Improved Distributed Expander Decomposition and Nearly Optimal Triangle Enumeration

An(ε,φ)-expander decomposition of a graph G=(V,E) is a clustering of the vertices V=V1∪…∪ Vx such that (1) each cluster Vi induces subgraph with conductance at least φ, and (2) the number of inter-cluster edges is at most ε|E|. In this paper, we give an improved distributed expander decomposition, and obtain a nearly optimal distributed triangle enumeration algorithm in the CONGEST model. Specifically, we construct an (ε,φ)-expander decomposition with φ=(ε/log n)2 O(k) in O(n2/k ⋅ poly (1/φ, log n))rounds for any ε ∈(0,1) and positive integer k. For example, a (1/no(1), 1/no(1))-expander decomposition only requires O(no(1)) rounds to compute, which is optimal up to subpolynomial factors, and a (0.01,1/poly log n)-expander decomposition can be computed in O(nγ) rounds, for any arbitrarily small constant γ > 0. Previously, the algorithm by Chang, Pettie, and Zhang can construct a (1/6,1/poly log n)-expander decomposition using Õ (n1-δ) rounds for any δ > 0, with a caveat that the algorithm is allowed to throw away a set of edges into an extra part which form a subgraph with arboricity at most nδ. 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 Õ(n1/3) rounds. This matches the lower bound by Izumi and LeGall [PODC'17] and Pandurangan, Robinson and Scquizzato [SPAA'18] of Ø(n1/3) which holds even in the CONGESTED-CLIQUE model. To the best of our knowledge, 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.