Small sample spaces with almost independent random variables are applied to design efficient sequential deterministic algorithms for two problems. The first algorithm, motivated by the attempt to design efficient algorithms for the All Pairs Shortest Path problem using fast matrix multiplication, solves the problem of computing witnesses for the Boolean product of two matrices. That is, if A and B are two n by n matrices, and C = AB is their Boolean product, the algorithm finds for every entry Cij = 1 a witness: an index k so that Aik = Bkj = 1. Its running time exceeds that of computing the product of two n by n matrices with small integer entries by a polylogarithmic factor. The second algorithm is a nearly linear time deterministic procedure for constructing a perfect hash function for a given n-subset of {1,..., m}.

We consider the problem of finding irredundant bases for inconsistent sets of equalities and disequalities. These are subsets of inconsistent sets which do not contain any literals which do not contribute to the unsatisfiability in an essential way, and can therefore be discarded. The approach we are pursuing here is to decorate derivations with proofs and to extract irredundant sets of assumptions from these proofs. This requires specialized operators on proofs, but the basic inference systems are otherwise left unchanged. In congruence closure, but our constructions can also be applied to other inference systems such as Gaussian elimination. 1

Let G = (V, E, w) be a directed graph, where w: V → R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c(u, v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n 2.575) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first sub-cubic algorithm for this problem. Unlike the sub-cubic algorithm for the all-pairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights. The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on

We considered the problem of processing queries that contain generalized quantifiers. We demonstrate that current relational systems are ill-equipped, both at the language and at the query processing level, to deal with such queries. We propose a boolean matrix approach which establishes the feasibility of building systems that can process queries with generalized quantifiers efficiently, and we provide insights into the intrinsic difficulties associated with processing such queries. 1 Introduction Numerous existing query languages (SQL [25], OQL [6], CORAL [24], RC/S [23] etc.) allow queries with embedded sub-queries as well as sub-query comparison statements. 1 It is often argued that these features enhance the declarativeness of the query language. In two recent papers, Hsu and Parker [17] and, independently, Badia, Van Gucht, and Gyssens [2], validated this argument by establishing a link between the phenomenon of sub-query syntax in query languages and the theory of generaliz...

Sets of permutations play an important role in the design of some efficient algorithms. In this paper we design two algorithms that manipulate sets of permutations. Both algorithms, each solving a different problem, use fast matrix multiplication techniques to achieve a significant improvement in the running time over the naive solutions. For a set of permutations P ⊂ Sn we say that i k-dominates j if the number of permutations π ∈ P for which π(i) < π(j) is k. The dominance matrix of P is the n × n matrix DP where DP (i, j) = k if and only if i k-dominates j. We give an efficient algorithm for computing DP using fast rectangular matrix multiplication. In particular, when |P | = n our algorithm runs in O(n2.684) time. Computing the dominance matrix of permutations is computationally equivalent to the dominance problem in computational geometry. Thus, our algorithm slightly improves upon a well-known O(n2.688) time algorithm of Matousek for the dominance problem. Permutation dominance is used, together with several other ingredients, to obtain a truly sub-cubic algorithm for the All Pairs Shortest Paths (APSP) problem in real-weighted directed graphs, where the number of distinct weights emanating from each vertex is O(n 0.338). A special case of this algorithm implies an O(n 2.842) time algorithm for real vertexweighted APSP, which slightly improves a recent result of Chan [STOC-07]. A set of permutations P ⊂ Sn is fully expanding if the product of any two elements of P yields a distinct permutation. Stated otherwise, |P 2 | = |P | 2 where P 2 ⊂ Sn is the set of products of two elements of P. We present a randomized algorithm that computes |P 2 | and hence decides if P is fully expanding. The algorithm also produces a table that, for any σ1, σ2, σ3, σ4 ∈ P, answers the query σ1σ2 = σ3σ4 in Õ(1) time. The algorithm uses, among other ingredients, a combination of fast matrix multiplication and polynomial identity testing. In particular, for |P | = n our algorithm runs in O(nω) time where ω < 2.376 is the matrix multiplication