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Bridging Convex Regions and Related Problems

Hee-kap Ahn

heekap@cs.uu.nl Otfried Cheong

otfried@cs.uu.nl Chan-Su Shin†

cssin@jupiter.kaist.ac.kr

1 Introduction

Let R1and R2be convex (not necessarily polyhedral) regions in a ﬁxed dimension d2. The minimum diameter

bridge problem (MDBP) for R1and R2is a geometric optimization problem to ﬁnd a bridge Bbetween R1and R2such

that the endpoints of Bare points in R1and R2, and the maximum length of the shortest paths connecting two points

in R1and R2passing through Bis minimized.

The MDBP has been ﬁrst considered in the literature [1] for the 2-dimensional case that the regions are convex

polygons, and a linear time algorithm was given for the 2-dimensional case [3]. For two convex polyhedra in 3-

dimension, a quadratic time algorithm is given [6]. For a ﬁxed dimension d2, Tokuyama [7] recently gave a linear

time algorithm for convex polytopes by employing the multidimensional parametric search.

For the MDBP in any ﬁxed dimension d2, we present a simple approximation algorithm computing a bridge

such that the diameter in the union is at most 2 times of the optimal one. If the regions are convex polytopes,

then the algorithm computes the 2-approximate bridge in linear time. More signiﬁcantly, we extend the MDBP

to the problems of bridging more than two convex regions by geometric networks such as a circuit (tour) like TSP

tour and a spanning tree : the Euclidean Traveling Salesman and Buyers Problem (TSBP), the Geometric Network-

Base Location Problem (GNLP), and the Minimum Diameter Spanning Tree Problem (MDSTP). These problems are

the generalizations of several classic geometry problems, in which the regions are assumed to be points in general.

Given a collection of convex regions in a ﬁxed dimension d2, we can obtain at most 3 2-approximation for these

generalized problems by applying the bridging technique used for the MDBP. For the case that the regions are convex

polytopes, the algorithms run in linear time.

Throughout this paper, we call the center of the minimum enclosing sphere of a region the center of the region.

Since our algorithms use only the centers of regions, they can be applied to the convex regions whose centers are

given or can be computed efﬁciently. For convex polytopes, the set of the centers can be computed in linear time [8].

For a convex region R, we denote by int Rand ∂R, the interior and the boundary of R, respectively. We deﬁne

Rcl Rint R∂R. We also use to denote the sum of lengths of the edges in a path, a tour or a tree. fRx

denotes the point in Rfarthest from x R. If understood in the context, we will use f x instead of fRxif x R.

2 Approximate bridge between two convex regions

The minimum diameter bridge problem (MDBP) is formally deﬁned as follows: Let R1and R2be two disjoint convex

regions. We want to build a bridge pq of R1and R2at the position where the following function is minimized:

max

p∂R1d p p d pq max

q∂R2d qq

where d pq for any point p R3denotes the Euclidean distance between pand q. If f p denotes the point in the

same region as pand farthest from p, then the function can be simpliﬁed as follows: df p p d pq d qf q

We shall refer to the bridge p q that minimizes the function as an optimal bridge of Rand Q.

We compute 2-approximation as follows: We compute the smallest enclosing spheres S1and S2of R1and R2,

respectively.

Let c1be the center and r1be the radius of S1. Similarly, let c2be the center and r2be the radius of S2. Since each re-

gion is convex, it is not difﬁcult to see that the center of a region lies in the closure of the region. We choose the line seg-

ment c1c2as the approximate bridge. Let πf c1c1c2f c2. We prove that the path πcomputed is 2-approximation

Institute of Information and Computing Sciences, Universiteit Utrecht

†Dept. Computer Science, Korea Advanced Inst. Science & Tech.

1

of the optimal path containing the optimal bridge. The length of πis πd f c1c1d c1c2d c2f c2

r1d c1c2r2Let p q be the optimal bridge where pcl R1and qcl R2. Let πf p p q f q be the

optimal path containing pq. Figure 1(b) shows a cross section of R1containing the bridge c1c2and p. We take a

R1R2

c1c2

S1S2

(a)

fc1f c2

(b)

y

c1

S1

p

y

α

p

S1

D

Figure 1: (a) The bridge c1c2, (b) the cross section containing pand the bridge c1c2.

disk Dof radius r1centered at c1and orthogonal to pc1. The disk Ddivides S1into two hemispheres S1(containing

p) and S1as in Figure 1(b). Let ybe any point on ∂D. Since S1is the smallest enclosing sphere, we can prove that

dp y d p f p .

