths in time O E n log n using Dijkstras algorithm with appropriate data

structures eg Fib onacci heaps or relaxed heaps The result of Dijkstras

algorithm is a shortest path tree SPTs In practice it may b e faster to apply

the A heuristic search algorithm eg see Pearl using the straightline

Euclidean distance as heuristic function h which is a lower b ound so it implies

an admissible algorithm

Since VG can b e computed in time O E n log n we conclude

that Euclidean shortest paths among obstacles in the plane can b e computed in

time O E n log n O n

Sp ecial cases of these results are p ossible when the obstacles are convex in which

case the quadratic term can b e written in terms of h the number of obstacles

rather than n the number of vertices see Mitchell and Rohnert

Another sp ecial case of relevance to VLSI routing problems see is

to compute shortest paths among obstacles of a given homotopy type Hershberger

and Sno eyink generalize the shortest path algorithm for simple p olygons to

show that one can compute a shortest path among obstacles of a particular thread

ing in time prop ortional to the size of the description of the homotopy type

Shortest Path Maps A shortest path map SPMs is an implicit represen

tation of the set of shortest paths from s to al l p oints of P The utility of SPMs is

22 J Mitchel l and S Suri

that it is a planar sub division of size O n such that once we p erform an O log n

time p oint lo cation query for t the map tells us the length of a shortest st path

and allows a path to b e rep orted in time prop ortional to its size number of b ends

The general concept of shortest path maps applies to all metrics here we mention

some facts relevant to Euclidean shortest paths among p olygonal obstacles in the

plane

If our nal goal is to compute a shortest path map SPMs then we can obtain

it in O n log n time given the shortest path tree obtained by searching VG with

Dijkstras algorithm

An alternative approach is to build the linearsize SPMs directly and avoid

altogether the construction of the quadraticsize VG Lee and Preparata

use this approach to construct a shortest path map in optimal O n log n time

for the case of obstacles that are parallel line segments implying monotonicity of

shortest paths with resp ect to the direction p erp endicular to the segments This

approach also leads Reif and Storer to an O hn n log n time O n space

algorithm for general p olygonal obstacles based on adding the obstacles oneata

time up dating the SPMs at each step using a shortest path algorithm for simple

p olygons without holes

Mitchell shows how the Euclidean SPMs can b e built in O k n log n time

O n space where k is a quantity called the illumination depth and is b ounded

ab ove by the number of obstacles touched by a shortest path This algorithm is

based on a general technique for solving geometric shortest path problems called

the continuous Dijkstra paradigm see The main

idea is to simulate the wavefront propagation that o ccurs when running Dijkstras

algorithm in the continuum The continuous Dijkstra paradigm has led to ecient

algorithms for a variety of shortest path problems as we mention later including

shortest paths on p olyhedral surfaces shortest paths through weighted re

gions maximum ows in the continuum and rectilinear paths among

obstacles in the plane

A ma jor op en question in planar computational geometry is to devise a

sub quadratictime algorithm for Euclidean shortest obstacleavoiding paths The

only known lower b ound is the trivial n h log h one

Op en Problem Given a polygonal domain with n vertices compute a Eu

clidean shortest path between two points in O n log n time

Other Notions of Short

Instead of measuring the length of a path as its Euclidean length several other

ob jective functions are p ossible as we describ e b elow

Rectilinear Metric

If we measure path length by the L or L metric d p q jp q j jp

x x y

q j or d p q maxfjp q j jp q jg or require that paths b e rectilinear

y x x y y

Ch A Survey of Computational Geometry 23

with edges parallel to the co ordinate axes then sub quadratictime algorithms for

shortest paths in the plane are known

For the general case of a p olygonal domain P Mitchell shows how to

apply the continuous Dijkstra paradigm to build the L or L shortest path map

in time O n log n and space O n

Clarkson Kap o or and Vaidya develop a metho d based on principles similar

the use of visibility graphs in searching for L optimal paths They construct a

sparse graph with O n log n no des and edges that is path preserving meaning

that it suces for searching for shortest paths This allows them to apply Dijkstras

