A Distributed Conductance Tester Without Global Information Collection
Tugkan Batu, Amitabh Trehan, Chhaya Trehan
TL;DR
This work tackles the problem of testing graph conductance in the CONGEST model, aiming for a time-efficient, distributed tester that does not rely on global information collection. The authors propose a simple, aggregation-free algorithm that performs $2m^2$ short lazy random walks of length $\ell=\frac{32}{\alpha^2}\log n$ from $\Theta(1/\epsilon)$ degree-weighted sources, in $O(\log n)$ rounds. A spectral-graph-theory framework with sticky vertices governs the analysis: good conductors mix quickly and bad conductors exhibit trapped, sticky regions that concentrate walk endpoints. They prove correctness with a two-sided error: Accept on all vertices for $\alpha$-conductors and Reject on at least one vertex with probability at least $2/3$ when the graph is $\epsilon$-far from any $(\alpha^2/2880)$-conductor, establishing near-optimal round complexity in the CONGEST model.
Abstract
We propose a simple and time-optimal algorithm for property testing a graph for its conductance in the CONGEST model. Our algorithm takes only $O(\log n)$ rounds of communication (which is known to be optimal), and consists of simply running multiple random walks of $O(\log n)$ length from a certain number of random sources, at the end of which nodes can decide if the underlying network is a good conductor or far from it. Unlike previous algorithms, no aggregation is required even with a smaller number of walks. Our main technical contribution involves a tight analysis of this process for which we use spectral graph theory. We introduce and leverage the concept of sticky vertices which are vertices in a graph with low conductance such that short random walks originating from these vertices end in a region around them. The present state-of-the-art distributed CONGEST algorithm for the problem by Fichtenberger and Vasudev [MFCS 2018], runs in $O(\log n)$ rounds using three distinct phases : building a rooted spanning tree (\emph{preprocessing}), running $O(n^{100})$ random walks to generate statistics (\emph{Phase~1}), and then convergecasting to the root to make the decision (\emph{Phase~2}). The whole of our algorithm is, however, similar to their Phase~1 running only $O(m^2) = O(n^4)$ walks. Note that aggregation (using spanning trees) is a popular technique but spanning tree(s) are sensitive to node/edge/root failures, hence, we hope our work points to other more distributed, efficient and robust solutions for suitable problems.