Talagrand’s Inequality and Locality in Distributed Computing

Abstract

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.

Work partly done while at BRICS, Department of Computer Science, University of Aarhus, Denmark. Partially supported by the ESPRIT Long Term Research program of the EU under contract No. 20244 (ALCOM-IT)