algorithm obtaining an O n log n time O n log n space algorithm for L short

est paths Alternatively this approach yields an O n log n time O n log n

space algorithm

Fixed Orientations and Approximations Metho ds for nding L short

est paths generalize immediately to the case of xed orientation metrics in which

distances are measured in terms of the length of the shortest p olygonal path

whose links are restricted to a set of k xed orientations see Widmayer Wu and

Wong The L and L metrics are sp ecial cases in which there are four xed

orientations equally spaced by degrees The result is an algorithm for nding

shortest obstacleavoiding paths in time O k n log n

We can apply the ab ove result to get an approximation algorithm for Euclidean

shortest paths by noting that the Euclidean metric is approximated to within ac

curacy O k by the xed orientation metric with k equally spaced orientations

log n to pro duce a path The result is an algorithm that runs in time O n

guaranteed to have length within factor of the Euclidean shortest path

length Clarkson also gives an approximation algorithm using a related

metho d that computes an optimal path in time O n n log n after sp ending

O n log n time to build a data structure of size O n

Minimum Link Paths

In some applications the number of edges in a path and not its length is a more

appropriate measure of the path complexity In real life for instance while traveling

in an unfamiliar territory we tend to prefer directions with fewer turns even at the

exp ense of a slight increase in travel time The technical motivation for minimizing

the number of edges in a path arises from applications in rob ot motion planning

graph layouts and telecommunication networks where straightline routing is often

cheaper and preferable while turning is an exp ensive op eration

One also encounters minimum link paths in solid mo deling where they are

used for curvecompression and the approximation of univariate functions

With this background let us now formally dene the notion of a minimum link

path We concentrate on two dimensions but extensions to higher dimensions will

b e obvious Given a p olygonal domain P a minimum link path b etween two p oints

24 J Mitchel l and S Suri

s and t is a p olygonal path with fewest p ossible number of edges that connects s

to t while staying inside P The link distance b etween s and t denoted d s t is

the number of edges in a minimum link path from s to t It is p ossible that there

is no path from s to t avoiding all the obstacles in which case the link distance is

dened to b e innite

Most of the results on minimum link paths are for the case of a simple p olygon

and so we discuss that rst

Minimum link paths in a simple polygon

Like other shortest path problems a considerable eort has b een sp ent on the

simple p olygon In this case the obstacle space consists of the b oundary of a simple

p olygon P and the freespace consists of the interior of the p olygon Evidently the

notion of link distance is closely related to the notion of visibility After all the

visibility p olygon V s consists of precisely the set of p oints whose link distance to

s is one Building up on this idea Suri introduced the concept of a window

partition of a p olygon The window partition of P with resp ect to s is a partition

of the interior of P into cells over which the link distance to s is constant Clearly

V s is the cell with link distance The cells with link distance are the regions

of P V s that are visible from the windows of V s a window of V s is an

edge that forms a b oundary b etween V s and P V s The cells with larger link

distance are obtained by iterating this pro cedure Figure shows an example of a window partition

2 2

2 1 3

Fig The window partition of a p olygon from p oint s Numbers in regions denote their link

distance from s A minimum link path from s to t has three links

Window partitions turn out to b e a p owerful to ol for solving a number of mini

mum link path problems b oth optimally as well as approximately In Suri

Ch A Survey of Computational Geometry 25

presents a linear time algorithm for computing the window partition of a triangu

lated simple p olygon Based on this construction he derives the following results

i The link distance from a xed p oint s to all the vertices of P can b e com

puted easily once the window partition from s has b een computed the link distance

of a vertex v is k if the cell containing v has lab el k

ii The window partition is a planar sub division which can b e prepro cessed

in linear additional time to allow p oint lo cation queries in logarithmic time

cf Section With this prepro cessing the link distance from s to a query p oint

t can b e determined in O log n time

iii The graphtheoretic dual of the window partition is a tree called the window

tree Suri observes that distances in the window tree nicely approximate the

link distance b etween p oints In particular he shows how to calculate the link

diameter of the p olygon within links in linear time the link diameter is the

maximum link distance b etween any two p oints of the p olygon More generally the

