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Improved Distributed Expander Decomposition and Nearly Optimal Triangle Enumeration

Abstract

An (ϵ,ϕ)(\epsilon,\phi)-expander decomposition of a graph G=(V,E)G=(V,E) is a clustering of the vertices V=V1VxV=V_{1}\cup\cdots\cup V_{x} such that (1) each cluster ViV_{i} induces subgraph with conductance at least ϕ\phi, and (2) the number of inter-cluster edges is at most ϵE\epsilon|E|. In this paper, we give an improved distributed expander decomposition. Specifically, we construct an (ϵ,ϕ)(\epsilon,\phi)-expander decomposition with ϕ=(ϵ/logn)2O(k)\phi=(\epsilon/\log n)^{2^{O(k)}} in O(n2/kpoly(1/ϕ,logn))O(n^{2/k}\cdot\text{poly}(1/\phi,\log n)) rounds for any ϵ(0,1)\epsilon\in(0,1) and positive integer kk. For example, a (0.01,1/polylogn)(0.01,1/\text{poly}\log n)-expander decomposition can be computed in O(nγ)O(n^{\gamma}) rounds, for any arbitrarily small constant γ>0\gamma>0. Previously, the algorithm by Chang, Pettie, and Zhang can construct a (1/6,1/polylogn)(1/6,1/\text{poly}\log n)-expander decomposition using O~(n1δ)\tilde{O}(n^{1-\delta}) rounds for any δ>0\delta>0, with a caveat that the algorithm is allowed to throw away a set of edges into an extra part which forms a subgraph with arboricity at most nδn^{\delta}. Our algorithm does not have this caveat. By slightly modifying the distributed algorithm for routing on expanders by Ghaffari, Kuhn and Su [PODC'17], we obtain a triangle enumeration algorithm using O~(n1/3)\tilde{O}(n^{1/3}) rounds. This matches the lower bound by Izumi and Le Gall [PODC'17] and Pandurangan, Robinson and Scquizzato [SPAA'18] of Ω~(n1/3)\tilde{\Omega}(n^{1/3}) which holds even in the CONGESTED CLIQUE model. This provides the first non-trivial example for a distributed problem that has essentially the same complexity (up to a polylogarithmic factor) in both CONGEST and CONGESTED CLIQUE. The key technique in our proof is the first distributed approximation algorithm for finding a low conductance cut that is as balanced as possible. Previous distributed sparse cut algorithms do not have this nearly most balanced guarantee.

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