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Relations Between Communication Complexity, Linear Arrangements, and Computational Complexity

  • Jürgen Forster
  • Matthias Krause
  • Satyanarayana V. Lokam
  • Rustam Mubarakzjanov
  • Niels Schmitt
  • Hans Ulrich Simon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2245)

Abstract

Recently, Forster [7] proved a new lower bound on probabilistic communication complexity in terms of the operator norm of the communication matrix. In this paper, we want to exploit the various relations between communication complexity of distributed Boolean functions, geometric questions related to half space representations of these functions, and the computational complexity of these functions in various restricted models of computation. In order to widen the range of applicability of Forster’s bound, we start with the derivation of a generalized lower bound. We present a concrete family of distributed Boolean functions where the generalized bound leads to a linear lower bound on the probabilistic communication complexity (and thus to an exponential lower bound on the number of Euclidean dimensions needed for a successful half space representation), whereas the old bound fails. We move on to a geometric characterization of the well known communication complexity class C-PP in terms of half space representations achieving a large margin. Our characterization hints to a close connection between the bounded error model of probabilistic communication complexity and the area of large margin classification. In the final section of the paper, we describe how our techniques can be used to prove exponential lower bounds on the size of depth-2 threshold circuits (with still some technical restrictions). Similar results can be obtained for read-k-times randomized ordered binary decision diagram and related models.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Jürgen Forster
    • 1
  • Matthias Krause
    • 2
  • Satyanarayana V. Lokam
    • 3
  • Rustam Mubarakzjanov
    • 4
  • Niels Schmitt
    • 1
  • Hans Ulrich Simon
    • 1
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Institut für InformatikUniversität MannheimMannheimGermany
  3. 3.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  4. 4.Fakultät für InformatikUniversität TrierTrierGermany

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