Short random walks on graphs

We study the short term behavior of random walks on graphs, in particular, the rate at which a random walk discovers new vertices and edges. We prove a conjecture by Linial that the expected time to find ~ distinct vertices is O (Af3). We also prove an upper bound of O(A42) on the time to traverse M edges, and O (MN) on the time to either visit ~ vertices or traverse M edges. For d-regular graphs, we prove bounds of O(N2 log ~) on the time to visit Af vertices, and (2(M + (M2 log M)/d) on the time to traverse M edges. We use these bounds to improve the randomized time-space tradeoff of Broder et al. [7] for undirected S-7 connectivity.

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