Topic
Communication complexity
About: Communication complexity is a research topic. Over the lifetime, 3870 publications have been published within this topic receiving 105832 citations.
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17 May 2008TL;DR: An entirely different proof of Razborov's result is given, using the original, one-dimensional discrepancy method, which refutes the commonly held intuition that the original discrepancy method fails for functions such as DISJOINTNESS and establishes a large new class of total Boolean functions whose quantum communication complexity is at best polynomially smaller than their classical complexity.
Abstract: In a breakthrough result, Razborov (2003) gave optimal lower bounds on the communication complexity of every function f of the form f(x,y)=D(|x AND y|) for some D:{0,1,...,n}->{0,1}, in the bounded-error quantum model with and without prior entanglement. This was proved by the multidimensional discrepancy method. We give an entirely different proof of Razborov's result, using the original, one-dimensional discrepancy method. This refutes the commonly held intuition (Razborov 2003) that the original discrepancy method fails for functions such as DISJOINTNESS. More importantly, our communication lower bounds hold for a much broader class of functions for which no methods were available. Namely, fix an arbitrary function f:{0,1}n/4->{0,1} and let A be the Boolean matrix whose columns are each an application of f to some subset of the variables x1,x2,...,xn. We prove that the communication complexity of A in the bounded-error quantum model with and without prior entanglement is Omega(d), where d is the approximate degree of f. From this result, Razborov's lower bounds follow easily. Our result also establishes a large new class of total Boolean functions whose quantum communication complexity (regardless of prior entanglement) is at best polynomially smaller than their classical complexity. Our proof method is a novel combination of two ingredients. The first is a certain equivalence of approximation and orthogonality in Euclidean n-space, which follows by linear-programming duality. The second is a new construction of suitably structured matrices with low spectral norm, the pattern matrices, which we realize using matrix analysis and the Fourier transform over (Z2)n. The method of this paper has recently inspired important progress in multiparty communication complexity.
92 citations
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21 Feb 2007TL;DR: An efficient communication-optimal tworound PSMT protocol for messages of length polynomial in n that is almost optimally resilient in that it requires a number of channels n ≥ (2 + ɚ)t, for any arbitrarily small constant ɛ > 0.
Abstract: Perfectly secure message transmission (PSMT), a problem formulated by Dolev, Dwork, Waarts and Yung, involves a sender S and a recipient R who are connected by n synchronous channels of which up to t may be corrupted by an active adversary. The goal is to transmit, with perfect security, a message from S to R. PSMT is achievable if and only if n > 2t.
For the case n >2t, the lower bound on the number of communication rounds between S and R required for PSMT is 2, and the only known efficient (i.e., polynomial in n) two-round protocol involves a communication complexity ofO(n3l) bits, wherel is the lengthof themessage. A recent solution by Agarwal, Cramer and de Haan is provably communication-optimal by achieving an asymptotic communication complexity of O(nl) bits; however, it requires the messages to be exponentially large, i.e., l=ω(2n).
In this paper we present an efficient communication-optimal tworound PSMT protocol for messages of length polynomial in n that is almost optimally resilient in that it requires a number of channels n ≥ (2 + ɛ)t, for any arbitrarily small constant ɛ > 0. In this case, optimal communication complexity is O(l) bits.
92 citations
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21 May 2002TL;DR: A tight characterization of multi-party one-way communication complexity for product distributions in terms of VC-dimension and shatter coefficients and a suite of lower bounds for specific functions in the simultaneous communication model.
Abstract: We use tools and techniques from information theory to study communication complexity problems in the one-way and simultaneous communication models. Our results include: (1) a tight characterization of multi-party one-way communication complexity for product distributions in terms of VC-dimension and shatter coefficients; (2) an equivalence of multi-party one-way 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 multi-party set disjointness problem of Alon et al. (1999) and for the generalized addressing function problem of Babai et al. (1996) 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.
91 citations
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22 Apr 2003TL;DR: A set of microbenchmarks is used to quantify the cost of this commoditization of supercomputers, measuring software overhead, latency, and bandwidth on five contemporary supercomputing networks and comparing the performance of the ubiquitous MPI layer to that of lower-level communication layers, and quantify the advantages of the latter for small message performance.
Abstract: High-end supercomputers are increasingly built out of commodity components, and lack tight integration between the processor and network. This often results in inefficiencies in the communication subsystem, such as high software overheads and/or message latencies. In this paper we use a set of microbenchmarks to quantify the cost of this commoditization, measuring software overhead, latency, and bandwidth on five contemporary supercomputing networks. We compare the performance of the ubiquitous MPI layer to that of lower-level communication layers, and quantify the advantages of the latter for small message performance. We also provide data on the potential for various communication-related optimizations, such as overlapping communication with computation or other communication. Finally, we determine the minimum size needed for a message to be considered 'large' (i.e., bandwidth-bound) on these platforms, and provide historical data on the software overheads of a number of supercomputers over the past decade.
91 citations
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TL;DR: A PIR protocol with the communication complexity of O(γ logn) bits and extended to a private block retrieval (PBR) protocol, a natural and more practical extension of PIR in which the user retrieves a block of bits, instead of retrieving single bit.
Abstract: Private Information Retrieval (PIR) allows a user to retrieve the ith bit of an n-bit database without revealing to the database server the value of i. In this paper, we present a PIR protocol with the communication complexity of O(γ logn) bits, where -y is the ciphertext size. Furthermore, we extend the PIR protocol to a private block retrieval (PBR) protocol, a natural and more practical extension of PIR in which the user retrieves a block of bits, instead of retrieving single bit. Our protocols are built on the state-of-the-art fully homomorphic encryption (FHE) techniques and provide privacy for the user if the underlying FHE scheme is semantically secure. The total communication complexity of our PBR is O(γ logm + γn/m) bits, where m is the number of blocks. The total computation complexity of our PBR is O(m logm) modular multiplications plus O(n=2) modular additions. In terms of total protocol execution time, our PBR protocol is more efficient than existing PBR protocols which usually require to compute O(n=2) modular multiplications when the size of a block in the database is large and a high-speed network is available.
91 citations