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URN: urn:nbn:de:0030-drops-24169
URL: http://drops.dagstuhl.de/opus/volltexte/2010/2416/

Datta, Samir ; Limaye, Nutan ; Nimbhorkar, Prajakta ; Thierauf, Thomas ; Wagner, Fabian

Planar Graph Isomorphism is in Log-Space

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Abstract

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. There is a significant gap between extant lower and upper bounds for planar graphs as well. We bridge the gap for this natural and important special case by presenting an upper bound that matches the known log-space hardness [JKMT03]. In fact, we show the formally stronger result that planar graph canonization is in log-space. This improves the previously known upper bound of AC1 [MR91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to log-space reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in log-space by [DLN08]. This is achieved by using the above decomposition, and by making significant modifications to Lindellís algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph.

BibTeX - Entry

@InProceedings{datta_et_al:DSP:2010:2416,
  author =	{Samir Datta and Nutan Limaye and Prajakta Nimbhorkar and Thomas Thierauf and Fabian Wagner},
  title =	{Planar Graph Isomorphism is in Log-Space},
  booktitle =	{Algebraic Methods in Computational Complexity},
  year =	{2010},
  editor =	{Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans},
  number =	{09421},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2010/2416},
  annote =	{Keywords: Planar Graphs, Graph Isomorphism, Logspace}
}

Keywords: Planar Graphs, Graph Isomorphism, Logspace
Seminar: 09421 - Algebraic Methods in Computational Complexity
Issue date: 2010
Date of publication: 19.01.2010


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