External Memory Computational Geometry
Preliminary Version
y z x
Michael T Go o drich Jyh Jong Tsay Darren Erik Vengro Je rey Scott Vitter
The Johns Hopkins National Chung Cheng Brown University Duke University
University University dev cs brown edu jsv cs duke edu
goodrich cs jhu edu jjt cs ccu edu tw
Abstract Large scale problems involving geometric data are
ubiquitous in spatial databases geo
graphic information systems GIS con
In this paper we give new techniques for designing
straint logic programming ob ject oriented
e cient algorithms for computational geometry prob
databases statistics virtual reality systems and
lems that are too large to be solved in internal memory
computer graphics As an example NASA s so on
and we use these techniques to develop optimal and
to b e p etabyte sized databases are designed to facili
practical algorithms for a number of important large
tate a variety of complex geometric queries Im
scale problems We discuss our algorithms primarily
p ortant op erations on geometric data include range
in the context of single processor single disk machines
queries constructing convex hulls nearest neighb or
a domain in which they are not only the rst known
calculations nding intersections and ray tracing to
optimal results but also of tremendous practical value
name but a few
Our methods also produce the rst known optimal al
gorithms for a wide range of two level and hierarchical
Our I O mo del
multilevel memory models including paral lel models
The algorithms are optimal both in terms of I O cost
In I O systems data are usually transferred in units
and internal computation
of blocks which may consist of several kilobytes This
blo cking takes advantage of the fact that the seek time
Intro duction
is usually much longer than the time needed to trans
fer a record of data once the disk read write head is
Input Output I O communication b etween fast
in place An increasingly p opular way to get further
internal memory and slower secondary storage is the
sp eedup is to use many disk drives and or many CPUs
b ottleneck in many large scale information pro cessing
working in parallel We mo del such
applications and its relative signi cance is increasing
systems examples of which are shown in Figure us
as parallel computing gains p opularity In this pap er
ing the following four parameters
we consider the imp ortant application area of compu
tational geometry and develop several paradigms for
M items that can t in internal memory
optimal geometric computation using secondary stor
B records p er blo ck
age
P CPUs internal pro cessors
This research was supp orted in part by the National Science
D disk drives
Foundation under Grants CCR and IRI and
by the NSF and DARPA under Grant CCR
y
For the problems we consider we de ne three general
This research was supp orted by the National Science Coun
cil under grant NSC E Republic of China parameters
z
This research was supp orted in part by Army Research Of
ce grant DAAL G and by National Science Foun
N items or up dates in the problem instance
dation grant DMR Author is currently visiting Duke
K query op erations in the problem instance
University e mail dev cs duke edu
x
This research was supp orted in part by National Science
T items in the solution to the problem
Foundation grant CCR and by Army Research O ce
grant DAAL G
Our results
D ּ D
e present a numb er of general tech a)ּ (b) In this pap er w)
niques for designing external memory algorithms for
computational geometry problems These techniques
include the following
distribution sweeping a generic metho d for ex
ternalizing plane sweep algorithms
persistent B trees an o line metho ds for con
structing an optimal space p ersistent version of
the B tree data structure For batched prob
this gives a factor of B improvement over
Internal Internal lemsInternal Internal
generic p ersistence techniques of Driscoll et
memory memory thememory memory
al
Figure a The parallel disk mo del Each of the
batch ltering a general metho d for p erform
D disks can simultaneously transfer B records to and
ing K simultaneous external memory searches in
internal memory in a single I O The internal
from CPU CPU CPU
data structures that can b e mo deled as a planar
memory can store M DB records b Multipro ces
layered dags
generalization of the I O mo del in a in which
sor CPU
each of the P D internal pro cessors controls one
external marriage before conquest an external
disk and has an internal memory of size M P The
memory analog to the well known technique of
P pro cessors are connected by some top ology such as
Kirkpatrick and Seidel for p erforming output
a hyp ercub e or an EREW PRAM and their memories
e hull constructions
“Thesensitiv Net”
collectively have size M
We apply these techniques to derive optimal
external memory algorithms for the following funda
We will assume that M N P M log M
mental problems in computational geometry com
and DB M The measures of p erformance
puting the pairwise intersection of N orthogonal seg
that we would like to minimize simultaneously are the
ments answering K range queries on N p oints con
numb er of I Os and the internal computation time
