This paper is dedicated to the memory of Ronald V. Book, 1937-1997.

We investigate two problems concerning the complexity of evaluating a function f at a k-tuple of unrelated inputs by k parallel decision tree algorithms. In the product problem, for some fixed depth bound d, we seek to maximize the fraction of input k-tuples for which all k decision trees are correct. Assume that for a single input to f , the best decision tree algorithm of depth d is correct on a fraction p of inputs. We prove that the maximum fraction of k-tuples on which k depth d algorithms are all correct is at most p k , which is the trivial lower bound. We show that if we replace the depth d restriction by "expected depth d", then this result fails. In the help-bit problem, we are permitted to ask k \Gamma 1 arbitrary binary questions about the k-tuple of inputs. For each possible k \Gamma 1-tuple of answers to these queries we will have a k-tuple of decision trees which are supposed to correctly compute all functions on k-tuples that are consistent with the part...

We examine the power of incremental evaluation systems that use an SQL-like language for maintaining recursively-defined views. We show that recursive queries such as transitive closure, and "alternating paths" can be incrementally maintained in a nested relational language, when some auxiliary relations are allowed. In the presence of aggregate functions, even more queries can be maintained, for example, the "same generation" query. In contrast, it is still an open problem whether such queries are maintainable in relational calculus. We then restrict the language so that no nested relations are involved (but wekeep the aggregate functions). Such a language captures the capability of most practical relational database systems. We prove that this restriction does not reduce the incremental computational power; that is, any query that can be maintained in a nested language with aggregates, is still maintainable using only flat relations. We also show that one does not need auxiliar...

In databases, as in other computational settings, one would like to efficiently update answers to queries while minor changes are being made to the data. Dynamic complexity asks what resources are required to perform such updates. In this paper our main focus will be a particular dynamic graph problem, tree isomorphism, and the efficiency of our update scheme will be measured in terms of how expressive a query language is required to express our update. Working in the framework developed by [DS93, PI97] for dynamic query evaluation, we show that dynamic tree isomorphism can be performed via first-order updates to a relational database (in [DS93] this framework is called a first-order incremental evaluation system, and in [PI97] it is called Dyn-FO). In [EI95] it was shown that tree-isomorphism can not be expressed in first-order logic augmented with a transitive closure operator and counting, (FO + TC + COUNT) (but without ordering). We thus obtain a first example of a graph problem...

This paper contains several results regarding the communication complexity model and the 2-prover games model, which are based on interaction between the two models: 1. We show how to improve the rate of exponential decrease in the parallel repetition theorem of [Ra] in terms of the communication complexity of the verifier's predicate. 2. We apply the improved parallel repetition theorem of 2-prover games to derive, for the first time, a direct product theorem for communication complexity. The second derivation uses a common generalization of the two models, which is independently interesting. We initiate a study of its power by considering the GCD problem, and some variations of it, which exhibit a power gap between the new model and the classical communication complexity model. This gap is partly based on the following upper bounds: Given n-bit inputs x and y to Alice and Bob respectively, they can achieve the tasks below with very high probability using only O(n= log n) communica...

Abstract. We establish a nonlinear lower bound for halfplane range searching over a group. Specifically, we show that summing up the weights of n (weighted) points within n halfplanes requires Ω(n log n) additions and subtractions. This is the first nontrivial lower bound for range searching over a group. By contrast, range searching over a semigroup (which forbids subtractions) is almost completely understood. Our proof has two parts. First, we develop a general, entropy-based method for relating the linear circuit complexity of a linear map A to the spectrum of A ⊤ A. In the second part of the proof, we design a “high-spectrum ” geometric set system for halfplane range searching and, using techniques from discrepancy theory, we estimate the median eigenvalue of its associated map. Interestingly, the method also shows that using up to a linear number of help gates cannot help; these are gates that can compute any bivariate function.

this paper by reducing 3k + 1 to k + 1, or k, or even a constant. We will prove this by modifying Cai's result [3] and by modifying the reduction used in [7]. Section 2 provides a brief review the notion of "first-order incremental evaluation systems ". Section 3 establishes a necessary technical lemma, which is a variant of Cai's theorem. Section 4 gives the proof of the above theorem. 2 First-Order Incremental Evaluation Systems

We prove that two-party randomized communication complexity satisfies a strong direct productproperty, so long as the communication lower bound is proved by a "corruption" or "one-sided discrepancy" method over a rectangular distribution. We use this to prove new n\Omega (1) lower bounds for number-on-the-forehead protocols in which the first player speaks once and then the other two players proceed arbitrarily. Using other techniques, we also establish an \Omega (n1/(k-1)/(k- 1)) lower bound for k-playerrandomized number-on-the-forehead protocols for the disjointness function in which all messages are broadcast simultaneously. A simple corollary of this is that general randomized number-on-the-foreheadprotocols require \Omega (log n/(k- 1)) bits of communication to compute the disjointness function.

It is common knowledge that relational calculus and even SQL are not expressive enough to express recursive queries such as the transitive closure. In a real database system, one can overcome this problem by storing a graph together with its transitive closure and maintaining the latter whenever updates to the former occur. This leads to the concept of an incremental evaluation system, or IES. Much is already known about the theory of IES but very little has been translated into practice. The purpose of this paper is to ll in this gap by providing a gentle introduction to and an overview of some recent theoretical results on IES. The introduction is through the translation into SQL of three interesting positive maintenance results that have practical importance { the maintenance of the transitive closure of acyclic graphs, of undirected graphs, and of arbitrary directed graphs. Interestingly, these examples also allow ustoshow the relationship between power and cost in the incremental maintenance of database queries. 1

We prove that corruption, one of the most powerful measures used to analyze 2-party randomized communication complexity, satisfies a strong direct sum property under rectangular distributions. This direct sum bound holds even when the error is allowed to be exponentially close to 1. We use this to analyze the complexity of the widelystudied set disjointness problem in the usual “number-onthe-forehead” (NOF) model of multiparty communication complexity. 1