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Testing Graph Isomorphism in Parallel by Playing a Game

  • Martin Grohe
  • Oleg Verbitsky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4051)

Abstract

Our starting point is the observation that if graphs in a class C have low descriptive complexity, then the isomorphism problem for C is solvable by a fast parallel algorithm. More precisely, we prove that if every graph in C is definable in a finite-variable first order logic with counting quantifiers within logarithmic quantifier depth, then Graph Isomorphism for C is in \({\rm TC^1}\subseteq\mbox{\rm NC\ensuremath{^{2}}}\). If no counting quantifiers are needed, then Graph Isomorphism for C is even in AC1. The definability conditions can be checked by designing a winning strategy for suitable Ehrenfeucht-Fraïssé games with a logarithmic number of rounds. The parallel isomorphism algorithm this approach yields is a simple combinatorial algorithm known as the Weisfeiler-Lehman (WL) algorithm.

Using this approach, we prove that isomorphism of graphs of bounded treewidth is testable in TC1, answering an open question from [9]. Furthermore, we obtain an AC1 algorithm for testing isomorphism of rotation systems (combinatorial specifications of graph embeddings). The AC1 upper bound was known before, but the fact that this bound can be achieved by the simple WL algorithm is new. Combined with other known results, it also yields a new AC1 isomorphism algorithm for planar graphs.

Keywords

Planar Graph Rotation System Order Logic Input Graph Tree Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Martin Grohe
    • 1
  • Oleg Verbitsky
    • 1
  1. 1.Institut für InformatikHumboldt Universität zu BerlinBerlinGermany

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