Multiplicative approximations of random walk transition probabilities

• Authors: Michael Kapralov;Rina Panigrahy
• Venue: APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
• Year: 2011

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 Puvl ≥ 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(n2/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 = Ω(sn2/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 Ω(ln3/2+ε) space such that all probabilities larger than n-10 can be approximated within a (1 ± ε) factor as long as l ≥ 40/ε2. 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 Puvl using a notion of bicriteria approximation based on approximate distance oracles of Thorup and Zwick [STOC'01].