In this paper we obtain the first superpolynomial lower bounds for monotone span programs computing explicit functions. The best previous lower bound was Ω(n 5/2 ) by Beimel, Gál, Paterson [BGP]; 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 Weil's character sum estimates. We prove an n Ω(log n/ log log n) 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. * Part of this work has been presented at the 28th ACM STOC'96. The second two authors wish to thank DIMACS for its hospitality and acknowledge its supporting agencies,

In the multiparty communication game (CFL-game) of Chandra, Furst, and Lipton (Proc. 15th ACM STOC, 1983, 94-99) k players collaboratively evaluate a function f (x 0 , . . . , x k−1 ) in which player i knows all inputs except x i . 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 (log n) 1− players, for any > 0. This separation is obtained by comparing the respective complexities of the generalized addressing function, GAF G,k , where G is a group of order n. We also combine our lower bounds on SM complexity with ideas of Håstad and Goldmann (Computational Complexity 1 (1991), 113-129) to derive superpolynomial lower bounds for certain depth-2 circuits computing a function related to the GAF function.We prove some counter-intuitive upper bounds on SM-complexity. We show that GAF Z t 2 ,3has SM-complexity O(n 0.92 ). When the number of players is at least c log n, for some constant c > 0, our SM protocol for GAF Z t 2 ,k has polylog(n) complexity. We also examine a class of functions defined by certain depth-2 circuits. This class includes the "Generalized Inner Product" function and "Majority of Majorities." When the number of players is at least 2+log n, we obtain polylog(n) upper bounds for this class of functions.

Abstract. We show lower bounds in the cell probe model for the redundancy/query time tradeoff of solutions to static data structure problems.

In the cell probe model with word size 1 (the bit probe model), a static data structure problem is given by a map f : {0,n is a set of possible data to be stored, {0, 1} m is a set of possible queries (for natural problems, we have m n) and f(x, y) is the answer to question y about data x.A solution is given by a representation φ : y). The time t of the query algorithm is the number of bits it reads in φ(x).In this paper, we consider the case of succinct representations where s = n + r for some redundancy r n. For a boolean version of the problem of polynomial evaluation with preprocessing of coefficients, we show a lower bound on the redundancy-query time tradeoff of the form (r + 1)t ≥ Ω(n/ log n).In particular, for very small redundancies r, we get an almost optimal lower bound stating that the query algorithm has to inspect almost the entire data structure (up to a logarithmic factor). We show similar lower bounds for problems satisfying a certain combinatorial property of a coding theoretic flavor. Previously, no ω(m) lower bounds were known on t in the general model for explicit functions, even for very small redundancies.By restricting our attention to systematic or index structures φ satisfying φ(x) = x·φ * (x) for some map φ * (where · denotes concatenation) we show similar lower bounds on the redundancy-query time tradeoff

In this paper we obtain the first superpolynomial lower bounds for monotone span programs computing explicit functions. The best previous lower bound was Ω(n 5/2 ) by Beimel, Gál, Paterson [BGP]; 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 Weil's character sum estimates. We prove an n Ω(log n/ log log n) 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. * Part of this work has been presented at the 28th ACM STOC'96. The second two authors wish to thank DIMACS for its hospitality and acknowledge its supporting agencies,

We prove that the reliable computation of any Boolean function with sensitivity s requires Ω(s log s) gates if the gates of the circuit fail independently with a fixed positive probability. This theorem was stated by Dobrushin and Ortyukov in 1977, but their proof was found by Pippenger, Stamoulis and Tsitsiklis to contain some errors. We save the original structure of the proof of Dobrushin and Ortyukov, correcting two points in the probabilistic argument.

Batch codes, introduced by Ishai, Kushilevitz, Ostrovsky and Sahai, represent the distributed storage of an n-element data set on m servers in such a way that any batch of k data items can be retrieved by reading at most one (or more generally, t) items from each server, while keeping the total storage over m servers equal to N . This paper considers a class of batch codes (for t = 1), called combinatorial batch codes (CBCs), where each server stores a subset of a database. A CBC is called optimal if the total storage N is minimal for given n, m, and k. A c-uniform CBC is a combinatorial batch code where each item is stored in exactly c servers. A c-uniform CBC is called optimal if its parameter n has maximum value for given m and k. Optimal c-uniform CBCs have been known only for c ∈ {2, k − 1, k − 2}. In this paper we present new constructions of optimal CBCs in both the uniform and general settings, for values of the parameters where tight bounds have not been established previously. In the uniform setting, we provide constructions of two new families of optimal uniform codes with c ∼ √ k. Our constructions are based on affine planes and transversal designs.

WC give a characterization of span program size by a combinatorial-algebraic measure defined on covers of pairs of O's and l's of the function computed. The measure we consider is a generalization of a measure on covers which bar; been used to prove lower bounds on formula size [K, Ry, Ra], and has also been studied with respect to communication complexity,In the monotone case our new methods yield 12*(*~s") lower bounds for the monotone span program complexity of explicit Boolean functions in TZ variables over arbitrary fields, improving the previous lower bounds on monotone span program size. Our characterization of span program ai7k implies that any matrix with superpolynomial separation between its rank and cover number can be used to obtain ouperpolynomial lower bounds on monotone span program aizc, We also identify a property of bipartite graphs that is sufticient for constructing Boolean functions with large monotone span program complexity.