A given Boolean function has its input distributed among many parties. The aim is to determine which parties to tMk to and what information to exchange with each of them in order to evaluate the function while minimizing the total communication. This paper shows that it is possible to obtain the Boolean answer deterministically with only a polynomial increase in communication with respect to the information lower bound given by the nondeterministic communication complexity of the function.

This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

In this paper we consider two party communication complexity when the input sizes of the two players differ significantly, the "asymmetric" case. Most of previous work on communication complexity only considers the total number of bits sent, but we study tradeoffs between the number of bits the first player sends and the number of bits the second sends. These

We also study the three party communication model, and exhibit an exponential gap in 3-round protocols that differ in the starting player.

(X; Y ) is a pair of random variables distributed over a support set S. Person PX knows X, Person P Y knows Y , and both know S. Using a predetermined protocol, they exchange binary messages in order for P Y to learn X. PX may or may not learn Y . The m-message complexity, Cm , is the number of information bits that must be transmitted (by both persons) in the worst case if only m messages are allowed. C1 is the number of bits required when there is no restriction on the number of messages exchanged. We consider a natural class of random pairs. ¯ is the maximum number of X values possible with a given Y value. j is the maximum number of Y values possible with a given X value. The random pair (X; Y ) is balanced if ¯ = j. The following hold for all balanced random pairs. One-way communication requires at most twice the minimum number of bits: C 1 2 C1 + 1. This bound is almost tight: for every ff, there is a balanced random pair for which C 1 2 C1 \Gamma 6 ff. Three...

One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with much fewer, even exponentially fewer, qubits than possible classically [1], [2], [3]. Moreover, some of these methods have a very simple structure|they involve only few message exchanges between the communicating parties. We consider the question as to whether every classical protocol may be transformed to a \simpler" quantum protocol|one that has similar eciency, but uses fewer message exchanges.

X and Y are random variables. Person PX knows X, Person P Y knows Y , and both know the joint probability distribution of the pair (X; Y ). Using a predetermined protocol, they communicate over a binary, error-free, channel in order for P Y to learn X. PX may or may not learn Y . How many information bits must be transmitted (by both persons) in the worst case if only m messages are allowed? C 1 (XjY ) is the number of bits required when at most one message is allowed, necessarily from PX to P Y . C 2 (XjY ) is the number of bits required when at most two messages are permitted: P Y transmits a message to PX , then PX responds with a message to P Y . C1 (XjY ) is the number of bits required when communication is unrestricted: PX and P Y can communicate back and forth. The maximum reduction in communication achievable via interaction is almost logarithmic. For all (X; Y ) pairs, C1 (XjY ) dlog C 1 (XjY )e + 1, whereas, for a class of (X; Y ) pairs, C1 (XjY ) = dlog C 1 (...

this paper can be generalized to the optimal model? 8 Acknowledgment I wish to thank L. Hemachandra for his invitation to me to spend the spring semester at the University of Rochester and for his permanent attention to my research and helpfulness in all my problems and J. Seiferas for extensive comments on an earlier draft of this paper. The results of section 4.1 of the paper are the realization of J. Seiferas's advice to investigate the probabilistic complexity properties of almost all functions in comparison with Yao's [Y1] results. I wish also to thank P. Dietz for his comments, which helped to simplify the proof of lemma 4.1

We prove upper bounds on the randomized communication complexity of evaluating a threshold gate (with arbitrary weights). For linear threshold gates this is done in the usual 2 party communication model, and for degree-d threshold gates this is done in the multiparty model. We then use these upper bounds together with known lower bounds for communication complexity in order to give very easy proofs for lower bounds in various models of computation involving threshold gates. This generalizes several known bounds and answers several open problems.

Abstract. We study a clean machine model for external memory and stream processing. We show that the number of scans of the external data induces a strict hierarchy (as long as work space is sufficiently small, e.g., polylogarithmic in the size of the input). We also show that neither joins nor sorting are feasible if the product of the number r(n) of scans of the external memory and the size s(n) of the internal memory buffers is sufficiently small, e.g., of size o ( 5 √ n). We also establish tight bounds for the complexity of XPath evaluation and filtering. 1