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An Approximation Algorithm for the Tree t-Spanner Problem on Unweighted Graphs via Generalized Chordal Graphs

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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:

  • a tree (2⌊log2 n⌋)-spanner in O(mlogn) time, if G is a chordal graph (i.e., every induced cycle of G has length 3);

  • a tree (2ρ⌊log2 n⌋)-spanner in O(mnlog2 n) time or a tree (12ρ⌊log2 n⌋)-spanner in O(mlogn) time, if G is a graph admitting a Robertson-Seymour’s tree-decomposition with bags of radius at most ρ in G; and

  • a tree (2⌈t/2⌉⌊log2 n⌋)-spanner in O(mnlog2 n) time or a tree (6t⌊log2 n⌋)-spanner in O(mlogn) 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⌉ and diameter at most t in G.

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Notes

  1. When G is an unweighted graph, t can be assumed to be an integer.

  2. We realize that it is perfectly possible that the authors of [27] did not try to optimize the constants in their analysis, and it may be the case that a more careful analysis of their algorithm may lead to an improved leading constant.

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Acknowledgements

We would like to thank reviewers for many useful comments.

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Authors and Affiliations

  1. Algorithmic Research Laboratory, Department of Computer Science, Kent State University, Kent, OH, 44242, USA

    Feodor F. Dragan

  2. Mathematisches Institut, Brandenburgische Technische Universität Cottbus, 03013, Cottbus, Germany

    Ekkehard Köhler

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Correspondence to Feodor F. Dragan.

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Dragan, F.F., Köhler, E. An Approximation Algorithm for the Tree t-Spanner Problem on Unweighted Graphs via Generalized Chordal Graphs. Algorithmica 69, 884–905 (2014). https://doi.org/10.1007/s00453-013-9765-4

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