We give a general complexity classification scheme for monotone computation, including monotone space-bounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic log-space) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = co-NL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for st-connectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...

The frequency moments of a sequence containing mi elements of type i, for 1 ≤ i ≤ n, are the numbers Fk = �n i=1 mki. We consider the space complexity of randomized algorithms that approximate the numbers Fk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, it turns out that the numbers F0, F1 and F2 can be approximated in logarithmic space, whereas the approximation of Fk for k ≥ 6 requires nΩ(1) space. Applications to data bases are mentioned as well.

We present a new method for proving strong lower bounds in communication complexity.

We show that any communication finding a Pareto efficient allocation in a private-information economy must also discover supporting Lindahl prices. In particular, efficient allocation of L indivisible objects requires naming a price for each of the 2 L ¡1 bundles. Furthermore, exponential communication in L is needed just to ensure a higher share of surplus than that realized by auctioning all items as a bundle, or even a higher expected surplus (for some probability distribution over valuations). When the valuations are submodular, efficiency still requires exponential communication (and fully polynomial approximation is impossible). When the objects are homogeneous, arbitrarily good approximation is obtained using exponentially less communication than that needed for exact efficiency.

Abstract. Massive data sets with terabytes of data are becoming commonplace. There is an increasing demand for algorithms and data structures that provide fast response times to queries on such data sets. In this paper, we describe a context for algorithmic work relevant to massive data sets and a framework for evaluating such work. We consider the use of "synopsis" data structures, which use very little space and provide fast (typically approximated) answers to queries. The design and analysis of effective synopsis data structures o er many algorithmic challenges. We discuss a number of concrete examples of synopsis data structures, and describe fast algorithms for keeping them up-to-date in the presence of online updates to the data sets.

We completely (that is, up to a logarithmic factor) characterize the bounded-error quantum communication complexity of every predicate f(x; y) (x; y [n]) depending only on jx\yj. Namely, for a predicate D on f0; 1; : : : ; ng let ` 0 (D) = max f` j 1 ` n=2 ^ D(`) 6 D(` 1)g and ` 1 (D) = max fn ` j n=2 ` < n ^ D(`) 6 D(` + 1)g. Then the bounded-error quantum communication complexity of f D (x; y) = D(jx \ yj) is equal (again, up to a logarithmic factor) to ` 1 (D). In particular, the complexity of the set disjointness predicate is n). This result holds both in the model with prior entanglement and without it.

We investigate the relative power of the common random string model vs. the private random string model in communication complexity. We show that the two model are essentially equal. Keywords: communication complexity, randomness, theory of computation. Communication complexity is a model of computation where two parties, each with an input, want to mutually compute a Boolean function that is defined on pairs of inputs. Formally, let f : X \Theta Y 7! f0; 1g be a Boolean function. The communication problem for f is the following two-player game. Player A gets x 2 X and player B gets y 2 Y . Their goal is to compute f(x; y). They have unlimited computational power and a full description of f , but they don't know each other's input. They determine the output value by exchanging messages. Let n, the length of the input, be log(jXjjY j). A protocol for computing f is a pair of algorithms (one for each player) according to which the players send binary messages. A protocol proceeds in ...

Any implementation of Carter-Wegman universal hashing from n-bit strings to m-bit strings requires a time-space tradeoff of TS = &Omega;(nm). The bound holds in the general boolean branching program model, and thus in essentially any model of computation. As a corollary, computing a+b*c in any field F requires a quadratic time-space tradeoff, and the bound holds for any representation of the elements of the field. Other lower bounds on the...

1 First, I would like to thank my advisor Noam Nisan. During the two years I have been working with Noam, he has been a most costructive in uence on me, teaching me how to think and write in a clear way. Iwould also liketothankmymyfellow students, who during this time had to su er hearing me lecturing my ideas � this goes especially to Amnon Tashma who hasn't recovered yet. My parents deserve a special thank, not only for getting me up to this point, but also for the help they have given me with the di cult task of writing this thesis in English. Finally I would like tothankmy wife Ruthie for working around my short \mental going to work &quot; periods, unlike the others she will have tocontinue living with me. 2

We present several results regarding randomized one-round communication complexity. Our results include a connection to the VCdimension, a study of the problem of computing the inner product of two real valued vectors, and a relation between \simultaneous" protocols and one-round protocols. Key words. Communication Complexity; One-round and simultaneous protocols; VC-dimension; Subject classications. 68Q25. 1.