Pseudo-Gilbert–Steiner Trees

D. Trietsch,1 J. F. Weng2

1 Department of Management Science and Information Systems, The University of Auckland,Auckland, New Zealand

2 Department of Mathematics and Statistics, The University of Melbourne, Victoria 3052,Australia

Received March 1994; accepted 16 June 1998

Abstract: The Gilbert network problem is a generalization of the Steiner minimal tree problem derivedby adding flow-dependent weights to the edges. In this paper, we define a special class of minimumGilbert networks, called pseudo-Gilbert–Steiner trees, and we show that it can be constructed by Gilbert’sgeneralization of Melzak’s method. Besides, a counterexample, a pseudo-Gilbert–Steiner tree, is con-structed to show that the condition given by Trietsch was misquoted by Weng. © 1999 John Wiley & Sons,Inc. Networks 33: 175–178, 1999

1. INTRODUCTION

A Gilbert network[1] is a network connecting a setA of ngiven pointsa1, a2, . . . , an (referred to asterminals) anda (potentially empty) setSof additional pointss1, s2, . . . inthe Euclidean plane. The network is designed to accommo-date a given set of bilateral flowsqij between pointsai andaj. Let f(q) be the cost per length unit of each edge in thenetwork, which is a nondecreasing function of the total flowassigned to the edge such that for anyq, r . 0,

f~q! 1 f~r ! . f~q 1 r ! $ max$ f~q!,f~r !%,

f~q! . f~0! 5 0.

The Gilbert network problem is to find the networkNwhich minimizes the total cost of the network:

Cost~N! 5 Oe[E

l e f~qe!,

whereE is the set of edges;l e, the length ofe; andqe, thetotal flow assigned to edgee. For simplicity,f(qe) is calledtheweightof edgee. Since this problem is a generalizationof the Steiner problem, the additional pointssi will also becalledSteiner points.However, a minimum Gilbert networkmay have cycles, which make the Gilbert problem muchmore complicated than the original Steiner problem. Gilbertproved a property of minimum Gilbert networks: Iff(q) isconcave, then there is no split routing between any twoterminalsai andaj, that is, the flow is through only one paththough the existence of cycles may provide more than onepath betweenai andaj. This property significantly reducesthe number of possible assignments of the flows in mini-mum Gilbert networks. Another contribution of Gilbert tothe problem was the generalization of the well-known Mel-zak construction of Steiner trees to find the optimal locationof degree 3 Steiner points in minimum Gilbert networks.Correspondence to:J. F. Weng

© 1999 John Wiley & Sons, Inc. CCC 0028-3045/99/030175-04

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We will refer to the construction asGilbert–Melzak’s con-struction.Suppose a Steiner points in a minimum Gilbertnetwork has three edgese1, e2, e3, with weightsw1, w2,w3, respectively. Then, the optimum location ofs satisfiesthe condition for mechanical equilibrium of weights. Let theangles ats oppositee1, e2, e3 be a1, a2, a3, respectively.Then, these angles are the external angles of the trianglewhose edge lengths are the weights (hence, referred to as theweight triangle) (Fig. 1).

In algebraic language, for example,a1, the angle be-tweene2 ande3, is

arccosSw12 2 w2

2 2 w32

2w2w3D .

By this formula, in [2], the first author of the present articleproposed an obviously sufficient condition to ensure that thedegree of each Steiner point in the minimum Gilbert net-work is three:

f 2~q! 1 f 2~r ! . f 2~q 1 r !, ; q,r . 0. (1)

In fact, this condition guarantees that the angles betweenadjacent edges incident to any Steiner point must be obtuse.Therefore, it is impossible to accommodate more than threeof them. The major benefit of the condition is that it rulesout any networks where two edges intersect each other.Under the condition, if two edges intersect, we can alwaysinsert a Steiner point to shorten the network. In this shortpaper, we study some special Gilbert networks based on theknown results above. First, we separate a class of Gilbertnetworks, referred to as pseudo-Gilbert–Steiner trees, whichcontain cycles but can be treated as trees using Gilbert–Melzak’s construction to locate Steiner points. Then, weconstruct a counterexample, a pseudo-Gilbert–Steiner tree,to show that condition (1) does not guarantee that theminimum Gilbert network is a tree as cited by the secondauthor of the present paper [6].

2. PSEUDO-GILBERT–STEINER TREES

In general, a Gilbert networkN for A may have cycles.Suppose there is a cycleC in N whose vertices are allSteiner points of degree three as shown in Figure 2.

We can shrink the cycle by parallelly shifting the edgeson the cycle. (The parallel-shift technique was used forinterconnecting existing networks in Trietsch [3].) Assumethats is a Steiner point whose incident edgese2, e3 are onC and whose third edge ise1. Let the weights of these edgesbe wi, i 5 1, 2, 3, respectively. Suppose thats moves tos9 when the cycle shrinks. Let the line throughs and parallelto e3 intersecte2 at p. Clearly, the external angles ofDss9pare just equal to those of the weight triangle. Therefore, bythe similarity of the two triangles,

uss9uuspu 5

w2

w1, i.e., uss9u z w1 5 uspu z w2.

