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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.
We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved in $tilde{O}(n^{1/2})$ rounds. In contrast, the previous state-of-the-ar
Minimum-weight cut (min-cut) is a basic measure of a networks connectivity strength. While the min-cut can be computed efficiently in the sequential setting [Karger STOC96], there was no efficient way for a distributed network to compute its own min-
Network decomposition is a central tool in distributed graph algorithms. We present two improvements on the state of the art for network decomposition, which thus lead to improvements in the (deterministic and randomized) complexity of several well-s
We present a randomized distributed algorithm that computes a $Delta$-coloring in any non-complete graph with maximum degree $Delta geq 4$ in $O(log Delta) + 2^{O(sqrt{loglog n})}$ rounds, as well as a randomized algorithm that computes a $Delta$-col
We present a simple deterministic distributed algorithm that computes a $(Delta+1)$-vertex coloring in $O(log^2 Delta cdot log n)$ rounds. The algorithm can be implemented with $O(log n)$-bit messages. The algorithm can also be extended to the more g