David A. Grable

Fast Distributed Algorithms for Brooks-Vizing Colorings

Let G be a Δ-regular graph with n vertices and girth at least 4 such that Δ » log n. We give very simple, randomized, distributed algorithms for vertex coloring G with Δ/k colors in O(k + log n) communication rounds, where k = O(log Δ). The algorithm may fail or exceed the above running time, but the probability that this happens is o(1), a quantity that goes to zero as n grows. The probabilistic analysis relies on a powerful generalization of Azumas martingale inequality that we dub the Method of Bounded Variances.

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.

A Large Deviation Inequality for Functions of Independent, Multi-Way Choices

Often when analysing randomized algorithms, especially parallel or distributed algorithms, one is called upon to show that some function of many independent choices is tightly concentrated about its expected value. For example, the algorithm might colour the vertices of a given graph with two colours and one would wish to show that, with high probability, very nearly half of all edges are monochromatic.The classic result of Chernoff [3] gives such a large deviation result when the function is a sum of independent indicator random variables. The results of Hoeffding [5] and Azuma [2] give similar results for functions which can be expressed as martingales with a bounded difference property. Roughly speaking, this means that each individual choice has a bounded effect on the value of the function. McDiarmid [9] nicely summarized these results and gave a host of applications. Expressed a little differently, his main result is as follows.

Nearly-perfect hypergraph packing is in NC

Answering a question of Rodl and Thoma, we show that the Nibble Algorithm for finding a collection of disjoint edges covering almost all vertices in an almost regular, uniform hypergraph with negligible pair degrees can be derandomized and parallelized to run in polylog time on polynomially many parallel processors. In other words, the nearly-perfect packing problem on this class of hypergraphs is in the complexity class NC.

Forbidden induced partial orders

Abstract For each finite partial order P , we consider the size of the set Forb ind n (P) of partial orders on n labelled points containing no induced copy of P . We show that | Forb ind n ( P )| = 2 o ( n 2 ) unless P has height at least 3, in which case | Forb ind n ( P )| = 2 n 2 /4+ o ( n 2 ) . We show that | Forb ind n ( P )| n ! c n for some constant c if and only if P is either an antichain or one of ten small partial orders. Between these extremes, we consider the question of which P have | Forb ind n ( P )| = n O ( n ) .

Random methods in design theory: A survey

In this article we review the results in Design Theory which have been obtained through the use of random methods, emphasizing the techniques employed, and pose a host of open problems which may be amenable to this approach.