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A measure of parallelization for the lexicographically first maximal subgraph problems

  • Ryuhei Uehara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1335)

Abstract

A maximum directed tree size (MDTS) is defined by the maximum number of the vertices of a directed tree on the directed acyclic graph of a given undirected graph. The MDTS of a graph measures the parallelization for the lexicographically first maximal subgraph (LFMS) problems. That is, the complexity of the problems on a graph family \(\mathcal{G}\)gradually increases as the value measured on each graph in the family grows; (1) if the MDTS of each graph in \(\mathcal{G}\)is O(log k n), the lexicographically first maximal independent set problem on \(\mathcal{G}\)is in NC k +1 and the LFMS problem for π is in NC k+S , where π is a property on graphs such that π is nontrivial, hereditary, and NCs−1 testable; (2) both problems above are P-complete if the MDTS of each graph in \(\mathcal{G}\)is cne. It is worth remarking that the problem to compute the MDTS is in NC2 this is important in the sense that a “measure” means only if measuring the complexity of a problem is easier than solving the problem.

Key words

Analysis of algorithms P-completeness NC algorithms the lexicographically first maximal independent set problem the lexicographically first maximal subgraph problems 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Ryuhei Uehara
    • 1
  1. 1.Center for Information ScienceTokyo Woman's Christian UniversityJapan

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