Faster Separators for Shallow Minor-Free Graphs via Dynamic Approximate Distance Oracles

  • Christian Wulff-Nilsen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)

Abstract

Plotkin, Rao, and Smith (SODA’97) showed that any graph with m edges and n vertices that excludes Kh as a depth O(ℓlogn)-minor has a separator of size O(n/ℓ + ℓh2logn) and that such a separator can be found in O(mn/ℓ) time. A time bound of O(m + n2 + ε/ℓ) for any constant ε > 0 was later given (W., FOCS’11) which is an improvement for non-sparse graphs. We give three new algorithms. The first two have the same separator size (the second having a slightly larger dependency on h) and running time O(poly(h)ℓn1 + ε) and \(O(\mbox{poly}(h)(\sqrt\ell n^{1+\epsilon} + n^{2+\epsilon}/\ell^{3/2}))\), respectively. The former is significantly faster than previous bounds for small h and ℓ. Our third algorithm has running time \(O(\mbox{poly}(h)\sqrt\ell n^{1+\epsilon})\). It finds a separator of size \(O(n/\ell) + \tilde O(\mbox{poly}(h)\ell\sqrt n)\) which is no worse than previous bounds when h is fixed and \(\ell = \tilde O(n^{1/4})\). A main tool in obtaining our results is a decremental approximate distance oracle of Roditty and Zwick.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Christian Wulff-Nilsen
    • 1
  1. 1.Department of Computer ScienceUniversity of CopenhagenDenmark

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