linkfarthest neighbor of each vertex of P can also b e computed within links in

linear time

In all the cases ab ove a minimum link path can always b e extracted in time

prop ortional to the link distance Suri and Ke prop ose O n log n time

algorithms for computing the link diameter exactly Another linkdistance related

concept that may have applications in shap e analysis is link center it is the set

of p oints from which the maximum link distance to any p oint of P is minimized

Lenhart et al prop osed an O n time algorithm based on window parti

tions for computing the link center This was subsequently improved to O n log n

indep endently by Ke and Djidjev Lingas and Sack

Recently Arkin Mitchell and Suri have developed an O n space data

structure for answering linkdistance queries in a simple p olygon when both s and t

are query p oints Their data structure stores O n window partitions The query

algorithm exploits information ab out the geo desic path b etween s and t If it detects

that the geo desic path has an inection edge ie the predecessor and the successor

edges lie on opp osite sides of the edge or a rotationally pinned edge ie the p olygon

touches the edges from b oth sides then the link distance d s t is computed by

searching the window partitions of the two p olygon vertices that are asso ciated with

the inection or pinned edge If the path has neither an inection nor a pinned

edge then it must b e a spiral path and this is the most complicated case The

query algorithm in this case uses projection functions which are fractional linear

forms to track the other endp oint of a constantturning path as its rst endp oint

moves linearly along an edge of P The query algorithm of works even if s

and t are convex p olygons instead of just p oints however the query time b ecomes

O log k log n if the two p olygons have a total of k edges In particular if the

p olygons have a xed number of edges the query time is asymptotically optimal

Op en Problem Devise a data structure to answer point link distance

queries in a simple polygon The data structure should use no more than O n

time and space for its construction and answer queries in O log n time

26 J Mitchel l and S Suri

Minimum link paths among obstacles

With multiple obstacles determining the homotopy class of an optimal path

b ecomes a critical problem Of course the basic idea b ehind the window partition

still holds rep eatedly compute visibility p olygons until the destination p oint t is

reached However unlike the simple p olygon where t is always separated from s

by a unique window there are multiple homotopically distinct paths in the general

case It requires a careful pruning technique to overcome a combinatorial explosion

There is essentially one result on link distance among general p olygonal obstacles

Mitchell Rote and Woginger present an O E log n time algorithm for

nding a minimum link path b etween two xed p oints among a set of p olygonal

obstacles with a total of n edges where E O n is the size of the visibility

graph The result of Mitchell et al is only a rst step the problem of computing

link distances among obstacles is far from solved The only lower b ound known is

n log n

Op en Problem Given a polygonal domain having n vertices compute a

minimumlink path between two given points in time O n log n or any subquadratic

bound

The assumption of orthogonal obstacles and rectilinear paths results in signicant

simplications In de Berg shows how to prepro cess a rectilinear simple p oly

gon in O n log n time and space to supp ort O log n time rectilinear link distance

b etween two arbitrary query p oints In de Berg et al develop an O n log n

space data structure for answering xedsource link distance queries among or

thogonal obstacles with a total of n edges The data structure requires O n

prepro cessing time and can answer a link distance query in O log n time In fact

their data structure allows for the minimization of a combined metric based on a

xed linear combination of the L length and the link length the cost of a rectilin

ear path is its L length plus C times the number of turns for some presp ecied

constant C Subsequently de Berg Kreveld and Nilsson generalized the

result of to arbitrary dimensions In d dimensions their data structure requires

d d

O n log n space O n log n prepro cessing time and supp orts xedsource

link distance queries in O log n time

For the general link distance problem in higher dimensions the only results known

are approximations Mitchell and Piatko show that one can get within a

constant factor of the link distance in p olynomial time for any xed d

Weighted Regions

A natural generalization of the standard shortest obstacleavoiding path problem

is to consider varied terrain in which each region of the plane is assigned a weight

that represents the cost p er unit distance of traveling in that region Clearly the

standard problem ts within this framework if we let obstacles have weight while

free space has weight

We can think of the weighted plane as b eing a network with an uncountably innite number of no des one p er p oint of the plane We join every pair of p oints

