Anna Gál

Lower bounds for the complexity of reliable Boolean circuits with noisy gates

Proves that the reliable computation of any Boolean function with sensitivity s requires /spl Omega/(s log s) gates if the gates fail independently with a fixed positive probability. This theorem was stated by Dobrushin and Ortyukov (1977), but their proof was found by Pippenger, Stamoulis, and Tsitsiklis (1991) to contain some errors. >

Superpolynomial Lower Bounds for Monotone Span Programs

monotone span programs computing explicit functions. The best previous lower bound was by Beimel, Gál, Paterson [7]; our proof exploits a general combinatorial lower bound criterion from that paper. Our lower bounds are based on an analysis of Paley-type bipartite graphs via Weils character sum estimates. We prove an lower bound for the size of monotone span programs for the clique problem. Our results give the first superpolynomial lower bounds for linear secret sharing schemes.We demonstrate the surprising power of monotone span programs by exhibiting a function computable in this model in linear size while requiring superpolynomial size monotone circuits and exponential size monotone formulae. We also show that the perfect matching function can be computed by polynomial size (non-monotone) span programs over arbitrary fields.

Communication Complexity of Simultaneous Messages

In the multiparty communication game (CFL game) of Chandra, Furst, and Lipton [Proceedings of the 15th Annual ACM Symposium on Theory of Computing, Boston, MA, 1983, pp. 94--99] k players collaboratively evaluate a function f(x0, . . . , xk-1) in which player i knows all inputs except xi. The players have unlimited computational power. The objective is to minimize communication. In this paper, we study the SIMULTANEOUS MESSAGES (SM) model of multiparty communication complexity. The SM model is a restricted version of the CFL game in which the players are not allowed to communicate with each other. Instead, each of the k players simultaneously sends a message to a referee, who sees none of the inputs. The referee then announces the function value. We prove lower and upper bounds on the SM complexity of several classes of explicit functions. Our lower bounds extend to randomized SM complexity via an entropy argument. A lemma establishing a tradeoff between average Hamming distance and range size for transformations of the Boolean cube might be of independent interest. Our lower bounds on SM complexity imply an exponential gap between the SM model and the CFL model for up to

A characterization of span program size and improved lower bounds for monotone span programs

(\log n)^{1-\epsilon}

Extremal bipartite graphs and superpolynomial lower bounds for monotone span programs

players for any

Lower bounds for monotone span programs

\epsilon > 0

Lower Bounds on Streaming Algorithms for Approximating the Length of the Longest Increasing Subsequence

. This separation is obtained by comparing the respective complexities of the Generalized Addressing Function, GAFG,k, where G is a group of order n. We also combine our lower bounds on SM complexity with the ideas of Hastad and Goldmann [Comput. Complexity, 1 (1991), pp. 113--129] to derive superpolynomial lower bounds for certain depth-2 circuits computing a function related to the GAF function. We prove some counterintuitive upper bounds on SM complexity. We show that {\sf GAF}

A note on monotone complexity and the rank of matrices

_{\mathbb{Z}_2^t,3}

On Arithmetic Branching Programs

has SM complexity

Boolean complexity classes vs. their arithmetic analogs

O(n^{0.92})