Derandomized Squaring of Graphs

We introduce a “derandomized ” analogue of graph squaring. This op-eration increases the connectivity of the graph (as measured by the second eigenvalue) almost as well as squaring the graph does, yet only increases the degree of the graph by a constant factor, instead of squaring the degree. One application of this product is an alternative proof of Reingold’s re-cent breakthrough result that S-T Connectivity in Undirected Graphs can be solved in deterministic logspace.

Deterministic Approximation of Random Walks in Small Space

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 [Eyal Rozenman and Salil Vadhan, 2005], which we recently used for solving Laplacian systems in nearly logarithmic space [Murtagh et al., 2017], with ideas from [Cheng et al., 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

Small-Bias Sets for Nonabelian Groups: Derandomizing the Alon-Roichman Theorem

In analogy with epsilon-biased sets over Z_2^n, we construct explicit epsilon-biased sets over nonabelian finite groups G. That is, we find sets S subset G such that | Exp_{x in S} rho(x)| <= epsilon for any nontrivial irreducible representation rho. Equivalently, such sets make G's Cayley graph an expander with eigenvalue |lambda| <= epsilon. The Alon-Roichman theorem shows that random sets of size O(log |G| / epsilon^2) suffice. For groups of the form G = G_1 x ... x G_n, our construction has size poly(max_i |G_i|, n, epsilon^{-1}), and we show that a set S \subset G^n considered by Meka and Zuckerman that fools read-once branching programs over G is also epsilon-biased in this sense. For solvable groups whose abelian quotients have constant exponent, we obtain epsilon-biased sets of size (log |G|)^{1+o(1)} poly(epsilon^{-1}). Our techniques include derandomized squaring (in both the matrix product and tensor product senses) and a Chernoff-like bound on the expected norm of the product of independently random operators that may be of independent interest.Comment: Our results on solvable groups have been significantly improved, giving eps-biased sets of polynomial (as opposed to quasipolynomial) siz

Singular Value Approximation and Sparsifying Random Walks on Directed Graphs

In this paper, we introduce a new, spectral notion of approximation between directed graphs, which we call singular value (SV) approximation. SV-approximation is stronger than previous notions of spectral approximation considered in the literature, including spectral approximation of Laplacians for undirected graphs (Spielman Teng STOC 2004), standard approximation for directed graphs (Cohen et. al. STOC 2017), and unit-circle approximation for directed graphs (Ahmadinejad et. al. FOCS 2020). Further, SV approximation enjoys several useful properties not possessed by previous notions of approximation, e.g., it is preserved under products of random-walk matrices and bounded matrices. We provide a nearly linear-time algorithm for SV-sparsifying (and hence UC-sparsifying) Eulerian directed graphs, as well as $\ell$-step random walks on such graphs, for any $\ell\leq \text{poly}(n)$. Combined with the Eulerian scaling algorithms of (Cohen et. al. FOCS 2018), given an arbitrary (not necessarily Eulerian) directed graph and a set $S$ of vertices, we can approximate the stationary probability mass of the $(S,S^c)$ cut in an $\ell$-step random walk to within a multiplicative error of $1/\text{polylog}(n)$ and an additive error of $1/\text{poly}(n)$ in nearly linear time. As a starting point for these results, we provide a simple black-box reduction from SV-sparsifying Eulerian directed graphs to SV-sparsifying undirected graphs; such a directed-to-undirected reduction was not known for previous notions of spectral approximation.Comment: FOCS 202

Derandomization Beyond Connectivity: Undirected Laplacian Systems in Nearly Logarithmic Space

We give a deterministic O˜(log n)-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and escape probabilities for undirected graphs. Previously, such systems were known to be solvable by randomized algorithms using O(log n) space (Doron, Le Gall, and Ta-Shma, 2017) and hence by deterministic algorithms using O(log3/2 n) space (Saks and Zhou, FOCS 1995 and JCSS 1999). Our algorithm combines ideas from time-efficient Laplacian solvers (Spielman and Teng, STOC ‘04; Peng and Spielman, STOC ‘14) with ideas used to show that UNDIRECTED S-T CONNECTIVITY is in deterministic logspace (Reingold, STOC ‘05 and JACM ‘08; Rozenman and Vadhan, RANDOM ‘05).Engineering and Applied Science

Entropy Waves, The Zig-Zag Graph Product, and New Constant-Degree Expanders and Extractors

The main contribution of this work is a new type of graph product, which we call the zig-zag product. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and its expansion properties from both! Iteration yields simple explicit constructions of constant-degree expanders of every size, starting from one constant-size expander. Crucial to our intuition (and simple analysis) of the properties of this graph product is the view of expanders as functions which act as "entropy wave" propagators --- they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated. In these terms, the graph product affords the constructive interference of two such waves. A variant of this product can be applied to extractors, giving the first explicit extractors whose seed length depends (poly)logarithmically on only the entropy deficiency of the source (rather than its length) and that extract almost all the entropy of high min-entropy sources. These high min-entropy extractors have several interesting applications, including the first constant-degree explicit expanders which beat the "eigenvalue bound."Engineering and Applied Science

Nonlinear spectral calculus and super-expanders

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.Comment: Typos fixed based on referee comments. Some of the results of this paper were announced in arXiv:0910.2041. The corresponding parts of arXiv:0910.2041 are subsumed by the current pape