2019/04/17 by Yi‐Jun Chang, Thatchaphol Saranurak, Chang, Yi-Jun +1 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #Distributed #FOS: Computer and information sciences #Markov Chains and Monte Carlo Methods #Parallel #and Cluster Computing (cs.DC)
paper · pdf · doi:10.48550/arxiv.1904.08037
openalex publication_date 2019/04/17 · openalex created_date 2019/04/25 · openalex updated_date 2026/07/28
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. Specifically, we construct an (ε,ϕ)-expander decomposition with ϕ=(ε/log n)^2O(k) in O(n2/k\cdotpoly(1/ϕ,log n)) rounds for any ε∈(0,1) and positive integer k. For example, a (0.01,1/polylog 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/polylog n)-expander decomposition using O(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 forms 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 O(n1/3) rounds. This matches the lower bound by Izumi and Le Gall [PODC'17] and Pandurangan, Robinson and Scquizzato [SPAA'18] of Ω(n1/3) 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.