Improved Distributed Expander Decomposition and Nearly Optimal Triangle Enumeration

TL;DR

This paper presents an improved distributed expander decomposition algorithm that enables nearly optimal triangle enumeration in distributed networks, matching theoretical lower bounds and advancing understanding of distributed graph algorithms.

Contribution

The authors develop a new distributed expander decomposition method with better parameters and no edge discard caveats, enabling near-optimal triangle enumeration.

Findings

Achieved an expander decomposition with conductance depending on epsilon and log n.

Developed a triangle enumeration algorithm with O(n^{1/3}) rounds matching lower bounds.

First example of a distributed problem with similar complexity in both CONGEST and CONGESTED CLIQUE models.

Abstract

An $(\epsilon,\phi)$-expander decomposition of a graph $G=(V,E)$ is a clustering of the vertices $V=V_{1}\cup\cdots\cup V_{x}$ 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 $O(n^{2/k}\cdot\text{poly}(1/\phi,\log n))$ rounds for any $\epsilon\in(0,1)$ 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 [PODC'17], we obtain a triangle enumeration algorithm using $\tilde{O}(n^{1/3})$ rounds. This matches the lower bound by Izumi and Le Gall [PODC'17] and Pandurangan, Robinson and Scquizzato [SPAA'18] 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.