Multipath spanners

  • Authors:

  • Cyril Gavoille;Quentin Godfroy;Laurent Viennot

  • Affiliations:

  • LaBRI, University of Bordeaux;LaBRI, University of Bordeaux;LIAFA, INRIA, University Paris 7

  • Venue:
  • SIROCCO'10 Proceedings of the 17th international conference on Structural Information and Communication Complexity
  • Year:
  • 2010

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Abstract

This paper concerns graph spanners that approximate multipaths between pair of vertices of an undirected graphs with n vertices. Classically, a spanner H of stretch s for a graph G is a spanning subgraph such that the distance in H between any two vertices is at most s times the distance in G. We study in this paper spanners that approximate short cycles, and more generally p edge-disjoint paths with p1, between any pair of vertices. For every unweighted graph G, we construct a 2-multipath 3-spanner of O(n3/2) edges. In other words, for any two vertices u,v of G, the length of the shortest cycle (with no edge replication) traversing u,v in the spanner is at most thrice the length of the shortest one in G. This construction is shown to be optimal in term of stretch and of size. In a second construction, we produce a 2-multipath (2,8)-spanner of O(n3/2) edges, i.e., the length of the shortest cycle traversing any two vertices have length at most twice the shortest length in G plus eight. For arbitrary p, we observe that, for each integer k≥1, every weighted graph has a p-multipath p(2k−1)-spanner with O(pn1+1/k) edges, leaving open the question whether, with similar size, the stretch of the spanner can be reduced to 2k−1 for all p1.