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.

翻译：给定一个无向图$G$和电导参数$\alpha$，测试$G$是否具有至少$\alpha$的电导或与具有至少$\Omega(\alpha^2)$的电导相距甚远的问题，在有界度图的经典属性测试模型中已被广泛研究。在最近几年，同一问题也在非顺序计算模型（如MPC和分布式CONGEST）中得到探讨。然而，这些模型中的所有算法，如同其经典对应算法一样，都将针对$G$上一组随机游走的某些统计量应用聚合函数作为测试标准。该问题唯一的分布式CONGEST算法（参见\cite{VasudevDistributed}）在无界度图模型中测试底层网络的电导。该算法构建一棵底层网络的根生成树，以便在根节点收集信息，然后将聚合函数应用于这些信息。我们提出一个问题：能否利用分布式计算提供的并行性来避免信息收集，并给出了肯定的答案。我们提出一种新算法，该算法也在$G$上执行一组随机游走，但不在中心节点收集任何统计量。事实上，我们证明对于适当的统计量，每个节点拥有足够的信息自行决定是否接受。给定一个具有$n$个顶点、$m$条边的无向无权图$G$，电导参数$\alpha$和距离参数$\epsilon$，我们的分布式电导测试器在$G$具有至少$\alpha$的电导时接受$G$，在$G$与具有$\Omega(\alpha^2)$的电导相距$\epsilon$时拒绝$G$，并且通信轮数为$O(\log n)$。与\cite{VasudevDistributed}的算法不同，我们的算法不依赖于浪费资源的生成树构建及其根节点的信息累积。