Deterministic Random Walks for Rapidly Mixing Chains

The rotor-router model, also known as the Propp machine, is a deterministic process analogous to a random walk on a graph. Instead of distributing tokens to randomly chosen neighbors, the rotor-router deterministically serves the neighbors in a fixed order. This paper is concerned with a generalized model, functional-router model. While the rotor-router is an analogy with a random walk consisting of only rational transition probabilities using parallel edges, the functional-router imitates a random walk containing irrational transition probabilities. For the functional-router model, we investigate the discrepancy on a single vertex between the number of tokens in the functional-router model and the expected number of tokens in a random walk. We present an upper bound of the discrepancy in terms of the mixing rate of the corresponding transition matrix. Using the result, we give a polynomial time deterministic sampler for particular #P-complete problems, such as 0-1 knapsack solutions, linear extensions, matchings, etc., for which rapidly mixing chains are known; Our deterministic algorithm provides samples from a “distribution” with a point-wise distance at most ε from the target distribution, in time polynomial in the input size and ε.