Many Random Walks Are Faster Than One

作者：Alon Noga^*; Avin Chen; Koucky Michal; Kozma Gady; Lotker Zvi; Tuttle Mark R

来源：Combinatorics Probability & Computing, 2011, 20(4): 481-502.

DOI：10.1017/S0963548311000125

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.

• 出版日期2011-7