New sparseness results on graph spanners

Abstract

Let G = (V,E) be an n-vertex connected graph with positive edge weights. A subgraph G′ = (V,E′) is a t-spanner of G if for all u, v is a member of V, the weighted distance between u and v in G′ is at most t times the weighted distance between u and v in G. We consider the problem of constructing sparse spanners. Sparseness of spanners is measured by two criteria, the size, defined as the number of edges in the spanner, and the weight, defined as the sum of the edge weights in the spanner. In this paper, we concentrate on constructing spanners of small weight. For an arbitrary positive edge-weighted graph G, for any t > 1, and any ε > 0, we show that a t-spanner of G with weight O(n2+ε/t-1) · wt (MST) can be constructed in polynomial time. We also show that (log2 n)-spanners of weight O(1) · wt(MST) can be constructed. We then consider spanners for complete graphs induced by a set of points in d-dimensional real normed space. The weight of an edge xy is the norm of the xy vector. We show that for these graphs, t-spanners with total weight O(log n) · wt(MST) can be constructed in polynomial time.