文章基本信息

标题：On Pairwise Spanners
本地全文：下载
作者：Marek Cygan ; Fabrizio Grandoni ; Telikepalli Kavitha 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2013
卷号：20
页码：209-220
DOI：10.4230/LIPIcs.STACS.2013.209
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：Given an undirected n-node unweighted graph G = (V, E), a spanner with stretch function f(.) is a subgraph H \subseteq G such that, if two nodes are at distance d in G, then they are at distance at most f(d) in H. Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function. In this paper we study pairwise spanners, where we require to approximate the u-v distance only for pairs (u,v) in a given set P \subseteq V x V. Such P-spanners were studied before [Coppersmith,Elkin'05] only in the special case that f(.) is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers). Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same P) and of the best known spanners (with the same f(.)). In more detail, for arbitrary P, we show that there exists a P-spanner of size O(n(|P|log n)^{1/4}) with f(d) = d + 4 log n. Alternatively, for any epsislon > 0, there exists a P-spanner of size O(n|P|^{1/4} sqrt{(log n) / epsilon}) with f(d) = (1 + epsilon)d + 4. We also consider the relevant special case that there is a critical set of nodes S \subseteq V, and we wish to approximate either the distances within nodes in S or from nodes in S to any other node. We show that there exists an (S x S)-spanner of size O(n sqrt{|S|}) with f(d) = d + 2, and an (S x V)-spanner of size O(n sqrt{|S| log n}) with f(d) = d + 2 log n. All the mentioned pairwise spanners can be constructed in polynomial time.
关键词：Undirected graphs; shortest paths; additive spanners; distance preservers