An approximation algorithm for the tree t-spanner problem on unweighted graphs via generalized chordal graphs

Authors:
Feodor F. Dragan;Ekkehard Köhler
Affiliations:
Algorithmic Research Laboratory, Department of Computer Science, Kent State University, Kent, OH;Mathematisches Institut, Brandenburgische Technische Universität Cottbus, Cottbus, Germany
Venue:
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Year:
2011

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Abstract

A spanning tree T of a graph G is called a tree t-spanner of G if the distance between every pair of vertices in T is at most t times their distance in G. In this paper, we present an algorithm which constructs for an n-vertex m-edge unweighted graph G: (1) a tree (2⌊log2 n⌋)-spanner in O(m log n) time, if G is a chordal graph; (2) a tree (2ρ⌊log2 n⌋)-spanner in O(mnlog2 n) time or a tree (12ρ⌊log2 n⌋)-spanner in O(m log n) time, if G is a graph admitting a Robertson-Seymour's tree-decomposition with bags of radius at most ρ in G; and (3) a tree (2⌈t/2⌉⌊log2 n⌋)-spanner in O(mn log2 n) time or a tree (6t⌊log2 n⌋)-spanner in O(m log n) time, if G is an arbitrary graph admitting a tree t-spanner. For the latter result we use a new necessary condition for a graph to have a tree t-spanner: if a graph G has a tree t-spanner, then G admits a Robertson-Seymour's tree-decomposition with bags of radius at most ⌈t/2⌉ in G.