Improved Distributed Expander Decomposition and Nearly Optimal Triangle Enumeration
share
An (ϵ,ϕ)-expander decomposition of a graph G=(V,E) is a clustering of the vertices V=V_1∪...∪ V_x such that (1) each cluster V_i 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 in the CONGEST model. Specifically, we construct an (ϵ,ϕ)-expander decomposition with ϕ=(ϵ/ n)^2^O(k) in O(n^2/k·poly(1/ϕ, n)) rounds for any ϵ∈(0,1) and positive integer k. For example, a (0.01,1/poly n)-expander decomposition can be computed in O(n^γ) rounds, for any constant γ>0. Previously, the algorithm by Chang, Pettie, and Zhang can construct a (1/6,1/poly n)-expander decomposition using Õ(n^1-δ) rounds for any δ>0, with a caveat that the algorithm is allowed to throw away a set of edges which forms a subgraph with arboricity at most n^δ. Our algorithm does not have this caveat. By modifying the distributed algorithm for routing on expanders by Ghaffari, Kuhn and Su [PODC'17], we obtain a triangle enumeration algorithm using Õ(n^1/3) rounds. This matches the lower bound by Izumi and Le Gall [PODC'17] and Pandurangan, Robinson and Scquizzato [SPAA'18] of Ω̃(n^1/3) which holds even in the CONGESTED-CLIQUE model. To the best of our knowledge, this provides the first non-trivial example for a 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.
READ FULL TEXT