structing the d and d convex hull of N p oints
The relevant terms that enter the formul for the
p erforming K p oint lo cation queries in a planar sub
I O b ounds are often in units of blo cks such as N B
division of size N nding all nearest neighb ors for a
M B and so on For that reason we de ne the fol
set of N p oints in the plane nding the pairwise in
lowing shorthand notation
tersections of N rectangles computing the measure of
M K T N
the union of N rectangles computing the visibility of
N segments from a p oint p erforming K ray sho oting
B B B B
queries in CSG mo dels of size N as well as several
In order to get across our techniques in the mini
geometric dominance problems Our results are sum
mum space we illustrate our results in this pap er to
marized in the following theorem individual parts of
the sp ecial case of the I O mo del in which P and
which are discussed in the remaining sections of the
D Even in this simpli ed mo del our results are
pap er
extremely signi cant as P and D accurately
mo dels the vast ma jority of I O systems currently in
Theorem Each of the problems mentioned in
stalled and b eing pro duced yet no optimal algorithms
the preceding paragraph can be solved in external mem
were previously known for the problems we discuss
ory using O log I Os If D disks are
Additionally our results are optimal in the general
used in paral lel the number of I Os required can be
I O mo del and in the parallel hierarchy mo dels In
reduced by a factor of D
particular in the I O mo del using P pro cessors re
For problems in which there are no queries as part of duces the internal computation time by a factor of P
the problem instance we use K and thus and using D disks reduces the numb er of I O steps by
if the output solution size is xed we use T and a factor of D This is discussed in greater detail in
thus B o
Section and in the full version of this pap er
we will use is the problem of rep orting all intersecting Distribution sweeping
pairs from a set of N orthogonal line segments This
The well known plane sweep paradigm is a p ow
problem is imp ortant in graphics and VLSI design sys
erful approach for developing computational geome
tems
try algorithms that are e cient in terms of internal
We de ne the sweep in terms of a vertical sweep of
computation In this section we develop a new plane
the plane by a horizontal line The x co ordinates of
sweep approach that for the rst time achieves opti
the vertical segments de ne the total order and the
mal I O p erformance and a subsequent improvement
y co ordinates of their endp oints de ne the sweep line
in practice for a large numb er of large scale o line
up date events in The y co ordinates of the hori
problems in computational geometry Sp eci c exam
zontal segments de ne the sweep line query events in
ples that we explore include orthogonal segment inter
section batched range queries computing all nearest
Initially we use an optimal sorting algorithm so
neighb ors rectangle intersection and union computa
that endp oints of all segments are in two sorted lists
tions visibility among a set of non intersecting line
one sorted by x and the other by y We now partition
segments and d maxima
the p oints into d e vertical strips Now the sweep
i
We assume we are given a sequence
ing b egins progressing from top to b ottom As we
such that each is an up date op eration
1 2 N i
sweep we will rep ort some of the segment intersections
of the form insert x or delete x where each such x
and distribute data to recursive subproblems that we
is taken from a known total order The sequence
can solve to nd the rest When the top endp oint of
corresp onds to the sequence of up date op erations dur
a vertical segment is encountered the segment is in
ing the sweep In addition we are given a sequence
serted into an active list A asso ciated with the strip
i
such that each is a query op eration
1 2 K i
in which the segment lies and later when the b ot
i
query j which is de ned as some kind of search in
tom endp oint is encountered the segment is deleted
a search tree de ned on by p erforming the op era
from A When we encounter a horizontal segment
i
tions We assume without loss of general
1 2 j
R we consider the strips that R passes completely
ity that the s are sorted by their j arguments The
i
through and rep ort all the vertical segments in the ac
problem is to determine the answer to each query
i
tive lists of those strips Note that horizontal segments
One obvious external memory solution to this prob
are only distributed to the two strips containing their
lem is to implement the search tree as a dynamic B
endp oints thus at each level of recursion each segment
tree and to p erform the queries in in an on line
is represented only twice Once the numb er of p oints
fashion while p erforming the up dates in Unfortu
in a recursive subproblem falls b elow M we simply
nately this requires N K log B
solve the problem in main memory This pro cess is
log I O op erations in the worst case which is
illustrated in Figure
prohibitive Previous work using lazy batched up dates
Insertions and vertical segments can b e pro cessed
