The birthday problem and Markov chain Monte Carlo

We study the problem of generating a sample from the stationary distribution nof a Markov chain, given a method to simulate the chain. We give an napproximation algorithm for the case of a random walk on a regular graph with n nvertices that runs in expected time O^*( sqrt{n} x L^2-mixing time). This is nclose to the best possible, since sqrt{n} is a lower bound on t..