A Distributed Conductance Tester Without Global Information Collection

Given an undirected graph $G$ and a conductance parameter $\alpha$, the problem of testing whether $G$ has conductance at least $\alpha$ or is far from having conductance at least $\Omega(\alpha^2)$ has been extensively studied for bounded-degree graphs in the classic property testing model. In the last few years, the same problem has also been addressed in non-sequential models of computing such as MPC and distributed CONGEST. However, all the algorithms in these models like their classic counterparts apply an aggregate function over some statistics pertaining to a set of random walks on $G$ as a test criteria. The only distributed CONGEST algorithm for the problem by~\cite{VasudevDistributed} tests conductance of the underlying network in the unbounded degree graph model. Their algorithm builds a rooted spanning tree of the underlying network to collect information at the root and then applies an aggregate function to this information. We ask the question whether the parallelism offered by distributed computing can be exploited to avoid information collection and answer it in affirmative. We propose a new algorithm which also performs a set of random walks on $G$ but does not collect any statistic at a central node. In fact, we show that for an appropriate statistic, each node has sufficient information to decide on its own whether to accept or not. Given an $n$-vertex, $m$-edge undirected, unweighted graph $G$, a conductance parameter $\alpha$, and a distance parameter $\epsilon$, our distributed conductance tester accepts $G$ if $G$ has conductance at least $\alpha$ and rejects $G$ if $G$ is $\epsilon$-far from having conductance $\Omega(\alpha^2)$ and does so in $O(\log n)$ rounds of communication. Unlike the algorithm of \cite{VasudevDistributed}, our algorithm does not rely on the wasteful construction of a spanning tree and information accumulation at its root.