Ch A Survey of Computational Geometry 27

with an edge assigning a weight equal to the line integral of the weight function

along the straight line segment joining the two p oints

More formally we consider the problem in which a planar p olygonal sub division

S is given with a weight f W g assigned to each face of the sub di

vision We let n denote the total number of vertices describing the sub division Our

ob jective is to nd a path from s to t that has minimum weighted length over all

paths from s to t The weighted length of a path is given by the path integral of

the weight function it equals the weighted sum of its Euclidean lengths within

each region This problem of nding an optimal path within a varied terrain is

called the Weighted Region Problem WRP and was introduced by Mitchell and

Papadimitriou

There are many p otential applications of the WRP The original motivation was

to solve the minimumtime path problem for a p oint rob ot without dynamic con

straints moving in a terrain of varied types grassland brushland blacktop marsh

land b o dies of water obstacles to overland travel and other types of terrain can

each b e assigned a weight according to the maximum sp eed at which a mobile rob ot

can traverse the region In this sense the weights denote a traversability index

or the recipro cal of maximum sp eed

Mitchell and Papadimitriou present a p olynomialtime solution to the WRP

based on the continuous Dijkstra paradigm that nds a path guaranteed to b e

within a factor of of the optimal weighted length where is any user

sp ecied degree of precision The time complexity of the algorithm is O E S where

E is the number of events in the simulation of Dijkstras algorithm and S is the

complexity of p erforming a numerical search to solve the following subproblem

Find a shortest path from s to t that go es through a given sequence of

k edges of S It is known that E O n and that there are examples where E

can actually achieve this upp er b ound so that no b etter b ound is p ossible

Mitchell and Papadimitriou also show that the numerical search can b e done with

a form of binary search that exploits the lo cal optimality condition that an optimal

path b ends according to Snells Law of Refraction when crossing a region b oundary

This leads to a b ound of S O k lognN W on the time needed to p erform a

search on a k edge sequence where N is the largest integer co ordinate of any vertex

of the sub division S Since one can show that k O n this yields an overall time

b ound of O n L where L lognN W can b e thought of as the bit complexity

of the problem instance Although the exp onent lo oks particularly bad we note

that these are truly worstcase b ounds in the average case we might exp ect that

E b ehaves like n or n and that k is eectively constant

Many other pap ers are written on the WRP problem and its sp ecial cases eg

see A recent pair of pap ers by Kindl Shing and Rowe

rep orts practical exp erience with a simulated annealing approach to the WRP

Papadakis and Perakis have generalized the WRP to the case of time

varying maps where b oth the weights and the region b oundaries may change over

time they obtain generalized lo cal optimality conditions for this case and prop ose a search algorithm to nd go o d paths

28 J Mitchel l and S Suri

Bicriteria Shortest Paths

The shortest path problem asks for paths that minimize some one ob jective

function that measures length or cost Frequently however our application

actually wants us to nd paths to minimize two or more dierent costs For example

in mobile rob otics applications we may wish to nd a path that simultaneously is

short in Euclidean length and has few turns

Multicriteria optimization problems tend to b e hard Even the bicriteria path

problem in a graph is NPhard Do es there exist a path from s to t whose

length is less than L and whose weight is less than W Pseudop olynomial time

algorithms are known and many heuristics have b een devised eg see

Several geometric versions of bicriteria shortest path problems have recently b een

investigated Various optimality criteria are of interest including any pair from the

following list Euclidean L length rectilinear L length other L metrics the

number of turns in a path its link length the total amount of integrated turning

done by a path etc

For example applications in rob ot motion planning problems may require us to

nd a shortest L path constrained to have at most k links To date no exact

metho d is known for this problem Part of the diculty is that a minimumlink

path will not in general lie on the visibility graph or any simple discrete graph