Let pbe the orthogonal projection of pon the segment c1c2. In some case, pmay not be deﬁned within c1c2. If

so, we set pto the closer one of c1and c2. Let y S1be a point in the cross section containing c1c2and psuch that

the angle p c1yis 90 . Since d c1p d c1pand d c1y d c1y, we have that d p y d p y . Combining

the fact that dp y d p f p , we have that d p y d p f p .

Let αbe an angle c1y p . Note that p c1c2should lie inside of or on the sphere S1, so 0 α45 . Then

d p c1d c1ysinαcosαd p y 2d p y. Thus d p c1d c1y2d p f p If yand qon ∂R2

are deﬁned as yand panalogously, then d q c1d c1y2d q f q . Since d c1y r1and d c1y r1,

the following lemma holds.

Lemma 1 d c1p r12d p f p and d c1q r12d q f q .

It is not difﬁcult to show that d p q d p q . As a result, we have that the following lemma.

Lemma 2 π2π.

The following observation immediately comes from the fact that the angle αis no more than 45 ; this will be used to

approximate the bridges among multiple convex regions in Section 3.

Observation 1 d c1p12dp f p and d c2q12dq f q .

Theorem 1 Given two convex regions R1andR2, the bridge connecting two centers of R1and R2gives 2-approximate

diameter of the MDBP.

For the case where regions are polytopes, we can compute two spheres S1and S2in linear time [8], so we have the

following result.

Corollary 1 Given two convex polytopes, we can compute the bridge giving 2-approximate diameter of the MDBP

in linear time.

If we consider a variant of the bridge problem in which the endpoints of a bridge are restricted to be on the boundaries,

then we can still get 2 approximate bridge by applying the same technique stated above and shortening the bridge

computed until both endpoints are on the boundaries. It should be noted that 2 approximation scheme can be easily

extended to 2-approximate the bridge problems in any higher dimension without modiﬁcations.

3 Connecting more than two convex regions

The approximation technique in Section 2 can be extended to several geometric optimization problems. We denote

by MST Xand MT Xaminimum spanning tree and a minimum tour, respectively, for a set Xof points in d2

dimension.

2

3.1 TSBP: Traveling Salesman and Buyers Problems

We will ﬁrst address a variant of the Euclidean TSP (with neighborhood), Traveling Salesman and Buyers Prob-

lem(TSBP): A salesman wants to meet all potential buyers who are scattered in a set of (connected and convex)

regions in ddimension. In a region, buyers are willing to travel to a market-point (a location) in the region where the

salesman visits. The salesman wants to ﬁnd a set of market-points, one for each region, and a tour of these points such

that not only the length of the tour but also the sum of the maximum length between a buyer and a market-point in a

region where the buyer lies is minimized. In this problem, no departure point is speciﬁed, and a set of points is to be

found. This TSBP is a generalization of the Euclidean TSP for the given points, so the TSBP is consequently NP-hard.

In this paper we consider two different versions of the TSBP, in a sense whether buyers take one-way trips or two-way

trips. The TSBP with two-way trips is formally deﬁned as follows:

Problem TSBP2: Given a set of convex regions Rifor i1 2 k, we choose a point piRisuch that

ΠT2p1p2pkMT p1p2pk

k

∑

i12d pif pi

is minimized.

Our strategy is to choose centers of the regions as market-points. We show that the set of the centers of gives

2-approximation to the TSBP2. Let Sp1p2pkbe the set of the optimal points, and Tbe the opti-

mal TSP tour of S. Let C c1c2ckbe the set of the centers of the regions, and Tbe a valid tour of

Chaving the same topology as T. For ease of proof, renumber the regions in certain order (clockwise or coun-

terclockwise) along the tour Tand also give the corresponding directions to the edges of T; we do the same

thing for Tas did for T. Let us consider two consecutive regions Riand Rjin the tour T(consecutive also in

T). Consider two paths πij f cicicjf cjand πij f pipipjfpj. By the Lemma 1, πi j 2πij . Then

T∑k

i12dcif ci∑cicjTπij 2∑pipjTπi j 2ΠT2MT S. Since MT C T ,ΠT2C

MT C∑k

i12dcif ci2ΠT2MT S. As a result, we can select kpoints in regions to give 2-approximation

to the TBLP2 in linear time.

For a ﬁxed dimension dand a small value ε0, the TSP tour of Cwithin 1 ε-factor can be computed in

klogktime [5], where the constant value hidden in the time bound depends on dand ε; as a result, it gives 2 1 ε-

approximation of ΠT2S. Hence, we have the result.

Lemma 3 Given a set of k convex regions in a ﬁxed dimension d 2, the centers of regions guarantee 2-approximate

tour to the TSBP2. If the set C of centers is given, then we can actually construct 2 1 ε-approximate tour to the

TSBP2 in O klogk time for a ﬁxed small ε0.

