Multiplicative Approximations of Random Walk Transition Probabilities

Abstract

We study the space and time complexity of approximating distributions of l-step random walks in simple (possibly directed) graphs G. While very efficient algorithms for obtaining additive ε-approximations have been developed in the literature, no non-trivial results with multiplicative guarantees are known, and obtaining such approximations is the main focus of this paper. Specifically, we ask the following question: given a bound S on the space used, what is the minimum threshold t > 0 such that l-step transition probabilities for all pairs u, v ∈ V such that $P_{uv}^l\geq t$ can be approximated within a 1±ε factor? How fast can an approximation be obtained?

We show that the following surprising behavior occurs. When the bound on the space is S = o(n ²/d), where d is the minimum out-degree of G, no approximation can be achieved for non-trivial values of the threshold t. However, if an extra factor of s space is allowed, i.e. $S=\tilde \Omega(sn^2/d)$ space, then the threshold t is exponentially small in the length of the walk l and even very small transition probabilities can be approximated up to a 1±ε factor. One instantiation of these guarantees is as follows: any almost regular directed graph can be represented in $\tilde O(l n^{3/2+{\epsilon}})$ space such that all probabilities larger than n ^− 10 can be approximated within a (1±ε) factor as long as l ≥ 40/ε ². Moreover, we show how to estimate of such probabilities faster than matrix multiplication time.

For undirected graphs, we also give small space oracles for estimating $P^l_{uv}$ using a notion of bicriteria approximation based on approximate distance oracles of Thorup and Zwick [STOC’01].