Talagrand’s Inequality and Locality in Distributed Computing

  • Devdatt P. Dubhashi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1518)


The aim of this paper is to advocate the use of Talagrand’s isoperimetric inequality [10] and an extension of it due to Marton [5, 6] as a tool for the analysis of distributed randomized algorithms that work in the locality paradigm. Two features of the inequality are crucially used in the analysis: first, very refined control on the influence of the underlying variables can be exercised to get signicantly stronger bounds by exploiting the non-uniform and asymmetric conditions required by the inequality (in contrast to previous methods) and second,the method, using an extension of the basic inequality to dependent variables due to Marton [6] succeeds in spite of lack of full independence amongst the underlying variables. This last feature especially makes it a particularly valuable tool in Computer Science contexts where lack of independence is omnipresent. Our contribution is to highlight the special relevance of the method for Computer Science applications by demonstrating its use in the context of a class of distributed computations in the locality paradigm.

We give a high probability analysis of a distributed algorithm for edgecolouring a graph [8]. Apart from its intrinsic interest as a classical combinatorial problem, and as a paradigm example for locality in distributed computing, edge colouring is also useful from a practical standpoint because of its connection to scheduling. In distributed networks or architectures an edge colouring corresponds to a set of data transfers that can be executed in parallel.




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  1. 1.
    Bollabas, B.: Graph Theory: An Introductory Course. Springer-Verlag 1980.Google Scholar
  2. 2.
    Dubhashi, D., Grable, D.A., and Panconesi, A.: Near optimal distributed edge colouring via the nibble method. Theoretical Computer Science 203 (1998), 225–251, a special issue for ESA 95, the 3rd European Symposium on Algorithms.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Grable, D., and Panconesi, A: Near optimal distributed edge colouring in O(log log n) rounds. Random Structures and Algorithms 10, Nr. 3 (1997) 385–405MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Luby, M.: Removing randomness in parallel without a processor penalty. J. Computer and Systems Sciences 47:2 (1993) 250–286.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Marton, K.Bounding l-d distance by informational divergence: A method to prove measure concentration. Annals of Probability 24 (1996) 857–866.Google Scholar
  6. 6.
    Marton, K.: On the measure concentration inequality of Talagrand for dependent random variables, submitted for publication. (1998).Google Scholar
  7. 7.
    Molloy, M. and Reed, B.:A bound on the strong chromatic index of a graph. J. Comb. Theory (B) 69 (1997) 103–109.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Panconesi, A. and Srinivasan, A.: Randomized distributed edge coloring via an extension of the Chernoff-Hoeffding bounds. SIAM J. Computing 26:2 (1997) 350–368.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Spencer, J.: Probabilistic methods in combinatorics. In proceedings of the International Congress of Mathematicians, Zurich, Birkhauser (1995).Google Scholar
  10. 10.
    Talagrand, M.: Concentration of measure and isoperimetric inequalities in product spaces. Publ. math. IHES, 81:2 (1995) 73–205.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Devdatt P. Dubhashi
    • 1
  1. 1.Department of Computer Science and Engg.Indian Institute of TechnologyNew DelhiIndia

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