Arkin et al show that in a simple p olygon one can always nd an st path

whose link length is within a factor of of the link distance from s to t while

of the Euclidean simultaneously having Euclidean length within a factor of

shortest path length A corresp onding result is not p ossible for p olygons with

holes

Mitchell et al study the problem of nding shortest k link paths in a simple

p olygon P They exploit the lo cal optimality condition on the turning angles at

consecutive b ends of a shortest k link path in order to devise a binary search scheme

tracing paths according to this lo cal optimality criterion in order to nd the turning

angle at the rst b end p oint The results of these searches are then combined via

dynamic programming recursions to yield an algorithm that pro duces a path whose

length is guaranteed to b e within a factor of the length of shortest k link

path for any usersp ecied tolerance The algorithm runs in time p olynomial in

n k and logarithmic in and the largest integer co ordinate of any vertex of P

For p olygons with holes we p ose an interesting op en question

Op en Problem Given a polygonal domain with holes what is the complexity

of computing a shortest k link path between two given points Is it NPcomplete to

decide if there exists a path with at most k links and Euclidean length at most L

Several recent pap ers have addressed the bicriteria path problem for a combi

nation of rectilinear link distance and L length in an environment of rectilinear

obstacles In de Berg et al ecient algorithms are given in two and higher

dimensions for computing optimal paths according to a combined metric which

takes a linear combination of rectilinear distance and L path length Note that

this is not the same as solving the problem of computing Paretooptimal solutions

Ch A Survey of Computational Geometry 29

Yang Lee and Wong give an O n log n algorithm for computing a

shortest k b end path a minimumbend shortest path or any combined ob jective

that uses a monotonic function of rectilinear link length and L length in a planar

rectilinear environment In all of these rectilinear problems there is an underlying

grid graph which can serve as a path preserving graph This immediately im

plies the existence of p olynomialtime solutions to the various problems studied by

the contributions of these pap ers lie in their clever metho ds to

solve the problems very eciently

Some lower b ounds on bicriteria path problems have b een established by

Arkin at al In particular they show that the following geometric versions

are NPhard Given a p olygonal domain nd a path whose L length is at most

L and whose total turn is at most T Given a p olygonal domain nd a path

whose L length is at most and whose L length is at most p q and

p p q q

Given a sub division of the plane into red and blue p olygonal regions nd a path

whose travel through blue resp red is b ound by B resp R

Higher Dimensions

While the shortest obstacleavoiding path problem is solved eciently in the

plane Canny and Reif show that the problem of nding shortest obstacle

avoiding paths according to any L p metric in three dimensions is

NPhard even when all of the obstacles are convex p olytop es

The diculty lies in the structure of shortest paths in three dimensions They

do not necessarily lie on any kind of discrete visibility graph In general shortest

paths in a threedimensional p olyhedral domain P will b e p olygonal with b end

p oints that lie interior to edges of obstacles The manner in which a shortest path

b ends at an edge is well constrained It must enter and leave at the same angle

to the edge This implies that any lo cally optimal subpath joining two consecutive

obstacle vertices can b e unfolded at each obstacle edge that it touches in such a

way that the subpath b ecomes a straight segment

The unfolding prop erty of optimal paths can b e exploited to yield p olynomial

time algorithms in the sp ecial case in which the path must stay on a p olyhedral

surface For the case of a convex surface Sharir and Schor give an O n log n

time algorithm for computing shortest paths Their algorithm has b een improved

by Mount who gives an O n log n time algorithm for the same problem

and shows how to use only O n log n space For the case of shortest paths on a

nonconvex p olyhedral surface ORourke Suri and Bo oth give an O n time

algorithm Mitchell Mount and Papadimitriou improved the time b ound

to O n log n giving an algorithm based on the continuous Dijkstra paradigm to

construct a shortest path map for any given source p oint on an arbitrary p olyhedral

surface having n facets Chen and Han improve the algorithm of obtaining

an O n time and O n space b ound See Aronov and ORourke for the pro of

of the nonoverlap of the star unfolding required by

For the case when the domain P has only a few convex obstacles Sharir

30 J Mitchel l and S Suri

O k

has given an n algorithm for shortest paths based on a careful analysis of the

structure of shortest paths and a b ound of O n on the number of distinct edge

sequences that corresp ond to shortest paths on the surface of a convex p olytop e

Mount has improved the b ound on edge sequences to O n which he shows to

b e tight Schevon and ORourke show a tight b ound of n on the number

of maximal edge sequences for shortest paths Agarwal et al give an O n log n

algorithm for computing all O n edge sequences that corresp ond to shortest paths

on a convex p olytop e

For general threedimensional p olyhedral domains P the b est algorithmic re

sults known are approximation algorithms Papadimitriou gives a fully p oly

nomial approximation scheme that pro duces a path guaranteed to b e no longer

than times the length of a shortest path His algorithm requires time

O n L logn where L is the number of bits in an integer co ordinate

of vertices of P Clarkson also gives a fully p olynomial approximation scheme

which improves up on that of in the case that n is large

While threedimensional shortest path problems are known already to b e hard

the pro of Canny and Reif is based up on a construction in which the size of

the SPM is exp onential This leaves op en an interesting algorithmic question of a

p otentially practical nature since we may hop e that in practice such huge SPMs

will not arise

Op en Problem Given a polyhedral domain in dimensions compute a short

est path map in outputsensitive time

Kinetics and Other Constraints

Minimum Time Paths Any real mobile rob ot has a b ounded acceleration

vector and a maximum sp eed If we include these constraints in our mo del for path

planning then an appropriate ob jective is to minimize the time necessary for a

p oint rob ot to travel from one p oint of free space to another with the velocity

vector known at the start and p ossibly constrained at the destination In general

this kino dynamic planning problem is a very dicult optimal control problem We

are no longer in the nice situation of having optimal paths that are taut string

paths lying on a visibility graph Instead the paths will b e complicated curves in

free space and the complexity of nding such optimal paths remains op en

In a rst step towards understanding the algorithmic complexity of computing

timeoptimal tra jectories under dynamic constraints Canny et al have pro

duced a p olynomialtime pro cedure for nding a provably good approximating time

optimal tra jectory that is within a factor of of b eing a minimumtime tra jec

tory Their metho d is fairly straightforward they discretize the fourdimensional

phase space that represents p osition and velocity Sp ecial care is needed however

to ensure that the size of the grid is b ounded by a p olynomial in and n and the

analysis to prove the eectiveness of the resulting paths is quite tedious

Canny Rege and Reif give an exact algorithm for computing an optimal

path when there is an upp er b ound on the L norm of the velocity and acceleration

Ch A Survey of Computational Geometry 31

vectors Their algorithm is based on characterizing a set of canonical solutions

related to bangbang controls in one dimension that are guaranteed to include

an optimal solution path Then by writing an appropriate expression in the rst

order theory of the reals they obtain an exp onential time but p olynomial space

algorithm It remains an op en question whether or not a p olynomialtime algorithm

exists

Bounded Turning Radius Related to the general problem of handling

dynamic constraints is the imp ortant problem of nding shortest paths sub ject to

a b ound on their curvature Placing a lower b ound on the curvature can b e thought

of as a means of handling an upp er b ound on the acceleration vector of a p oint rob ot

whose sp eed is constant or can b e thought of as the realistic constraint imp osed

by the fact that many mobile rob ots have a b ounded steering angle

Fortune and Wilfong gave an exp onentialtime decision pro cedure to deter

mine whether or not it was p ossible for a rob ot to move from a start to a goal

among a set of given obstacles while ob eying a lower b ound on the curvature of

its path and not allowing reversals If the p oint following the path is allowed to