Considering such equalities at all Steiner points onC, weconclude that the cost ofN does not change. Obviously, theexpansion of the cycle by a parallel shift does not changethe cost ofN either. The limit of a parallel shift is that eithera Steiner point of degree more than three appears or aSteiner point onC collapses into a terminal. A Gilbertnetwork without cycles consisting of Steiner points is de-fined to bebasic [2].

Theorem 1. If the flow function f(q) satisfies condition (1),then there is a minimum Gilbert network on A that is basic.

Proof. Suppose thatN is a minimum Gilbert networkand contains a cycle consisting of Steiner points of degree 3by (1). If the parallel shift causes the appearance of Steinerpoints of degree more than three, then condition (1) iscontradicted. Hence, we can shrink or expand all cyclesuntil each of them contains at least one terminal. Theresulting network is a basic minimum Gilbert network.■

By this theorem, a class of Gilbert network can beseparated. For example, look at the point setA in Figure3(1) which consists of two adjacent squares sharing a sidea1a4. Suppose that the flowsqij are qij 5 1 if j 5 i1 1(mod 6) andq13 5 q24 5 q46 5 q51 5 1; otherwise,

Fig. 2.

Fig. 1.

176 TRIETSCH AND WENG

qij 5 0. (The numbers on the edges in the figure are theflows through the edges.) If the flow function satisfiescondition (1), sayf(q) 5 q(0.52e) (0 , e # 0.5), then theminimum Gilbert networkN on A should be as depicted inthe figure. Once the topology and then the flows are deter-mined, the Steiner points still can be located by Gilbert–Melzak’s construction even though there are cycles in thenetwork. The technique is splitting certain points to open thecycles.

In Figure 3(2), the network becomes a tree (consisting oftwo full subtrees) aftera1 is split to two points on the samelocation. The process of Gilbert–Melzak’s construction forthe left full subtree is shown in the figure. Such Gilbertnetworks that might contain cycles but can be constructedusing the Gilbert–Melzak method are calledpseudo-Gil-bert–Steiner treesfollowing Trietsch [2]. Note that Gilbert–Steiner trees are also pseudo-Gilbert–Steiner trees.

Theorem 2. Suppose that the flow function satisfies condi-tion (1); then, there exists a minimum pseudo-Gilbert–Steiner tree on A.

Proof. By Theorem 1, there is a basic minimum GilbertnetworkN onA. Hence, we can gradually open all cycles bysplitting a terminal on the cycles. Hence,N is a pseudo-Gilbert–Steiner tree. ■

3. A COUNTEREXAMPLE

Clearly, condition (1) only involves the angles at Steinerpoints and does nothing with the global structure of aminimum Gilbert network. Hence, it cannot guaranteethat the minimum Gilbert network is a tree as misquotedin [6]. Below, we construct a counterexample. LetP5 a0a1 . . . a4 be a regular pentagon and the flow beq(i , j )5 1 if j 5 i (mod5); otherwise,q(i , j ) 5 0. Suppose thatthe flow function isf(q) 5 q(0.52e) (0 , e # 0.5), whichsatisfies condition (1). We claim that the minimum GilbertnetworkN cannot be a tree. Suppose to the contrary thatNis a tree. Then,N has a full Steiner topology by condition(1), and, moreover, by [4], only full Steiner topologies need

Fig. 4.

Fig. 3.

PSEUDO-GILBERT–STEINER TREES 177

to be considered. It is easy to see that up to symmetry thereis only one full Steiner topology as shown in Figure 4.

Since the flow on each edge equals two,N is a Steinertree. Hence, we easily know that the length of the treeapproximates 4.630 [5] and the cost is the length times=2,that is, 6.548. However, the cost of the closed polygonPobviously approximates 5. Hence,T is not the minimumGilbert network.

REFERENCES

[1] E.N. Gilbert, Minimum cost communication networks, BellSyst Tech J 46 (1967), 2209–2227.

[2] D. Trietsch, Minimal Euclidean networks with flow-depen-dent cost—the generalized Steiner case, discussion paper#655 (monograph), The Center for Mathematical Studies inEconomics and Management Science, Northwestern Uni-versity, 1985.

[3] D. Trietsch, Interconnecting networks in the plane: theSteiner case, Networks 20 (1990), 93–108.

[4] D. Trietsch and F.K. Hwang, An improved algorithm forSteiner trees, SIAM J Appl Math 50 (1990), 244–263.

[5] J.F. Weng, Steiner minimal trees on vertices of regularpolygons, Acta Math Appl Sin 8 (1985), 129–141. (Chi-nese)

[6] J.F. Weng, Degenerate Gilbert–Steiner trees, Networks 22(1992), 335–348.

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