Skip to main content
SpringerLink
Log in

The “log rank” conjecture for modular communication complexity

  • Conference paper
  • First Online:
STACS 96 (STACS 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1046))

Included in the following conference series:

Abstract

The “log rank” conjecture consists of the question how exactly the deterministic communication complexity of a problem can be determined in terms of algebraic invariants of the communication matrix of this problem. In the following, we answer this question in the context of modular communication complexity. We show that the modular communication complexity can be characterised precisely in terms of the logarithm of a certain rigidity function of the communication matrix. Thus, we are able to determine precisely the modular communication complexity of several problems, such as, e.g., set disjointness, comparability, and undirected graph connectivity. From the obtained bounds for the modular communication complexity, we can conclude exponential lower bounds on the size of depth two circuits having arbitary symmetric gates at the bottom level and a MODm-gate at the top.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Borodin, A. Razborov, R. Smolensky, On Lower Bounds for Read-k-Times Branching Programs, Computational Complexity 3(1993), pp. 1–18.

    Article  Google Scholar 

  2. C. Damm, M. Krause, Ch. Meinel, St. Waack, Separating Counting Communication Complexity Classes, in: Proc. 9th STACS, Lecture Notes in Computer Science 577, Springer Verlag 1992, pp. 281–293.

    Google Scholar 

  3. A. Hajnal, W. Maass, G. Turan, On the Communication Complexity of Graph Problems, in: Proc. 20th ACM STOC 1988, pp. 186–191.

    Google Scholar 

  4. M. Krause, St. Waack, Variation Ranks of Communication Matrices and Lower Bounds for Depth Two Circuits Having Symmetric Gates with Unbounded Fanin, in: Proc. 32nd IEEE FOCS 1991, pp. 777–782.

    Google Scholar 

  5. S. Lokam, Spectral Methods for Matrix Rigidity with Applications to Size-Depth Tradeoffs and Communication Complexity, in: Proc. 36th IEEE FOCS 1995.

    Google Scholar 

  6. L. Lovasz, Communication complexity: A survey, in: Paths, Flows and VLSI-Layouts, Springer-Verlag 1990, pp. 235–266.

    Google Scholar 

  7. L. Lovasz, M. Saks, Communication Complexity and Combinatorial Lattice Theory, Journal of Computer and System Sciences 47(1993), pp. 322–349.

    Google Scholar 

  8. Ch. Meinel, St. Waack, The M” obius Function, Variation Ranks, and Θ(n) — Bounds on the Modular Communication Complexity of the Undirected Graph Connectivity Problem, ECCC-Report TR94-022.

    Google Scholar 

  9. K. Mehlhorn, E. M. Schmidt, Las Vegas is Better than Determinism in VLSI and Distributed Computing, in: Proc. 14th ACM STOC 1982, pp. 330–337.

    Google Scholar 

  10. P. Pudlak, Large Communication in Constant Depth Circuits, Combinatorica 14(2)(1994), pp. 203–216.

    Article  Google Scholar 

  11. A.A. Razborov, Lower bounds for the size of circuits of bounded depth with basis {∧, ⊕}, Journ. Math. Zametki 41(1987), pp. 598–607.

    Google Scholar 

  12. A.A. Razborov, On Rigid Matrices, Manuscript.

    Google Scholar 

  13. R. Raz, B. Spieker, On the “log rank”— Conjecture in Communication Complexity, in: Proc. 34th IEEE FOCS 1993, pp. 168–176.

    Google Scholar 

  14. L. Skyum, L. V. Valiant, A Complexity Theory Based on Boolean Algebra, in: Proc. 22th IEEE FOCS, pp. 244–253.

    Google Scholar 

  15. V. Shoup, R. Smolensky, Lower Bounds for Polynomial Evaluation and Interpolation, in: Proc. 32nd IEEE FOCS (1991), pp. 378–383.

    Google Scholar 

  16. R. Smolensky, Algebraic Methods in the Theory of Lower Bounds for Boolean Circuit Complexity, in: Proc. 19th ACM STOC (1987), pp. 77–82.

    Google Scholar 

  17. A. Wigderson, The Complexity of Graph Connectivity, TR 92-19, Leibnitz Center for Research in Computer Science, Institute of Computer Science, Hebrew University, Jerusalem.

    Google Scholar 

  18. A. C.-C. Yao, Some Complexity Questions Related to Distributed Computing, in: Proc. 11st ACM STOC 1979, pp. 209–213.

    Google Scholar 

  19. A. C.-C. Yao, On ACC and Threshold Circuits, in: Proc. 31st IEEE FOCS 1990, pp. 619–627.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich IV - Informatik, Universität Trier, D-54286, Trier

    Christoph Meinel

  2. Inst. für Num. und Angew. Mathematik, Georg-August-Univ., D-37083, Göttingen

    Stephan Waack

Authors
  1. Christoph Meinel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephan Waack
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Claude Puech Rüdiger Reischuk

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Meinel, C., Waack, S. (1996). The “log rank” conjecture for modular communication complexity. In: Puech, C., Reischuk, R. (eds) STACS 96. STACS 1996. Lecture Notes in Computer Science, vol 1046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60922-9_50

Download citation

  • DOI: https://doi.org/10.1007/3-540-60922-9_50

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60922-3

  • Online ISBN: 978-3-540-49723-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation