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57
On Data Structures and Asymmetric Communication Complexity
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1994
"... 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 ..."
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Cited by 84 (9 self)
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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
Graph distances in the streaming model: the value of space
 In ACMSIAM Symposium on Discrete Algorithms
, 2005
"... We investigate the importance of space when solving problems based on graph distance in the streaming model. In this model, the input graph is presented as a stream of edges in an arbitrary order. The main computational restriction of the model is that we have limited space and therefore cannot stor ..."
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Cited by 57 (10 self)
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We investigate the importance of space when solving problems based on graph distance in the streaming model. In this model, the input graph is presented as a stream of edges in an arbitrary order. The main computational restriction of the model is that we have limited space and therefore cannot store all the streamed data; we are forced to make spaceefficient summaries of the data as we go along. For a graph of n vertices and m edges, we show that testing many graph properties, including connectivity (ergo any reasonable decision problem about distances) and bipartiteness, requires Ω(n) bits of space. Given this, we then investigate how the power of the model increases as we relax our space restriction. Our main result is an efficient randomized algorithm that constructs a (2t + 1)spanner in one pass. With high probability, it uses O(t · n 1+1/t log 2 n) bits of space and processes each edge in the stream in O(t 2 · n 1/t log n) time. We find approximations to diameter and girth via the log n constructed spanner. For t = Ω (), the space log log n requirement of the algorithm is O(n·polylog n), and the peredge processing time is O(polylog n). We also show a corresponding lower bound of t for the approximation ratio achievable when the space restriction is O(t · n1+1/t log 2 n). We then consider the scenario in which we are allowed multiple passes over the input stream. Here, we investigate whether allowing these extra passes will compensate for a given space restriction. We show that ∗This work was supported by the DoD University Research Initiative (URI) administered by the Office of Naval Research
Interactive Communication of Balanced Distributions and of Correlated Files
, 1993
"... (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 mmessage complexity, Cm , is the number of ..."
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Cited by 40 (1 self)
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(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 mmessage 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. Oneway 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...
Interaction in Quantum Communication and the Complexity of Set Disjointness
, 2001
"... 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 ..."
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Cited by 34 (7 self)
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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 structurethey 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 protocolone that has similar eciency, but uses fewer message exchanges.
The BNSChung Criterion for MultiParty Communication Complexity
 Computational Complexity
, 2000
"... The "Number on the Forehead" model of multiparty communication complexity was first suggested by Chandra, Furst and Lipton. The best known lower bound, for an explicit function (in this model), is a lower bound of \Omega\Gamma n=2 k ), where n is the size of the input of each player, an ..."
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Cited by 33 (0 self)
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The "Number on the Forehead" model of multiparty communication complexity was first suggested by Chandra, Furst and Lipton. The best known lower bound, for an explicit function (in this model), is a lower bound of \Omega\Gamma n=2 k ), where n is the size of the input of each player, and k is the number of players (first proved by Babai, Nisan and Szegedy). This lower bound has many applications in complexity theory. Proving a better lower bound, for an explicit function, is a major open problem. Based on the result of BNS, Chung gave a sufficient criterion for a function to have large multipartycommunication complexity (up to \Omega\Gamma n=2 k )). In this paper, we use some of the ideas of BNS, and Chung, together with some new ideas, resulting in a new (easier and more modular) proof for the results of BNS and Chung. This gives a simpler way to prove lower bounds for the multipartycommunicationcomplexity of a function. 1 MultiParty Communication Complexity Multiparty co...
The Communication Complexity of Threshold Gates
 In Proceedings of “Combinatorics, Paul Erdos is Eighty
, 1994
"... 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 degreed threshold gates this is done in the multiparty model. We then use these upp ..."
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Cited by 30 (1 self)
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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 degreed 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.
On quantum and probabilistic communication: Las Vegas and oneway protocols
 in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000
, 2000
"... We investigate the power of quantum communication protocols compared to classical probabilistic protocols. In our first result we describe a total Boolean function that has a quantum Las Vegas protocol communicating at most O(N^{10/11+ epsilon}) qubits for all epsilon > 0, while any classical pro ..."
