Information Processing Letters 74 (2000) 253–255

A lower bound for approximating thegeometric minimum weight matching

Gautam Dasa,∗, Michiel Smidb,1

a Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USAb Department of Computer Science, University of Magdeburg, D-39106 Magdeburg, Germany

Received 30 January 1999Communicated by F. Dehne

Abstract

Given a setS of 2n points inRd , a perfect matching ofS is a set ofn edges such that each point ofS is a vertex of exactly oneedge. The weight of a perfect matching is the sum of the Euclidean lengths of all edges. Rao and Smith have recently shownthat there is a constantr > 1, that only depends on the dimensiond, such that a perfect matching whose weight is less thanor equal tor times the weight of a minimum weight perfect matching can be computed in O(n logn) time. We show that thisalgorithm is optimal in the algebraic computation tree model. 2000 Published by Elsevier Science B.V. All rights reserved.

Keywords:Minimum weight matching; Lower bounds; Computational geometry

1. Introduction

Let S be a set of 2n points inRd , whered > 1 isa (small) constant. We consider sets of edges havingthe points ofS as vertices. Such a setM is called aperfect matchingof S, if each point ofS is a vertexof exactly one edge inM. In other words, a perfectmatching is a partition ofS into n subsets of size two.Theweight wt(M) of a perfect matchingM is definedas the sum of the Euclidean lengths of all edges inM.The minimum weight matching MWM(S) of S is theperfect matching ofS that has minimum weight.

The best-known algorithm that computes a mini-mum weight matching is due to Vaidya [6]; its run-ning time is bounded by O(n5/2(logn)4) if d = 2, andO(n3−1/cd ) if d > 2, for some constantc > 1.

∗ Corresponding author. Email: dasg@msci.memphis.edu.1 Email: michiel@isg.cs.uni-magdeburg.de.

Rao and Smith [5] considered the easier problemof approximating the minimum weight matching. Letr > 1 be a real number. A perfect matchingM of S iscalled anr-approximate MWM, if

wt(M)6 r ·wt(MWM(S)

).

Rao and Smith have shown that anr-approximateMWM, for

r = c · exp(8 · 21−1/(d−1)√d ),

wherec is a constant, can be computed in O(n logn)time.

In this paper, we will show that Rao and Smith’salgorithm is optimal in the algebraic computation treemodel. That is, we will prove the following theorem.

Theorem 1. Let d > 1 be an integer. Every algebraiccomputation tree algorithm that, when given a set of2n points inRd and a real numberr > 1, computes

0020-0190/00/$ – see front matter 2000 Published by Elsevier Science B.V. All rights reserved.PII: S0020-0190(00)00062-4

254 G. Das, M. Smid / Information Processing Letters 74 (2000) 253–255

an r-approximate MWM, has worst-case running time�(n logn).

Note that this lower bound even holds for dimensiond = 1. Moreover, it holds foranyapproximation factorr, even one that depends onn. For example, computinga 22n-approximateMWM has worst-case running time�(n logn).

Our proof of Theorem 1 uses Ben-Or’s theorem [1].The proof technique that we use is related to thoseused in Chen et al. [2], and Das et al. [3].

2. The proof of Theorem 1

In this section, we prove Theorem 1 for the casewhend = 1. Clearly, this implies an�(n logn) lowerbound for any dimensiond > 1.

We assume that the reader is familiar with thealgebraic computation tree model. (See Ben-Or [1],and Preparata and Shamos [4].) Our lower bound willuse the following well-known result.

Theorem 2 (Ben-Or [1]). Let V be any set inRnand let B be any algorithm that belongs to thealgebraic computation tree model and that acceptsV .Let #V denote the number of connected componentsof V . Then the worst-case running time ofB is�(log#V − n).

Let A be an arbitrary algebraic computation treealgorithm that, when given as input a sequence of 2n

real numbersx1, x2, . . . , x2n and a real numberr > 1,computes anr-approximateMWM for the xi ’s. Wewill use Theorem 2 to prove thatA has worst-caserunning time�(n logn).

Note that algorithmA solves a computation prob-lem. In order to apply Theorem 2, we need a decisionproblem, i.e., a problem having values YES and NO.Below, we will define such a decision problem; in fact,we will define the corresponding subsetV ⊆ R2n ofYES-inputs.

Fix the integern and the real numberr > 1. Wedefine an algorithmB that takes as input any sequenceof 2n real numbers. On input sequencex1, x2, . . . , x2n,algorithmB does the following.

Step 1. Check if xi = i, for all i, 16 i 6 n. If not,output NO, and terminate. Otherwise, go to Step 2.

Step 2. Let ε := 1/(2rn). Run algorithmA on theinputx1, x2, . . . , x2n, r. LetM be ther-approximateMWM that is computed byA. Check if all edgesof M have lengthε. If so, output YES. Otherwise,output NO.