on the B tree yielded algorithms with O log
2
e ciently using blo cks With the exception of deleting
I Os
segments from active lists the total numb er of I Os
In Sections and we show how to p erform
p erformed by this metho d is optimal O log
the queries using only O log I Os which is
where T B and T is the numb er of intersections
optimal The lower b ound follows by a simple reduc
rep orted If vigilant deletion is used to delete each
tion from the sorting problem which has the same I O
segment as so on as the sweepline reaches the b ot
b ound as a lower b ound Our new metho d uses an
tom endp oint a nonoptimal O N O B term
o line top down implementation of the sweep which
is added to the I O b ound Instead we use the fol
in turn is based up on a novel application of the sub
lowing lazy approach For each strip we maintain
division technique used by in the distribution sort
a stack of insertions pro cessed so far When a new
algorithms of It is for this reason that we
segment is inserted we simply add it to the stack
refer to our technique as distribution sweeping
We keep all but the most recently added B elements
of this stack in blo cks of size B in external memory
An example orthogonal segment in
When we are asked to output the active list we scan
tersection rep orting
the entire stack outputting the segments still current
Before discussing the distribution sweeping as a and removing the segments whose deletion time has
general technique let us b egin with a simple exam passed A simple amortization argument shows that
ple illustrating the main ideas involved The example this metho d achieves the b ound of Theorem
distribution sorting algorithms Asymptotically par
titioning the input in this manner do es not a ect the
overall running time of an algorithm since it increases
d
e
the depth of recursion by only a constant factor of
g
b
We make use of this lemma to pro cess the up dates
c
f
and the queries as follows We rst merge and
a
into a single list ordered by index This can b e done in
O I Os by a simple merging pro cedure Moreover
during this same merging pro cedure we can construct
the set X of all elements referenced in p ossibly with
duplicates We then apply Lemma to partition X
into X X X by the ordering Since the op
1 2 s
erations in and are de ned for a tree structure or
dered by this distribution immediately implies that
each and can b e decomp osed into sub op erations
i i
and such that
i 1 i 2 i s i 1 i 2 i s i j
resp is de ned on X The sp eci c de nitions of
i j j
these sub op erations will of course dep end up on the
Figure Distribution sweeping for orthogonal seg
application Note that in the example given in Sec
ment intersection Supp ose the sweep line moving
tion they were recursively solving the problem in
down from the top has reached horizontal segment
the strips Since and are de ned for a tree struc
i
a At this p oint the active lists for the four strips are
ture ordered by only O sub op erations resp
i j
A fbg A fc dg A fe f g and A fg g
involve a recursive search in X for any i The
1 2 3 4
i j j
Note that this assumes lazy deletion At this p oint
other sub op erations for X involve an up date or query
j
the intersections of a and c d and f are rep orted
that can b e p erformed by a single scan ordered by i
b ecause a fully spans and Note that the in
through these non recursive and op erations
2 3
i j i j
p
tersection of a and b will not b e rep orted until the
we may p erform these non recursive Since s is O
problem is solved recursively on e and g are now
sub op erations as well as construct an ordered list of
1
deleted from A and A resp ectively Finally we dis
each sequence of recursive sub op erations by a single
3 4
tribute the p oints to the subproblems corresp onding
pass through the combined list We form all the re
to the strips
cursive sublists simultaneously in main memory using
p
MB B As so on as a bucket s buckets of size
lls we empty its contents out to external memory
General distribution sweeping
During this pass for each j we also maintain an active
bucket in main memory that stores information such
In this section we present the distribution sweeping
as counts sums or p ointers into external memory
technique at a high level Our metho d is based up on
for the elements in X present after p erforming the
j
the following distribution lemma which we will use
op erations on the elements of X up to the current
j
to distribute input data into recursive subproblems
one a or The active buckets are used to
i j i j
answer the non recursive sub op erations Having p er
Lemma Let S be a set of N elements
formed all the non recursive sub op erations we then
taken from some total order and let S be stored
recursively p erform each sequence of recursive sub in N B blocks in external memory Then using
op erations in turn A single pass requires only O only O I O operations S can be partitioned into
p
p
I Os and results in s subsequences made up of O s
e d M B e subsets S S S such s d
1 2 s
N N 3 1
blo cks each which requires only O blo cks for all
jS j and each element in that for al l i
i
2 s 2 s
the subsequences combined Thus the total numb er
