Many Random Walks Are Faster Than One

This paper is about searching a graph with a random walk. We show that several random walks running (or walking) in parallel can often search a graph much faster than a single random walk on its own. We measure the cover time, which is the expected amount of time required to visit every node in the graph, and we show that for a large collection of interesting graphs running many random walks in parallel yields a nearly linear speed up in the cover time. We demonstrate that an exponential speed up is sometimes possible, and that there are natural examples of graphs that allow only a logarithmic speed up.