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Cited by 29 (5 self)
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We investigate the power of quantum communication protocols compared to classical probabilistic protocols. In our first result we describe a total Boolean function that has a quantum Las Vegas protocol communicating at most O(N^{10/11+ epsilon}) qubits for all epsilon > 0, while any classical probabilistic protocol (with bounded error) needs Omega(N/log N) bits. Then we investigate quantum oneway communication complexity. First we show that the VCdimension lower bound on oneway probabilistic communication of [26] holds for quantum protocols, too. Then we prove that for oneway protocols computing total functions quantum Las Vegas communication is asymptotically as efficient as exact quantum communication, which is exactly as efficient as deterministic communication. We describe applications of the lower bounds for oneway communication complexity to quantum finite automata and quantum formulae.
Information Theory Methods in Communication Complexity
 In Proceedings of the 17th Annual IEEE Conference on Computational Complexity
, 2002
"... We use tools and techniques from information theory to study communication complexity problems in the oneway and simultaneous communication models. Our results include: (1) A tight characterization of multiparty oneway communication complexity for product distributions in terms of VCdimension an ..."
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Cited by 28 (7 self)
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We use tools and techniques from information theory to study communication complexity problems in the oneway and simultaneous communication models. Our results include: (1) A tight characterization of multiparty oneway communication complexity for product distributions in terms of VCdimension and shatter coefficients; (2) An equivalence of multiparty oneway and simultaneous communication models for product distributions; (3) A suite of lower bounds for specific functions in the simultaneous communication model, most notably an optimal lower bound for the multiparty set disjointness problem of Alon et al. [AMS99] and for the generalized addressing function problem of Babai et al. [BGKL96] for arbitrary groups. Methodologically, our main contribution is rendering communication complexity problems in the framework of information theory. This allows us access to the powerful calculus of information theory and the use of fundamental principles such as Fano's inequality and the Maximum Likelihood Estimate Principle.
Simultaneous Messages vs. Communication
"... In the multiparty communication game introduced by Chandra, Furst, and Lipton [CFL] (1983), k players wish to evaluate collaboratively a function f(x0 , ..., xk\Gamma1 ) for which player i sees all inputs except x i : The players have unlimited computational power. The objective is to minimize the a ..."
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Cited by 27 (3 self)
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In the multiparty communication game introduced by Chandra, Furst, and Lipton [CFL] (1983), k players wish to evaluate collaboratively a function f(x0 , ..., xk\Gamma1 ) for which player i sees all inputs except x i : The players have unlimited computational power. The objective is to minimize the amount of communication. We consider a restricted version of the multiparty communication game which we call the simultaneous messages model. The difference is that in this model, each of the k players simultaneously sends a message to a referee, who sees none of the input. The referee then announces the function value. We demonstrate an exponential gap between the Simultaneous Messages and the Communication models for up to (log n) 1\Gammaffl players, for any ffl ? 0: The separation is obtained by comparing the respective complexities of the generalized addressing function, GAFG;k , in each model. In addition, we give a nontrivial protocol for GAFG;k for G = Z t 2 ; which is very eff...
Boolean Circuits, Tensor Ranks, And Communication Complexity
 SIAM J. ON COMPUTING
, 1997
"... We investigate two methods for proving lower bounds on the size of small depth circuits, namely the approaches based on multiparty communication games and algebraic characterizations extending the concepts of the tensor rank and rigidity of matrices. Our methods are combinatorial, but we think that ..."
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Cited by 25 (2 self)
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We investigate two methods for proving lower bounds on the size of small depth circuits, namely the approaches based on multiparty communication games and algebraic characterizations extending the concepts of the tensor rank and rigidity of matrices. Our methods are combinatorial, but we think that the main contribution concerns the algebraic concepts used in this area (tensor ranks and rigidity). Our main results are following. (i) An o(n) bit protocol for a communication game for computing shifts, which also gives an upper bound of o(n 2 ) on the contact rank of the tensor of multiplication of polynomials; this disproves some earlier conjectures. A related probabilistic construction gives o(n) upper bound for computing all permutations and O(n log log n) upper bound on the communication complexity of pointer jumping with permutations. (ii) A lower bound on certain restricted circuits of depth 2 which are related to the problem of proving a superlinear lower bound on the size of ...