Probabilistic analysis of an enhanced partitioning algorithm for the steiner tree problem in R^d

Abstract

The Geometric Steiner Minimum Tree problem (GSMT) is to connect at minimum cost n given points (called terminals) in d-dimensional Euclidean space. We generalize a GSMT approximation partitioning algorithm by Komlbs and Shing (KS) and analyze its performance under more relaxed conditions. These generalizations have practical applications for multilayer VLSl routing. Whereas KS assumed d = 2, our algorithm works for any dimension d 2 2. Moreover, whereas their analysis assumed the rectilinear norm and a uniform distribution of input points, our analysis holds for any norm on Rd and whenever the terminals are any independent identically distributed random variables taking on values in any bounded subset of R d . Both algorithms depend on a parameter t through which the user can trade off time for solution quality. We evaluate our algorithm in terms of its performance ratio-the ratio of the cost of the Steiner tree computed by the algorithm divided by the cost of a Steiner minimum tree. Applying a probability theorem on subadditive Euclidean functionals by Steele, we prove the following: Under the aforementioned distribution of inputs, the limit as n + co of the supremum of the performance ratio of our algorithm is 1 + O(t-'/d(d-')), almost surely. This result generalizes the corresponding 1 + O(t-'/') bound proven by KS. Along the way, we prove a useful combinatorial lemma about d-dimensional rectangle slicings. We prove that the worst-case time and space complexity of our algorithm is o(n lg ( n / t ) + TMsT(U, u ) + n T s M T ( t ) / t ) and O(SMsT(u, u ) + S s M T ( t ) ) , respectively, where u I n t 2 d + ' ( n / t )~( t )is the number of vertices in the resulting Steiner tree. Here, T S M T ( f ) and SsMT(t) are the time and space required to solve exactly any GSMT problem of size less than t ; T, (n, rn) and S,T(n, m) are the time and space required to find a minimum spanning tree of a graph with n nodes and m edges; and .(t) is the maximum number of Steiner points for any Steiner minimum tree with t terminals. For example, for R2 and the rectilinear norm, the time is O(n Ig(n/t) + n Ig*n + nTSMT(f)/t) and the space is O(n Ig*n + S,(t)). 0 1994 John Wi/ey& Sons, Inc.

Smoothed Analysis of Partitioning Algorithms for Euclidean Functionals

“…Kalpakis and Sherman [17] proposed a partitioning algorithm for the Euclidean minimum Steiner tree problem analogous to Karp's partitioning scheme for Euclidean TSP. The solution produced by their algorithm converges to the optimal value with probability 1 − o (1).…”

“…The solution produced by their algorithm converges to the optimal value with probability 1 − o (1). Also, their algorithm [17] is known to produce near-optimal solutions in practice too [24]. Let us now describe Kalpakis and Sherman's algorithm [17].…”

“…Also, their algorithm [17] is known to produce near-optimal solutions in practice too [24]. Let us now describe Kalpakis and Sherman's algorithm [17]. The running-time of this algorithm is polynomial for the choice of s = n/ log n [8].…”

“…Let KS(X) denote the cost of the Steiner tree computed Kalpakis and Sherman's algorithm [17]. For the analysis of the approximation performance, let ST(X) denote the cost of a minimum Steiner tree for the point set X, and let MST(X) denote the cost of a minimum-length spanning tree of X. Kalpakis and Sherman [17] have shown that As KP for the traveling salesman problem, KS comes with a worst-case approximation ratio of α(n) = O(n).…”

“…For the analysis of the approximation performance, let ST(X) denote the cost of a minimum Steiner tree for the point set X, and let MST(X) denote the cost of a minimum-length spanning tree of X. Kalpakis and Sherman [17] have shown that As KP for the traveling salesman problem, KS comes with a worst-case approximation ratio of α(n) = O(n). The reason is that, for any two points x, y ∈ X, we have x − y ≤ ST(X).…”