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Bridging Convex Regions and Related Problems

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Abstract

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(R) and R, the interior and the boundary of R, respectively. We define R = cl(R) = int(R) [ R. We also use j j to denote the sum of lengths of the edges in a path, a tour or a tree. f R (x) denotes the point in R farthest from x 2 R. If understood in the context, we will use f (x) instead of f R (x) if x 2 R. 2 Approximate bridge between two convex regions The minimum diameter bridge problem (MDBP) is formally defined as follows: Let R 1 and R 2 be two disjoint convex regions. We want to build a bridge pq of R 1 and R 2 at the position where the following function is minimized: max p 2R 1 d(p 0 p) +d(pq)+ max q 2R 2 d(qq 0 ); where d(pq) for any point p 2 R 3 denotes the Euclidean distance between p and q. If f (p) denotes the point in the same region as p and farthest from p, then the function can be simplified as follows: d( f (p)p) + d(pq) + d(q f (q)): We shall refer to the bridge (p;q) that minimizes the function as an optimal bridge of R and Q . We compute p 2-approximation as follows: We compute the smallest enclosing spheres S 1 and S 2 of R 1 and R 2 , respectively. Let c 1 be the center and r 1 be the radius of S 1 . Similarly, let c 2 be the center and r 2 be the radius of S 2 . Since each region 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 segment c 1 c 2 as the approximate bridge. Let p = f (c 1 )c...
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 RR. 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
pR1d p p d pq max
qR2d 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
Tk
i12dcif cicicjTπij 2pipjTπi j 2ΠT2MT S. Since MT C T ,ΠT2C
MT Ck
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 xed 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.
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4
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