reverse direction then Laumond has shown that it is always p ossible to obtain

a b ounded curvature path if a feasible path exists

Since the general problem seems to b e extremely dicult a restricted version

has b een studied Wilfong considers the case in which the rob ot is to

follow a given network of lanes with the rob ot allowed to turn from one segment

to another along a b ounded curvature circular arc if the two lanes intersect In

Wilfong a p olynomialtime algorithm is given for pro ducing some feasible

path in Wilfong the problem of nding a shortest feasible path is shown to

b e NPcomplete while a p olynomialtime metho d is given for deforming a given

feasible path into a shortest equivalent feasible path The time b ound is O k n

where n is the number of vertices describing the obstacles and k is the number of

turns in the path

Optimal Robot Motion

Most of our discussion is fo cused on the case of p oint rob ots When the rob ot is

not a p oint the problem usually b ecomes much harder An exception is the case of

a circular rob ot which is often a very go o d assumption anyhow or a nonrotating

convex rob ot In the case of a circular rob ot the problem of nding a shortest

path among obstacles is solved almost as in the p oint rob ot case we simply

must grow the obstacles by the radius of the rob ot and shrink the rob ot to

a p oint This is the standard conguration space approach in motion planning

and leads to shortest path algorithms with time b ounds comparable to the p oint

rob ot case

Optimal motion of rotating noncircular rob ots is a very hard problem Consider

the simplest case of moving a line segment ladder in the plane The motion

32 J Mitchel l and S Suri

planning problem which ignores any measure of cost of motion is solvable in time

O n log n A natural denition of cost of motion for a ladder is to consider

the work necessary to move the ladder from one place to another assuming that

there is a uniform co ecient of kinetic friction Optimal motion of a ladder is an

op en problem at this p oint Papadimitriou and Silverberg and ORourke

give solutions for restricted cases of moving a ladder among obstacles and Icking

et al have characterized the solution for the general case without obstacles

Op en Problem Given a polygonal domain compute an optimal motion of a

ladder from one position to another

Online Algorithms and Navigation Without Maps

In all of the path planning problems we have discussed so far we have assumed

that we know in advance the exact layout of the environment in which the rob ot

moves ie we assume we are given a p erfect map In most real problems we

cannot make this assumption Indeed if we are given a map or o orplan of where

walls and obstacles are lo cated the map will invariably contain inaccuracies and

we may b e interested also in b eing able to navigate around obstacles that may not

b e in the map For example for a rob ot moving in an oce building while the

o orplan and desk layouts may b e considered accurate and xed the lo cation of

a chair or a trashcan is something that we usually cannot assume to b e known in

advance

When planning paths in the absence of p erfect map information we must have

some mo del of sensory inputs that enable the rob ot to sense the lo cal structure of

its environment Many dierent assumptions are p ossible here visual sensors range

sensors p erhaps from sonar or computed from stereo imagery touch sensors etc

While numerous heuristic metho ds have b een devised for sensorbased autonomous

vehicle navigation see only recently has there b een interest in these questions

from the theory of algorithms community

Necessarily the theoretical results require stringent assumptions b efore any

thing can b e claimed and proven One of the rst pap ers was by Lumelsky and

Stepanov who show that if a p oint rob ot is endowed only with a contact

tactile sensor which can determine when it is in contact with an obstacle then

there is a strategy for feeling ones way from a start to goal such that the resulting

path length is at most times the total p erimeter length of the set of obstacles

The strategy called BUG is closely related to the strategy of keeping ones

hand on the wall when going through a maze No assumptions have to b e made

ab out the shap es of the obstacles Lumelsky and Stepanov show that this ratio

is essentially b est p ossible for this mo del see for some further work on an

extension of the LumelskyStepanov mo del

An obvious complaint with the mo del of is that it do es not b ound the

competitive ratio the worstcase ratio of the length of the actual path to that

of an optimal path Among the rst results that b ound the comp etitive ratio is

Ch A Survey of Computational Geometry 33

that of Papadimitriou and Yannakakis who show that if the obstacles are

assumed to b e squares one can achieve a comp etitive ratio of and no strategy

can achieve a ratio b etter than Further by an adversary argument they show

that for arbitrary eg thin aligned rectangular obstacles and a rob ot that has

p erfect lineofsight vision there is no strategy with a b ounded comp etitive ratio