Additive Tree O(ρlog n)-Spanners from Tree Breadth ρ
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The tree breadth tb(G) of a connected graph G is the smallest non-negative integer ρ such that G has a tree decomposition whose bags all have radius at most ρ. We show that, given a connected graph G of order n and size m, one can construct in time O(mlog n) an additive tree O( tb(G)log n)-spanner of G, that is, a spanning subtree T of G in which d_T(u,v)≤ d_G(u,v)+O( tb(G)log n) for every two vertices u and v of G. This improves earlier results of Dragan and Köhler (Algorithmica 69 (2014) 884-905), who obtained a multiplicative error of the same order, and of Dragan and Abu-Ata (Theoretical Computer Science 547 (2014) 1-17), who achieved the same additive error with a collection of O(log n) trees.
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