S is less than each element in S
i i+1
of I Os needed is O log O log
s
p
Note that this lemma divides the input into d e
subsets whereas in the example in the previous sec
Other applications of distribution
tion we divided the input into d e strips The di er
sweeping
ence is that this lemma do es not require the input to
Though space precludes a full exp osition the distri b e sorted by the ordering b efore it is partitioned
in fact this lemma was originally proven for use in
bution sweeping metho d can b e used to solve a num
b er of other o line problems in computational ge sth element in each list as a bridge element and we
ometry that are traditionally solved by plane sweep merge these bridge elements into a single list Y We
techniques The resulting algorithms use an optimal store p ointers from each element y Y to all its bridge
O log I Os Problems in this cate predecessors in the recursively constructed lists The
gory include batched range queries nding all nearest list Y together with these p ointers de nes the ro ots
neighb ors computing the visibility from a p oint in the of the p ersistent structure Since we only cho ose every
plane nding pairwise rectangle intersections com sth element from each list as a bridge it is easy to see
puting the measure of a union of rectangles and the that total space needed is O blo cks and the depth
d maxima problem These problems are discussed of the resulting layered dag p ersistent structure is
in greater detail in the full version of this pap er O log
A search in the past say at time p osition i b egins
by lo cating the ro ot active for time i and searching
Persistent B trees
down in the structure from there always searching in
The B tree data structure is a fundamental
no des whose time stamp is the largest time stamp less
structure for maintaining a dynamically changing dic
than or equal to i Performing only one such search
tionary in external memory In some cases however
would not b e an e cient strategy however unless s
p
it may b e advantageous to b e able to access previous
is O B Nevertheless as we show in the next
versions of the data structure Being able to access
section this is a very e cient data structure e g
such previous versions is known as persistence and
for p oint lo cation if it is searched using the batched
there exist very general techniques for making most
ltering technique
data structures p ersistent Persistence can b e im
plemented either in an on line fashion i e where the
Batch ltering
tree up dates are coming on line or in an o line fash
ion i e where one is given the sequence of tree up
In this section we demonstrate how for many query
dates in advance
problems in computational geometry we can represent
For the on line case the metho d of Driscoll et
a data structure of size N in disk blo cks in such a
al can b e applied to hysterical B trees as de
way that K constant sized output queries of the data
structure can b e answered in O log I O scrib ed by Maier and Salveter Since it is on line
op erations Because we represent the data structure this structure requires O N log I Os to construct
which is optimal in an on line setting Unfortunately
as a dag through which the K queries lter down from
this is a factor of B away from optimal for the sort of
source to sinks we call this technique batch ltering
batch geometric problems we would like to consider
Given a data structure that supp orts queries we
For these we need an o line strategy that requires
can often mo del the pro cessing of a query as the
only O log I Os In the following section we de
traversal of a decision dag isomorphic to the data
scrib e just such a metho d
structure We b egin at a source no de in the dag and
at each no de we visit we make a decision based on
O line p ersistence
the outcome of comparisons b etween the query value
and some numb er d of values stored at the no de We
In the o line case we can build a p ersistent tree
then make a decision as to which of the no de s O d
by the distribution sweep metho d We slightly mo dify
children to visit next This pro cess continues until we
our application of distribution sweeping for this con
reach a sink in the dag at which p oint we rep ort the
struction however in that we follow the recursive calls
outcome of the query
on the sequences of sub op erations by a non recursive
By restricting the class of such dags we are willing
merge step
to consider we are able to prove the following lemma
We b egin by applying the Lemma to divide the
which will serve as a building blo ck for optimal al
set X of elements mentioned in into s groups of
gorithms to solve a numb er of imp ortant geometric
p
e This of size roughly N s each where s d
problems
course divides into s subsequences one for each
Lemma Let G V E be a planar layered de group We then recursively construct a p ersistent data
cision dag with a single source such that the maximum structure for each subsequence Each such recursive
out degree of any node is Let the graph be repre call returns a list of ro ots of s way trees each of
sented in blocks with the nodes ordered by level and which is marked with a time stamp that represents the
