Multiplicative approximations of random walk transition probabilities

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 ɛ-approximations have been developed in the literature, non 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 l uv ≥ t can be approximated within a 1 ± ɛ 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 = ˜ Ω(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±ɛ factor. One instantiation of these guarantees is as follows: any almost regular directed graph can be represented in Õ(ln3/2+ɛ) space such that all probabilities larger than n −10 can be approximated within a (1 ± ɛ) factor as long as l ≥ 40/ɛ 2. Moreover, we show ho