Near-optimal compression for the planar graph metric

Amir Abboud, Pawel Gawrychowski, Shay Mozes, Oren Weimann

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


The Planar Graph Metric Compression Problem is to compactly encode the distances among k nodes in a planar graph of size n. Two naïve solutions are to store the graph using O(n) bits, or to explicitly store the distance matrix with O(k2 log n) bits. The only lower bounds are from the seminal work of Gavoille, Peleg, Prennes, and Raz [SODA'01], who rule out compressions into a polynomially smaller number of bits, for weighted planar graphs, but leave a large gap for unweighted planar graphs. For example, when k/p n, the upper bound is O(n) and their constructions imply an (n3/4) lower bound. This gap is directly related to other major open questions in labeling schemes, dynamic algorithms, and compact routing. Our main result is a new compression of the planar graph metric into ~O(min(k2; p k/n)) bits, which is optimal up to log factors. Our data structure circumvents an (k2) lower bound of Krauthgamer, Nguyen, and Zondiner [SIDMA'14] for compression using minors, and the lower bound of Gavoille et al. for compression of weighted planar graphs. This is an unexpected and decisive proof that weights can make planar graphs inherently more complex. Moreover, we design a new Subset Distance Oracle for planar graphs with ~O( √k/n) space, and ~O(n3/4) query time. Our work carries strong messages to related fields. In particular, the famous O(n1/2) vs. (n1/3) gap for distance labeling schemes in planar graphs cannot be resolved with the current lower bound techniques. On the positive side, we introduce the powerful tool of unit-monge to planar graph algorithms.

Original languageEnglish
Title of host publication29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
EditorsArtur Czumaj
PublisherAssociation for Computing Machinery
Number of pages20
ISBN (Electronic)9781611975031
StatePublished - 2018
Event29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States
Duration: 7 Jan 201810 Jan 2018

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
Country/TerritoryUnited States
CityNew Orleans

Bibliographical note

Publisher Copyright:
© Copyright 2018 by SIAM.

ASJC Scopus subject areas

  • Software
  • General Mathematics


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