Problem TSBP1: Given a set of convex regions Rifor i1 2 k, we choose a point piRisuch that

ΠT1p1p2pkMT p1p2pk

k

∑

i11d pif pi

is minimized.

In this problem, we consider the path πi j f cicicj. By Lemma 1 and Observation 1, we can show that πij

2d pifpi12dpjfpj.

ΠT1C T k

∑

i1d cif ci∑

cicjTπij

21

2Tk

∑

i1dpifpi21

2ΠT1S

Lemma 4 Given a set of k convex regions in a ﬁxed dimension d 2, the centers of regions guarantee 212-

approximate tour to the TSBP1. If the set C of k centers is given, then we can actually construct 2121ε-

approximate tour to the TSBP1 in O klogk time for a ﬁxed small ε0.

3.2 GNLP: Geometric Network-Base Location Problems

The optimal bridge problem can be extended to the following Geometric Network-Base Location Problem(GNLP):

3

Problem GNLP: Given a set of regions Rifor i1 2 k, choose a point piRisuch that

ΠNp1p2pkMST p1p2pk

k

∑

i1dpif pi

is minimized.

Lemma 5 Given a set of k convex regions in a ﬁxed dimension d 2, the set C of the centers of regions gives

32approximation to the GNLP. If the set C of k centers is given, then we can actually construct 3 2 1 εapproximate

spanning tree to the GNLP in O klogk time for a ﬁxed small ε0.

3.3 MDSTP: Minimum Diameter Spanning Tree Problems

We are given a set SR1Rkof k(disjoint) convex regions in the Euclidean space, and are interested in ﬁnding

a spanning tree connecting these regions so that the maximum length of shortest path between two points in k

i1Ri

is minimized. The minimum diameter spanning tree problem(MDSTP) is formally deﬁned as follows. Refer to the

paper [2] on minimum diameter spanning trees for the points.

Problem (MDSTP): Given a set Sof regions Rifor i1k, ﬁnd a spanning tree TDfor Ssuch that

max

shortest path πFπ

is minimized, where Fk

i1RiTD. In this problem, a node in the tree corresponds to a region, and an edge in

the tree connects two regions. Note that endpoints of edges in a region do not necessarily share a common point, that

is, there can be at most deg Ridisjoint endpoints, where deg Ridenotes the degree of the node Ri.

A spanning tree with minimum diameter is denoted TD, and the length DTDSis denoted Dfor short. Let TDbe a

spanning tree of centers c1ckof Ri’s. We build the spanning tree TDwith a very simple topology, i.e., monopolar.

A spanning tree of kpoints, k3, is said to be monopolar if there exists a point called the monopole such that all

the remaining points are connected to it. We construct TDas follow: Simply choose one arbitrary center point cmas

monopole and connect all the remaining points to cm. Let πf cicicmcjf cjbe the longest path, that is, πis the

(approximate) diameter of F.πd f cicid cicmd cmcjd cjf cj.

Let πim be the longest of shortest paths connecting Riand Rmand πmj be the the longest of shortest paths connecting

Riand Rmin TD. Now consider the MDBP between Riand Rm, and let pim pmi be the optimal bridge of it, pim cl Ri,

pmi cl Rm. Similarly, let pm j pjm be the optimal bridge of the MDBP between Rmand Rj. Then, by Lemma 1, we

have that

πim d f pim pim d pimpmi d pmi f pmi 1

2d f cicid cicmd cmf cm

πmj d f pm j pmj d pm j pjm d pjm f pjm 1

2d f cmcmd cmcjd cjf cj

Then the diameter Dmax πim πm j . We have π2πim πm j 2 2 max πim πm j 22D, which

simply gives 2 2-approximation to the MDSTP.

Lemma 6 Given a set S of k convex regions and k centers of regions in a ﬁxed dimension d 2, we can construct

22-approximate minimum diameter spanning tree of S in time O k .

Corollary 2 Given a set S of convex polytopes in a ﬁxed dimension d 2, we can construct 2 2-approximate mini-

mum diameter spanning tree of S in linear time.

References

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[2] J.-M. Ho, D. T. Lee, C.-H. Chang, and C. K. Wong. Minimum diameter spanning trees and related problems. SIAM J. Comput., 20:987–997,

1991.

[3] S. K. Kim and C. S. Shin. Computing the optimal bridge between two polygons. Research report tcsc-99-14, HKUST, 1999.

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[8] Emo Welzl. Smallest enclosing disks (balls and ellipsoids). In H. Maurer, editor, New Results and New Trends in Computer Science, volume

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