| Abstract: | Motivated by applications in Markov estimation and distributed computing, we define the blanket time of an undirected graph G to be the expected time for a random walk to hit every vertex of G within a constant factor of the number of times predicted by the stationary distribution. Thus the blanket time is, essentially, the number of steps required of a random walk in order that the observed distribution reflect the stationary distribution. We provide substantial evidence for the following conjecture: that the blanket time of a graph never exceeds the cover time by more than a constant factor. In other words, at the cost of a multiplicative constant one can hit every vertex often instead of merely once. We prove the conjecture in the case where the cover time and maximum hitting time differ by a logarithmic factor. This case includes almost all graphs, as well as most “natural” graphs: the hypercube, k-dimensional lattices for k ≥ 2, balanced k-ary trees, and expanders. We further prove the conjecture for perhaps the most natural graphs not falling in the above case: paths and cycles. Finally, we prove the conjecture in the case of independent stochastic processes. © 1996 John Wiley & Sons, Inc. Random Struct. Alg., 9 , 403–411 (1996) |
