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New Pairwise Spanners

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Abstract

Let G=(V,E) be an undirected unweighted graph on n vertices. A subgraph H of G is called an (all-pairs) purely additive spanner with stretch β if for every (u,v)∈V×V, d i s t H (u,v)≤d i s t G (u,v) + β. The problem of computing sparse spanners with small stretch β is well-studied. Here we consider the following variant: we are given \(\mathcal {P} \subseteq V \times V\) and we seek a sparse subgraph H where d i s t H (u,v)≤d i s t G (u,v) + β for each \((u,v) \in \mathcal {P}\). That is, distances for pairs outside \(\mathcal {P}\) need not be well-approximated in H. Such a subgraph is called a pairwise spanner with additive stretch β and our goal is to construct such subgraphs that are sparser than all-pairs spanners with the same stretch. We show sparse pairwise spanners with additive stretch 4 and with additive stretch 6. We also consider the following special cases: \(\mathcal {P} = S \times V\) and \(\mathcal {P} = S \times T\), where S⊆V and T⊆V, and show sparser pairwise spanners for these cases.

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Notes

  1. The problem of constructing a \(\mathcal {P}\)-spanner in G=(V,E) of size \(O(n|\mathcal {P}|^{1/4})\) and additive stretch k can be reduced to the problem of constructing a \(\mathcal {P}^{\prime }\)-spanner of size \(O(n|\mathcal {P}|^{1/4})\) and additive stretch k in a bipartite graph G ′=(V∪V ′,E ′) where V ′={v ′:v∈V}, E ′={(u,v ′),(v,u ′):(u,v)∈E}, and \(\mathcal {P}^{\prime } = \{(a,b^{\prime }): (a,b) \in \mathcal {P}\ \text {and}\ \delta _{G}(a,b)\ \mathrm {is\ odd}\} \cup \{(a,b): (a,b) \in \mathcal {P}\ \text {and}\ \delta _{G}(a,b)\ \mathrm {is\ even}\}\). Since additive stretch in G ′ is an even number, the correct stretch k is even.

  2. This construction was suggested by the anonymous reviewer.

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Acknowledgments

I am grateful to the reviewers for their very helpful comments and suggestions.

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  1. Mumbai, India

    Telikepalli Kavitha

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Correspondence to Telikepalli Kavitha.

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A preliminary version of this paper appeared in STACS 2015 [25].

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Kavitha, T. New Pairwise Spanners. Theory Comput Syst 61, 1011–1036 (2017). https://doi.org/10.1007/s00224-016-9736-7

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