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Distributed Discovery of Large Near-Cliques

Zvika Brakerski
Boaz Patt-Shamir
Abstract

Given an undirected graph and 0ϵ10\le\epsilon\le1, a set of nodes is called ϵ\epsilon-near clique if all but an ϵ\epsilon fraction of the pairs of nodes in the set have a link between them. In this paper we present a fast synchronous network algorithm that uses small messages and finds a near-clique. Specifically, we present a constant-time algorithm that finds, with constant probability of success, a linear size ϵ\epsilon-near clique if there exists an ϵ3\epsilon^3-near clique of linear size in the graph. The algorithm uses messages of O(logn)O(\log n) bits. The failure probability can be reduced to nΩ(1)n^{-\Omega(1)} in O(logn)O(\log n) time, and the algorithm also works if the graph contains a clique of size Ω(n/logαlogn)\Omega(n/\log^{\alpha}\log n) for some α(0,1)\alpha \in (0,1).

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