Improved Compression of the Okamura-Seymour Metric

Shay Mozes, Nathan Wallheimer, Oren Weimann

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Let G = (V, E) be an undirected unweighted planar graph. Let S = {s1, . . ., sk} be the vertices of some face in G and let T ⊆ V be an arbitrary set of vertices. The Okamura-Seymour metric compression problem asks to compactly encode the S-to-T distances. Consider a vector storing the distances from an arbitrary vertex v to all vertices S = {s1, . . ., sk} in their cyclic order. The pattern of v is obtained by taking the difference between every pair of consecutive values of this vector. In STOC’19, Li and Parter used a VC-dimension argument to show that in planar graphs, the number of distinct patterns, denoted p#, is only O(k3). This resulted in a simple Õ(min{k4 + |T|, k · |T|}) space compression of the Okamura-Seymour metric. We give an alternative proof of the p# = O(k3) bound that exploits planarity beyond the VC-dimension argument. Namely, our proof relies on cut-cycle duality, as well as on the fact that distances among vertices of S are bounded by k. Our method implies the following: (1) An Õ(p# + k + |T|) space compression of the Okamura-Seymour metric, thus improving the compression of Li and Parter to Õ(min{k3 + |T|, k · |T|}). (2) An optimal Õ(k + |T|) space compression of the Okamura-Seymour metric, in the case where the vertices of T induce a connected component in G. (3) A tight bound of p# = Θ(k2) for the family of Halin graphs, whereas the VC-dimension argument is limited to showing p# = O(k3).

Original languageEnglish
Title of host publication33rd International Symposium on Algorithms and Computation, ISAAC 2022
EditorsSang Won Bae, Heejin Park
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772587
DOIs
StatePublished - 1 Dec 2022
Event33rd International Symposium on Algorithms and Computation, ISAAC 2022 - Virtual, Online, Korea, Republic of
Duration: 19 Dec 202221 Dec 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume248
ISSN (Print)1868-8969

Conference

Conference33rd International Symposium on Algorithms and Computation, ISAAC 2022
Country/TerritoryKorea, Republic of
CityVirtual, Online
Period19/12/2221/12/22

Bibliographical note

Publisher Copyright:
© Shay Mozes, Nathan Wallheimer, and Oren Weimann.

Keywords

  • Shortest paths
  • distance oracles
  • metric compression
  • planar graphs

ASJC Scopus subject areas

  • Software

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