Devdatt P. Dubhashi

Near-optimal, distributed edge colouring via the nibble method

Abstract We give a distributed randomized algorithm for graph edge colouring. Let G be a Δ-regular graph with n nodes. Here we prove: • • If e>0 is fixed and Δ⪢log n, the algorithm almost always colours G with (1 + e)Δ colours in time O(log n). • • If s>0 is fixed, there exists a positive constant k such that if Δ⪢logk n, the algorithm almost always colours G with Δ + Δ/logs n colours in time O(log n + logs n log log n). By “almost always” we mean that the algorithm may either use more than the claimed number of colours or run longer than the claimed time, but that the probability that either of these sorts of failure occurs can be made arbitrarily close to 0. The algorithm is based on the nibble method, a probabilistic strategy introduced by Vojtěch Rodl. The analysis makes use of a powerful large deviation inequality for functions of independent random variables.

Near-Optimal Distributed Edge Coloring

We give a distributed randomized algorithm to edge color a network. Given a graph G with n nodes and maximum degree Δ, the algorithm, For any fixed λ>0, colours G with (1+λ)Δ colours in time O(log n). For any fixed positive integer s, colours G with Δ+Δ/(log Δ)s=(1+o(1))Δ colours in time O(log n+logsΔ loglog Δ).

A lower bound for area-universal graphs

Abstract We stablish a lower bound on the efficiency of area-universal circuits. The area Au of every graph H that can host any graph G of area (at most) A with dilation d, and congestion c⩽√A/log log A satisfies the tradeoff A u =Ω(A log A/(c 2 log(2d))) . In particular, if Au=O(A) then max (c,d)=Ω(√log A/log log A) .

The fourth moment in Luby's distribution

Luby (1988) proposed a way to derandomize randomized computations which is based on the construction of a small probability space whose elements are 3-wise independent. In this paper we prove some new properties of Lubys space. More precisely, we analyze the fourth moment and prove an interesting technical property which helps to understand better Lubys distribution. As an application, we study the behavior of random edge cuts in a weighted graph.

Probabilistic) Recurrence Realtions Revisited

The performance attributes of a broad class of randomised algorithms can be described by a recurrence relation of the form T(x)=a(x)+T(H(x)), where a is a function and H(x) is a random variable. For instance, T(x) may describe the running time of such an algorithm on a problem of size x. Then T(x) is a random variable, whose distribution depends on the distribution of H(x). To give high probability guarantees on the performance of such randomised algorithms, it suffices to obtain bounds on the tail of the distribution of T(x). Karp derived tight bounds on this tail distribution, when the distribution of H(x) satisfies certain restrictions. However, his proof is quite difficult to understand. In this paper, we derive bounds similar to Karps using standard tools from elementary probability theory, such as Markovs inequality, stochastic dominance and a variant of Chernoff bounds applicable to unbounded variables. Further, we extend the results, showing that similar bounds hold under weaker restrictions on H(x). As an application, we derive performance bounds for an interesting class of algorithms that was outside the scope of the previous results.

Quantifier elimination in p-adic fields

We present a tutorial survey of quantifier elimination and decision procedures in p-adic fields. The p-adic fields are studied in the (so-called) P n -formalism of Angus Macintyre, for which motivation in provided through a rich body of analogies with real-cloned fields. Quantifier elimination and decidion procedures are described proceeding via a Cylindrical Algebraic Decomposition of affine p-adic space. effective complexity analyses are also provided

Concentration of Measure for the Analysis of Randomized Algorithms: Bibliography

Randomized algorithms have become a central part of the algorithms curriculum based on their increasingly widespread use in modern applications. This book presents a coherent and unified treatment of probabilistic techniques for obtaining high- probability estimates on the performance of randomized algorithms. It covers the basic tool kit from the Chernoff-Hoeffding (CH) bounds to more sophisticated techniques like Martingales and isoperimetric inequalities, as well as some recent developments like Talagrands inequality, transportation cost inequalities, and log-Sobolev inequalities. Along the way, variations on the basic theme are examined, such as CH bounds in dependent settings. The authors emphasize comparative study of the different methods, highlighting respective strengths and weaknesses in concrete example applications. The exposition is tailored to discrete settings sufficient for the analysis of algorithms, avoiding unnecessary measure-theoretic details, thus making the book accessible to computer scientists as well as probabilists and discrete mathematicians.