the nodes within a level ordered from left to right Let index in when this ro ot was created We mark every
N jV j and let h be the height of G We can lter K like a ary tree augmented with catalogs Clearly we
inputs through G in O h I O operations can apply the technique of Lemma to the main
tree but the bridge p ointers connecting the catalogs
Pro of Sketch We traverse the levels one by one
make the dag non planar To get around this we note
sending all K inputs to the i th level b efore any are
that as queries traverse the edges b etween no des in
sent to the i st We do this by maintaining two
the main tree they are ordered by the catalog val
FIFO queues one for the current level and one for
ues they query This ordering is established at the
the next level Each such queue is left to right list of
ro ot of the data structure where a ary tree is used
edges b etween its level and the next one If less than B
to lo cate the queries in the rst catalog By relying
inputs traverse an edge then they are explicitly stored
on this ordering we can e ciently pro cess the queries
in the queue If B or more traverse the edge then
that arrive at each no de of the main tree The over
the queue contains a p ointer to a linked list of blo cks
all complexity of this technique is thereby maintained
storing them Since the graph is planar there exists
at O log Full details of the construction
an e ciently blo cked metho d of pro ducing one queue
app ear in the full version of the pap er
form the previous queue 2
Luckily the restrictions imp osed on the typ e of de
cision dags we can handle with batch ltering is not
The convex hull problem is that of computing the
to o severe In particular many computations use de
smallest convex p olytop e completely enclosing a set
cision trees which clearly constitute a sp ecial case of
of N p oints in d dimensional space This problem
the lemma Often these trees are binary but we can
has imp ortant applications ranging from statistics to
divide a binary tree into layers of height O log and
graphics to metallurgy In this section we will ex
then store each no de on a layer b oundary along with
amine the problem in external memory for two and
all its descendants in the layer b elow it as a single no de
three dimensions The three dimensional case is par
with branching factor This allows us to reduce h
ticularly interesting b ecause of the numb er of two
by a factor of O log yet still meet the conditions of
dimensional geometric structures closely related to it
the lemma We will see this approach used in solving
such as Voronoi diagrams and Delaunay triangula
subproblems of the d convex hull problem in Sec
tions In fact by well known reductions our
tion
d convex hull algorithm immediately gives external
Another way of using batch ltering as is to b e
memory algorithms for planar Voronoi diagrams and
seen in Section is by structuring more complicated
Delaunay triangulations with the same I O p erfor
decision dags as recursive constructions to get around
mance
the planarity restrictions of the lemma
In main memory the lower b ound for computing
the convex hull of N p oints in dimension d is
Application multiple p oint planar
N log N in the worst case In secondary mem
p oint lo cation
ory this b ound b ecomes log In this section we
Planar p oint lo cation is one of the fundamental
give optimal algorithms that match this lower b ound
problems of computational geometry In the version of
For the two dimensional case we show how to b eat
the problem considered here we are given a monotone
this lower b ound for the case when the output size T is
planar decomp osition having N vertices and a series
much smaller than N in the extreme case T O
of K query p oints For each query p oint we are to
We develop an output sensitive algorithm based up on
return the identi er of the region in which it lies In
an external memory version of the marriage b efore
main memory this problem can b e solved in optimal
conquest paradigm of Kirkpatrick and Seidel
time O N K log N using fractional cascading
Our d convex hull is somewhat esoteric so we
O N log N is sp ent on prepro cessing and O K log N
also describ e a simpli ed version which although not
is needed to p erform the queries
optimal asymptotically is simpler to implement and
Tamassia and Vitter have demonstrated a
will b e faster for the vast ma jority of practical cases
technique by which the fractional cascading used to
solve this problem can b e implemented in parallel
A worst case optimal two dimen
Their technique can solve our problem in O N p
sional convex hull algorithm
K log N time on a PRAM with p pro cessors We
p
For the two dimensional case a numb er of main can use a metho d based on their construction but us
memory algorithms are known that op erate in optimal
ing in place of p to get a data structure that lo oks
time O N log N A simple way to solve the prob sort to more advanced divide and conquer approaches
lem optimally in external memory is to mo dify one One idea is to use a plane to partition the p oints
of the main memory approaches namely Graham s into equally sized sets recursively construct the con