LetTA(n) andTB(n) denote the worst-case runningtimes of algorithmsA andB, respectively. Then, it isclear that

TB(n)6 TA(n)+ cn,for some constantc. Therefore, if we can show that

TB(n)=�(n logn),

then it follows immediately that

TA(n)=�(n logn).

LetV be the set of all points(x1, x2, . . . , x2n) inR2n

that are accepted by algorithmB. We will show thatV has at leastn! connected components. As a result,Theorem 2 implies the�(n logn) lower bound on therunning time ofB.

Lemma 1. Let π be any permutation of1,2, . . . , n,and letε = 1/(2rn). Then the point

P := (1,2, . . . , n,π(1)+ ε,π(2)+ ε, . . . , π(n)+ ε)is contained in the setV .

Proof. LetM∗ be theMWM of the elements 1,2, . . . ,n,π(1) + ε,π(2) + ε, . . . , π(n) + ε. Since 0< ε <

1/2, it is easy to see thatM∗ consists of the edges(i, i + ε), 16 i 6 n.

Consider what happens when algorithmB is run oninputP . Clearly, this input “survives” Step 1. LetMbe ther-approximateMWM that is computed in Step2. We will show below thatM =M∗. Having provedthis, it follows that algorithmB accepts the inputP ,i.e.,P ∈ V .

Suppose thatM 6=M∗. ThenM contains an edgeof the form (i, j), (i, j + ε), or (i + ε, j + ε), forsome integersi and j , i 6= j . (We consider edges tobe undirected.) Since 0< ε < 1/2, it follows that thisedge and, hence, also the matchingM, has weightmore than 1/2. Clearly, the optimal matchingM∗has weightnε = 1/(2r). Therefore,wt(M) > 1/2=r · wt(M∗). This is a contradiction, becauseM is anr-approximateMWM. 2

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Lemma 2. The setV has at leastn! connectedcomponents.

Proof. Let π andρ be two different permutations of1,2, . . . , n. Consider the points

P := (1,2, . . . , n,π(1)+ ε,π(2)+ ε, . . . , π(n)+ ε)and

R := (1,2, . . . , n, ρ(1)+ ε,ρ(2)+ ε, . . . , ρ(n)+ ε)in R2n. By Lemma 1, both these points are containedin the setV . We will show that they are in differentconnected components ofV .

Let C be an arbitrary curve inR2n that connectsP andR. Sinceπ and ρ are distinct permutations,there are indicesi and j such thatπ(i) < π(j) andρ(i) > ρ(j). Hence, the curveC contains a pointQ,

Q= (p1,p2, . . . , pn, q1, q2, . . . , qn),

such thatqi = qj . We claim thatQ is not containedin V . This will prove thatP andR are in differentconnected components ofV .

To prove the claim, first assume that there is anindexk, 16 k 6 n, such thatpk 6= k. Then pointQ isrejected by algorithmB and, therefore,Q /∈ V . Hence,we may assume that

Q= (1,2, . . . , n, q1, q2, . . . , qn).

Let us see what happens if we run algorithmB oninputQ. This input “survives” Step 1. LetM be ther-approximateMWM that is constructed in Step 2.

If M contains an edge of the form(pk,p`)= (k, `),then algorithmB rejects pointQ, because such anedge has length more thanε. Hence, we may assumethat each edge ofM has the form(pk, q`) = (k, q`).Let a and b be the integers, 16 a, b 6 n, such that(a, qi) and(b, qj ) are edges ofM. Since(i) a andb are distinct integers,(ii) qi = qj , and(iii) 0 < ε < 1/2,one of these two edges must have length more thanε.Hence, algorithmB rejects pointQ. This completesthe proof. 2

References

[1] M. Ben-Or, Lower bounds for algebraic computation trees, in:Proc. 15th Annual ACM Symp. on Theory of Computing, 1983,pp. 80–86.

[2] D.Z. Chen, G. Das, M. Smid, Lower bounds for computinggeometric spanners and approximate shortest paths, in: Proc. 8thCanadian Conf. Computational Geometry, 1996, pp. 155–160.

[3] G. Das, S. Kapoor, M. Smid, On the complexity of approximat-ing Euclidean traveling salesman tours and minimum spanningtrees, Algorithmica 19 (1997) 447–460.

[4] F.P. Preparata, M.I. Shamos, Computational Geometry: AnIntroduction, Springer, New York, 1985.

[5] S.B. Rao, W.D. Smith, Approximating geometrical graphs via“spanners” and “banyans”, in: Proc. 30th Annual ACM Symp.on Theory of Computing, 1998, pp. 540–550.

[6] P.M. Vaidya, Geometry helps in matching, SIAM J. Comput. 18(1989) 1201–1225.