Many Random Walks Are Faster Than One

TL;DR

This paper investigates how multiple parallel random walks can significantly reduce the time needed to cover all nodes in a graph, revealing linear and exponential speed-ups depending on the graph structure.

Contribution

It introduces a new analysis of parallel random walks' cover times, showing potential for substantial speed-ups and improving bounds in related probabilistic algorithms.

Findings

Parallel walks can achieve linear speed-up in cover time.

Exponential speed-up is possible on some graphs.

Natural graphs may only allow logarithmic speed-up.

Abstract

We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time - the expected time required to visit every node in a graph at least once - and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s-t connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.