Distributed Discovery of Large Near-Cliques

摘要

Given an undirected graph and 0 ≤ ∈ ≤ 1, a set of nodes is called ∈-near clique if all but an ∈ 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 ∈-near clique if there exists an ∈~3-near clique of linear size in the graph. The algorithm uses messages of O(log n) bits. The failure probability can be reduced to n~(-Ω)~(1) in O(log n) time factor, and the algorithm also works if the graph contains a clique of size Ω(n/ log~α log n) for some α ∈ (0, 1). Our approach is based on a new idea of adapting property testing algorithms to the distributed setting.