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Register a new userHigh-precision Estimation of Random Walks in Small Space
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Added by
Jack Murtagh
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We provide a deterministic $tilde{O}(log N)$-space algorithm for estimating random walk probabilities on undirected graphs, and more generally Eulerian directed graphs, to within inverse polynomial additive error ($epsilon=1/mathrm{poly}(N)$) where $N$ is the length of the input. Previously, this problem was known to be solvable by a randomized algorithm using space $O(log N)$ (following Aleliunas et al., FOCS 79) and by a deterministic algorithm using space $O(log^{3/2} N)$ (Saks and Zhou, FOCS 95 and JCSS 99), both of which held for arbitrary directed graphs but had not been improved even for undirected graphs. We also give improvements on the space complexity of both of these previous algorithms for non-Eulerian directed graphs when the error is negligible ($epsilon=1/N^{omega(1)}$), generalizing what Hoza and Zuckerman (FOCS 18) recently showed for the special case of distinguishing whether a random walk probability is $0$ or greater than $epsilon$. We achieve these results by giving new reductions between powering Eulerian random-walk matrices and inverting Eulerian Laplacian matrices, providing a new notion of spectral approximation for Eulerian graphs that is preserved under powering, and giving the first deterministic $tilde{O}(log N)$-space algorithm for inverting Eulerian Laplacian matrices. The latter algorithm builds on the work of Murtagh et al. (FOCS 17) that gave a deterministic $tilde{O}(log N)$-space algorithm for inverting undirected Laplacian matrices, and the work of Cohen et al. (FOCS 19) that gave a randomized $tilde{O}(N)$-time algorithm for inverting Eulerian Laplacian matrices. A running theme throughout these contributions is an analysis of cycle-lifted graphs, where we take a graph and lift it to a new graph whose adjacency matrix is the tensor product of the original adjacency matrix and a directed cycle (or variants of one).
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We give a deterministic, nearly logarithmic-space algorithm that given an undirected graph $G$, a positive integer $r$, and a set $S$ of vertices, approximates the conductance of $S$ in the $r$-step random walk on $G$ to within a factor of $1+epsilon$, where $epsilon>0$ is an arbitrarily small constant. More generally, our algorithm computes an $epsilon$-spectral approximation to the normalized Laplacian of the $r$-step walk. Our algorithm combines the derandomized square graph operation (Rozenman and Vadhan, 2005), which we recently used for solving Laplacian systems in nearly logarithmic space (Murtagh, Reingold, Sidford, and Vadhan, 2017), with ideas from (Cheng, Cheng, Liu, Peng, and Teng, 2015), which gave an algorithm that is time-efficient (while ours is space-efficient) and randomized (while ours is deterministic) for the case of even $r$ (while ours works for all $r$). Along the way, we provide some new results that generalize technical machinery and yield improvements over previous work. First, we obtain a nearly linear-time randomized algorithm for computing a spectral approximation to the normalized Laplacian for odd $r$. Second, we define and analyze a generalization of the derandomized square for irregular graphs and for sparsifying the product of two distinct graphs. As part of this generalization, we also give a strongly explicit construction of expander graphs of every size.
In this work, we achieve gap amplification for the Small-Set Expansion problem. Specifically, we show that an instance of the Small-Set Expansion Problem with completeness $epsilon$ and soundness $frac{1}{2}$ is at least as difficult as Small-Set Expansion with completeness $epsilon$ and soundness $f(epsilon)$, for any function $f(epsilon)$ which grows faster than $sqrt{epsilon}$. We achieve this amplification via random walks -- our gadget is the graph with adjacency matrix corresponding to a random walk on the original graph. An interesting feature of our reduction is that unlike gap amplification via parallel repetition, the size of the instances (number of vertices) produced by the reduction remains the same.
We establish a relationship between the Small-World behavior found in complex networks and a family of Random Walks trajectories using, as a linking bridge, a maze iconography. Simple methods to generate mazes using Random Walks are discussed along with related issues and it is explained how to interpret mazes as graphs and loops as shortcuts. Small-World behavior was found to be non-logarithmic but power-law in this model, we discuss the reason for this peculiar scaling
We prove that random walks in random environments, that are exponentially mixing in space and time, are almost surely diffusive, in the sense that their scaling limit is given by the Wiener measure.
We study the problem of extracting random bits from weak sources that are sampled by algorithms with limited memory. This model of small-space sources was introduced by Kamp, Rao, Vadhan and Zuckerman (STOC06), and falls into a line of research initiated by Trevisan and Vadhan (FOCS00) on extracting randomness from weak sources that are sampled by computationally bounded algorithms. Our main results are the following. 1. We obtain near-optimal extractors for small-space sources in the polynomial error regime. For space $s$ sources over $n$ bits, our extractors require just $kgeq scdot$polylog$(n)$ entropy. This is an exponential improvement over the previous best result, which required $kgeq s^{1.1}cdot2^{log^{0.51} n}$ (Chattopadhyay and Li, STOC16). 2. We obtain improved extractors for small-space sources in the negligible error regime. For space $s$ sources over $n$ bits, our extractors require entropy $kgeq n^{1/2+delta}cdot s^{1/2-delta}$, whereas the previous best result required $kgeq n^{2/3+delta}cdot s^{1/3-delta}$ (Chattopadhyay, Goodman, Goyal and Li, STOC20). To obtain our first result, the key ingredient is a new reduction from small-space sources to affine sources, allowing us to simply apply a good affine extractor. To obtain our second result, we must develop some new machinery, since we do not have low-error affine extractors that work for low entropy. Our main tool is a significantly improved extractor for adversarial sources, which is built via a simple framework that makes novel use of a certain kind of leakage-resilient extractors (known as cylinder intersection extractors), by combining them with a general type of extremal designs. Our key ingredient is the first derandomization of these designs, which we obtain using new connections to coding theory and additive combinatorics.
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