An $(epsilon,phi)$-expander decomposition of a graph $G=(V,E)$ is a clustering of the vertices $V=V_{1}cupcdotscup V_{x}$ such that (1) each cluster $V_{i}$ induces subgraph with conductance at least $phi$, and (2) the number of inter-cluster edges is at most $epsilon|E|$. In this paper, we give an improved distributed expander decomposition. Specifically, we construct an $(epsilon,phi)$-expander decomposition with $phi=(epsilon/log n)^{2^{O(k)}}$ in $O(n^{2/k}cdottext{poly}(1/phi,log n))$ rounds for any $epsilonin(0,1)$ and positive integer $k$. For example, a $(0.01,1/text{poly}log n)$-expander decomposition can be computed in $O(n^{gamma})$ rounds, for any arbitrarily small constant $gamma>0$. Previously, the algorithm by Chang, Pettie, and Zhang can construct a $(1/6,1/text{poly}log n)$-expander decomposition using $tilde{O}(n^{1-delta})$ rounds for any $delta>0$, 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 $n^{delta}$. Our algorithm does not have this caveat. By slightly modifying the distributed algorithm for routing on expanders by Ghaffari, Kuhn and Su [PODC17], we obtain a triangle enumeration algorithm using $tilde{O}(n^{1/3})$ rounds. This matches the lower bound by Izumi and Le Gall [PODC17] and Pandurangan, Robinson and Scquizzato [SPAA18] of $tilde{Omega}(n^{1/3})$ 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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