scan Graham s scan requires that we sort the vex hull for each set and then stitch the recursive
p oints which can b e done in O log I O op era solutions together in linear time Unfortunately we
tions and then scan linearly through them at times know no way of implementing an algorithm of this typ e
backtracking but only over each input p oint at most in secondary memory the problem is that we cannot
once Clearly this scanning stage can b e accomplished adequately anticipate all p ossible paths through the
in O I O op erations so the overall complexity of data that might b e traversed during the combining
the algorithm is O log phase Another obstacle is that we need to b e able
to stitch together O recursive solutions in linear
An output sensitive two dimensional
time rather than just two If we use any fewer then
convex hull algorithm
the depth of the recursion will not b e small enough to
give us an optimal algorithm
If the output size T is signi cantly smaller than N
In order to get around the problems asso ciated with
for example T can b e O then we can do b etter
a merging approach we use a novel formulation of the
than the Graham scan approach In this section we
distribution metho d We consider the dual of the con
show how to construct a two dimensional convex hull
vex hull problem namely that of computing the inter
using a numb er of I Os that is output size sensitive in
section of a set of N half spaces all of which contain
a stronger sense than any of the algorithms discussed
the origin Standard geometric duality transforma
thus far Note that when T o N we actually do
tions are used to show the equivalence of convex
b etter than Theorem indicates Our results are
hull and halfspace intersection Once we are dealing
optimal as stated in the following theorem
with the dual problem we can use a distribution based
Theorem Let S be a set of N points in the plane
approach along the lines of that prop osed by Reif and
whose convex hul l has T extreme points If is O B
Sen for computing d convex hulls in parallel
for some constant then the convex hul l of S can
Let S b e a set of N halfspaces all of which contain
be computed in O log I Os which is optimal
the origin Let the b oundary of a halfspace h S
i
We omit details in this preliminary version but the
b e denoted P Supp ose we have a subset S S
i 0
T
main idea of our metho d is as follows First we ob
such that jS j N Let I h A face of
0 0 j
h 2S
j 0
serve that we may restrict our attention to the up
I might have up to N edges We can reduce this
0
p er hull i e edges with normals with p ositive y
complexity by trangulating each face which can b e
comp onents without loss of generality We use the
done by sorting the vertices of I along a vector not
0
Lemma to divide the set of input p oints into s
p erp endicular to any face and then sweeping a plane
p
e buckets divided by vertical lines We then use d
along this sorted order By Euler s law the size of the
an external memory implementation of a metho d of
resulting set of faces is at most O N We can now
Go o drich for combining prune and search bridge
decomp ose I into O N cones C each of which has
0 i
nding with the Graham scan technique to
one of these faces as a base and the origin as an ap ex
nd all the upp er hull edges intersecting our given
An obvious way of distributing the halfspaces into sub
vertical lines Our implementation uses O I Os
problems is to create a subproblem for each cone C
i
Given these hull edges we may then recurse on any
consisting of nding the intersection of all halfspaces
buckets that are not completely covered by the hull
h S nS whose b ounding planes P intersect C Un
j 0 j i
edges we just discovered Our analysis is based on the
fortunately a given P may intersect many cones so it
j
fact that in any such divide step we either reduce the
is not clear that we can continue to work through the
numb er of p oints under consideration by a constant
O log log N required levels of recursion without caus
fraction or we will discover s hull edges p ossibly
ing a very large blow up in the total size of the sub
b oth If is O B for some constant this
problems Luckily using a form of random sampling
implies that the total numb er of I Os is O log
called p olling and eliminating redundant planes from
which is optimal for any value of T
within a cone prior to recursion we can with high
probability get around this problem In this discus
Three dimensional convex hulls
sion the phrase with high probability means with
probability N for some constant
Even in main memory sweep plane algorithms fail
to solve the d convex hull problem and we must re
Algorithm is the resulting distribution algo
Halfspace Intersection
are actually random variables It su ces to use Karp s
Input A set S of N halfspaces in d space
metho d for solving probabilistic recurrence relations
Output The set of all halfspaces h S whose b ounding
i
T
to get the optimal solution T n O log
planes lie on the b oundary of the intersection h
j
h S
j 0
with high probability
For j to log take a random sample S of
j
The distribution approach used here is di erent
