We consider the problem of estimating the average of a huge set of values. That is, given oracle access to an arbitrary function $f:\{0,1\}^n\mapsto[0,1]$, we need to estimate $2^{-n} \sum_{x\in\{0,1\}^n} f(x)$ upto an additive error of $\epsilon$. We are allowed to employ a randomized algorithm which may err with probability ... more >>>

We give explicit constructions of extractors which work for a source of any min-entropy on strings of length n. These extractors can extract any constant fraction of the min-entropy using O(log^2 n) additional random bits, and can extract all the min-entropy using O(log^3 n) additional random bits. Both of these ... more >>>

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 ... more >>>

We study the compression of polynomially samplable sources. In particular, we give efficient prefix-free compression and decompression algorithms for three classes of such sources (whose support is a subset of {0,1}^n). 1. We show how to compress sources X samplable by logspace machines to expected length H(X)+O(1). Our next results ... more >>>

Motivated by Reingold's recent deterministic log-space algorithm for Undirected S-T Connectivity (ECCC TR 04-94), we revisit the general RL vs. L question, obtaining the following results. 1. We exhibit a new complete problem for RL: S-T Connectivity restricted to directed graphs for which the random walk is promised to have ... more >>>

Optimal dispersers have better dependence on the error than optimal extractors. In this paper we give explicit disperser constructions that beat the best possible extractors in some parameters. Our constructions are not strong, but we show that having such explicit strong constructions implies a solution to the Ramsey graph construction ... more >>>

We introduce a "derandomized" analogue of graph squaring. This operation 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 ... more >>>

We highlight a common theme in four relatively recent works that establish remarkable results by an iterative approach. Starting from a trivial construct, each of these works applies an ingeniously designed sequence of iterations that yields the desired result, which is highly non-trivial. Furthermore, in each iteration, the construct is ... more >>>

We construct a randomness-efficient averaging sampler that is computable by uniform constant-depth circuits with parity gates (i.e., in AC^0[mod 2]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within NC^1. For example, we obtain the following results: ... more >>>

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP--hard problems are easier to solve. In particular, whether there exist algorithms that solve correctly ... more >>>

Using $\epsilon$-bias spaces over F_2 , we show that the Remote Point Problem (RPP), introduced by Alon et al [APY09], has an $NC^2$ algorithm (achieving the same parameters as [APY09]). We study a generalization of the Remote Point Problem to groups: we replace F_n by G^n for an arbitrary fixed ... more >>>

The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: (1) Constant degree dimension expanders in finite ... more >>>