Results 1  10
of
93
Monotone Complexity
, 1990
"... We give a general complexity classification scheme for monotone computation, including monotone spacebounded 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 ..."
Abstract

Cited by 2350 (12 self)
 Add to MetaCart
We give a general complexity classification scheme for monotone computation, including monotone spacebounded 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 logspace) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = coNL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for stconnectivity. 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 space complexity of approximating the frequency moments
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1996
"... 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, ..."
Abstract

Cited by 704 (12 self)
 Add to MetaCart
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.
An Information Statistics Approach to Data Stream and Communication Complexity
, 2003
"... We present a new method for proving strong lower bounds in communication complexity. ..."
Abstract

Cited by 153 (8 self)
 Add to MetaCart
We present a new method for proving strong lower bounds in communication complexity.
The communication requirements of efficient allocations and supporting prices
 Journal of Economic Theory
, 2006
"... We show that any communication finding a Pareto efficient allocation in a privateinformation 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 communicat ..."
Abstract

Cited by 113 (15 self)
 Add to MetaCart
We show that any communication finding a Pareto efficient allocation in a privateinformation 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.
Synopsis Data Structures for Massive Data Sets
"... 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 f ..."
Abstract

Cited by 108 (13 self)
 Add to MetaCart
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 uptodate in the presence of online updates to the data sets.
Quantum Communication Complexity of Symmetric Predicates
 Izvestiya of the Russian Academy of Science, Mathematics
, 2002
"... We completely (that is, up to a logarithmic factor) characterize the boundederror 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) = ..."
Abstract

Cited by 87 (1 self)
 Add to MetaCart
We completely (that is, up to a logarithmic factor) characterize the boundederror 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 boundederror 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.
Private vs. Common Random bits in Communication Complexity
 Information Processing Letters
, 1995
"... 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 comp ..."
Abstract

Cited by 84 (0 self)
 Add to MetaCart
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 twoplayer 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 ...
Quantum communication
, 1995
"... 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 lecturi ..."
Abstract

Cited by 62 (0 self)
 Add to MetaCart
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 " periods, unlike the others she will have tocontinue living with me. 2
Communication complexity lower bounds by polynomials
 In Proc. of the 16th Conf. on Computational Complexity (CCC
, 2001
"... The quantum version of communication complexity allows Alice and Bob to communicate qubits and/or to make use of prior entanglement (shared EPRpairs). Some lower bound techniques are available for qubit communication [17, 11, 2], but except for the inner product function [11], no bounds are known f ..."
Abstract

Cited by 60 (12 self)
 Add to MetaCart
The quantum version of communication complexity allows Alice and Bob to communicate qubits and/or to make use of prior entanglement (shared EPRpairs). Some lower bound techniques are available for qubit communication [17, 11, 2], but except for the inner product function [11], no bounds are known for the model with unlimited prior entanglement. We show that the “log rank ” lower bound extends to the strongest model (qubit communication + prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the “logrank conjecture ” and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for boundederror protocols. 1
The Computational Complexity of Universal Hashing
 Theoretical Computer Science
, 2002
"... Any implementation of CarterWegman universal hashing from nbit strings to mbit strings requires a timespace tradeoff of TS = Ω(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 ..."
Abstract

Cited by 58 (3 self)
 Add to MetaCart
Any implementation of CarterWegman universal hashing from nbit strings to mbit strings requires a timespace tradeoff of TS = Ω(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 timespace tradeoff, and the bound holds for any representation of the elements of the field. Other lower bounds on the...