Many Random Walks Are Faster Than One
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.
Cite
@article{arxiv.0705.0467,
title = {Many Random Walks Are Faster Than One},
author = {Noga Alon and Chen Avin and Michal Koucky and Gady Kozma and Zvi Lotker and Mark R. Tuttle},
journal= {arXiv preprint arXiv:0705.0467},
year = {2007}
}