S where jS j N for a constant
j
from those of the distribution sort algorithms for the
Recursively solve the halfspace intersection problem
various I O and memory hierarchy mo dels
on each sample S giving a set of solutions I
j j
but has the same asymptotic I O complexity In the
distribution sorting there are two sets of recursive Use p olling to estimate the size of the partition
of S S that each sample solution I will induce
calls rather than one but the time to partition is
j j
Let S b e the sample whose solution I generates the
r r
faster
smallest such partition
If desired the randomization in our algorithm can
For each cone C of I compute R the set of halfs
i r i
b e removed by an external memory implementation of
paces in S S whose b oundaries intersect C
r i
the technique in Details are omitted for brevity
Eliminate redundant planes from each R yielding
i
In the full version of this pap er we demonstrate
R
i
how for problems of any reasonable practical size we
Recursively solve the halfspace intersection problem
can improve up on this algorithm by using samples of
on each set R
i size instead of N The result is an algorithm that
2
has asymptotic I O p erformance of O log but
is far simpler to implement than Algorithm and
Algorithm An algorithm for computing the d
convex hull of a set of p oints will generally p erform b etter in practice The main
reason for the increase in p erformance is that to do
p olling e ciently the algorithm requires see
rithm for computing the intersection of all h S
i
and thus in most practical situations N
Step can b e completed with O log I Os by
making a linear pass through S for each sample as
Parallel and multi level extensions
suggested by Knuth Step consists of recursive
calls that will b e considered later In Step we de
comp ose each S into cones using a plane sweep This
j
Up to this p oint our discussion has centered on the
takes O jS j B log jS j B I Os We then take a
j j
case where D and P As has b een men
random sample from S S for each S This takes
j j
tioned even in this restricted mo del the results pre
O log I Os Finally we solve a tree structured
sented here are of tremendous practical imp ortance
p oint lo cation problem on all elements of the sam
Our results are even more signi cant however b e
ple This is done by batch ltering as describ ed in
cause the paradigms describ ed in this pap er continue
Section The numb er of I O op erations needed by
to work even when parallelism is added and D and P
P
r r
Step is O jR j In Step log where r
i
i
B B
increase Furthermore they can b e made to work op
redundant planes are eliminated using a variant of the
timally on hierarchical mo dels having more than two
d maxima algorithm from Section and a d convex
levels these include the well known HMM BT
r r
log I O op era hull algorithm Both require O
B B
and UMH pictured in Figure and their paral
tions Finally Step recursively solves the subprob
lelizations pictured in Figure
lems
Details of the algorithms for these mo dels are dis
By metho ds analogous to the approach of Reif and
cussed in the full version of this pap er To a large
Sen for the parallel case we can develop the fol
extent they are based on mo di ed versions of two of
lowing recurrence for the running time of our algo
the main paradigms discussed ab ove namely distribu
rithm
tion sweeping and batch ltering We can also rely on
X
the many way divide and conquer approach of Atallah
T jR j T n O log T N log
i
and Tsay which can b e extended to the I O mo del
i
To implement distribution sweeping in these mo dels
The rst term on the right hand side is the I O cost we take advantage of deterministic distribution tech
for sampling and partitioning the second term is the niques recently develop ed by No dine and Vitter
I O cost for sorting the samples and the last term is for optimal deterministic sorting To implement batch
for the recursive calls In the recurrence the jS j terms
ltering we can use disk striping i H (a) (b) (c)
}
Figure Multilevel hierarchy mo dels a The HMM mo del access to memory lo cation x takes f x time We
ypical access functions like f x lg x and f x x for constant b The BT mo del access to the
consider t CPU CPU CPU CPUCPU CPU
memory lo cations x x x takes f x time c The UMH mo del
Nevertheless there are a numb er of interesting
problems that remain op en
Is there a data structure for d on line range
queries that achieves O log I Os for up
“The Net” B
dates and range queries using O blo cks of
space The o line version of the problem is
solved optimally in this pap er Up dates and
three sided range queries can b e handled by
metablo ck trees in O log log B I Os
B
using O space Two sided range queries an
chored on the diagonal can b e done in O log
B
2
I Os p er query and O log log B
B B
I Os p er semidynamic insert
Can an N vertex p olygon b e triangulated using
O N B I Os Under what conditions
Can we nd all intersecting pairs of N non
orthogonal segments using O log I Os
Figure Parallel multilevel memory hierarchies The
H hierarchies of any of the typ es listed in Figure
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