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 tspanner of G if for all u, v E 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 c >0, we show that a t-spanner of G with weight O(n* ) . wt(MST) can be constructed in polynomial time. We also show that (logz 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 Zy is the norm of the~y vector. We show that for these graphs, t-spanners with total weight O(log n) . wt (MST) can be constructed in polynomial time.

(1 + εΒ) -spanner constructions for general graphs

“…The best known bound for the stretch i s 2 ; 1 1]. This bound is very close to the best possible in view of the lower bounds shown in 20,1,4], by which for every odd 3 there exist (in nitely many) n-vertex graphs G = ( V E) f o r w h i c h e v ery ( ; 2)-spanner requires (n 1+4=3 ) edges. In fact, for 3, 5 and 9-spanners, even better lower bounds are known 22].…”

“…Formally, g i v en an unweighted graph G = ( V E), we s a y that the subgraph H (where E(H) E) is an -spanner of G if dH(u w) dG(u w) for every u w 2 V , w h e r e d G 0(u v) denotes the distance between two vertices u and v in G 0 , namely, the minimum length of a path in G 0 connecting them. There exists a well-understood tradeo between the size of a spanner (namely, t h e n umber of edges it uses) and its stretch 20,1,4]. Generally speaking, for an integer parameter , a stretch o f O( ) can be guaranteed by a spanner using O(n 1+1= ) edges.…”

Mixed Spanning Trees in Theory and Practice

“…is a n-spanner of G if for all u, v € V, the distance dG>{u,v) between u and v in G' is at most K times the distance do{u,v) between u and v in G. There has been much recent activity on spanner graphs, especially spanners for complete Euclidean graphs; Chandra et al 7 contains a good early bibliography. Originally work centered on reducing the size of the spanner, the number of edges in the spanner graph.…”

“…Chandra et al proved that any set of edges having the weak gap property has bounded weight. Theorem 2 (Chandra et al 7 ) If E has the weak gap property, w(E) = 0(L log n).…”

A Geometric Spanner of Segments

“…Extensive research has been done on this structure and a number of interesting results have been obtained. [2][3][4][6][7][8][9][10][11][12][13][14]17 Almost all previous results consider the case in which the objects are points and seek to minimize the construction time, size, weight, maximum degree of vertex, diameter, or combinations of them.…”