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37
Protocols and impossibility results for gossipbased communication mechanisms
, 2002
"... In recent years, gossipbased algorithms have gained prominence as a methodology for designing robust and scalable communication schemes in large distributed systems. The premise underlying distributed gossip is very simple: in each time step, each node v in the system selects some other node w as a ..."
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Cited by 55 (3 self)
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In recent years, gossipbased algorithms have gained prominence as a methodology for designing robust and scalable communication schemes in large distributed systems. The premise underlying distributed gossip is very simple: in each time step, each node v in the system selects some other node w as a communication partner — generally by a simple randomized rule — and exchanges information with w; over a period of time, information spreads through the system in an “epidemic fashion”. A fundamental issue which is not well understood is the following: how does the underlying lowlevel gossip mechanism — the means by which communication partners are chosen — affect one’s ability to design efficient highlevel gossipbased protocols? We establish one of the first concrete results addressing this question, by showing a fundamental limitation on the power of the commonly used uniform gossip mechanism for solving nearestresource location problems. In contrast, very efficient protocols for this problem can be designed using a nonuniform spatial gossip mechanism, as established in earlier work with Alan Demers. We go on to consider the design of protocols for more complex problems, providing an efficient distributed gossipbased protocol for a set of nodes in Euclidean space to construct an approximate minimum spanning tree. Here too, we establish a contrasting limitation on the power of uniform gossip for solving this problem. Finally, we investigate gossipbased packet routing as a primitive that underpins the communication patterns in many protocols, and as a way to understand the capabilities of different gossip mechanisms at a general level.
Coding for Interactive Communication
 IN PROCEEDINGS OF THE 25TH ANNUAL SYMPOSIUM ON THEORY OF COMPUTING
, 1996
"... Let the input to a computation problem be split between two processors connected by a communication link; and let an interactive protocol ß be known by which, on any input, the processors can solve the problem using no more than T transmissions of bits between them, provided the channel is noiseless ..."
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Cited by 37 (4 self)
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Let the input to a computation problem be split between two processors connected by a communication link; and let an interactive protocol ß be known by which, on any input, the processors can solve the problem using no more than T transmissions of bits between them, provided the channel is noiseless in each direction. We study the following question: if in fact the channel is noisy, what is the effect upon the number of transmissions needed in order to solve the computation problem reliably? Technologically this concern is motivated by the increasing importance of communication as a resource in computing, and by the tradeoff in communications equipment between bandwidth, reliability and expense. We treat a model with random channel noise. We describe a deterministic method for simulating noiselesschannel protocols on noisy channels, with only a constant slowdown. This is an analog for general interactive protocols of Shannon's coding theorem, which deals only with data transmission, ...
New Lower Bounds for Hopcroft's Problem
, 1996
"... We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst cas ..."
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Cited by 33 (6 self)
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We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time #(n log m+n 2/3 m 2/3 +m log n) in two dimensions, or #(n log m+n 5/6 m 1/2 +n 1/2 m 5/6 + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2 O(log # (n+m)) of the best known upper bound, due to Matousek. Previously, the best known lower bound, in any dimension, was #(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive low...
List Partitions
 Proc. 31st Ann. ACM Symp. on Theory of Computing
, 2003
"... List partitions generalize list colourings and list homomorphisms. Each symmetric matrix M over 0; 1; defines a list partition problem. Different choices of the matrix M lead to many wellknown graph theoretic problems including the problem of recognizing split graphs and their generalizations, ..."
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Cited by 27 (11 self)
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List partitions generalize list colourings and list homomorphisms. Each symmetric matrix M over 0; 1; defines a list partition problem. Different choices of the matrix M lead to many wellknown graph theoretic problems including the problem of recognizing split graphs and their generalizations, finding homogeneous sets, joins, clique cutsets, stable cutsets, skew cutsets and so on. We develop tools which allow us to classify the complexity of many list partition problems and, in particular, yield the complete classification for small matrices M . Along the way, we obtain a variety of specific results including: generalizations of Lov'asz's communication bound on the number of cliqueversus stableset separators; polynomialtime algorithms to recognize generalized split graphs; a polynomial algorithm for the list version of the Clique Cutset Problem; and the first subexponential algorithm for the Skew Cutset Problem of Chv'atal. We also show that the dichotomy (NP complete versus polynomialtime solvable), conjectured for certain graph homomorphism problems would, if true, imply a slightly weaker dichotomy (NP complete versus quasipolynomial) for our list partition problems 1 . Email: tomas@theory.stanford.edu. y School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada, V5A1S6. Email: pavol@cs.sfu.ca. Supported by a Research Grant from the National Sciences and Engineering Research Council. z Departamento da Ciencia da Computac~ao  I.M., COPPE/Sistemas, Universidade Federal do Rio de Janeiro, RJ, 21945970, Brasil. Email: sula@cos.ufrj.br. Supported by CNPq and PRONEX 107/97. x Department of Computer Science, Stanford University, CA 943059045. Email: rajeev@cs.stanford.edu. Supported by an ARO MURI Grant DAAH04961...
On Frege and Extended Frege Proof Systems
, 1993
"... We propose a framework for proving lower bounds to the size of EF  proofs (equivalently, to the number of proofsteps in Fproofs) in terms of boolean valuations . The concept is motivated by properties of propositional provability in models of bounded arithmetic and it is a finitisation of a parti ..."
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Cited by 21 (2 self)
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We propose a framework for proving lower bounds to the size of EF  proofs (equivalently, to the number of proofsteps in Fproofs) in terms of boolean valuations . The concept is motivated by properties of propositional provability in models of bounded arithmetic and it is a finitisation of a particular forcing construction explained also in the paper. It reduces the question of proving a lower bound to the question of constructing a partial boolean algebra and a map of formulas into that algebra with particular properties. We show that in principle one can obtain via this method optimal lower bounds (up to a polynomial increase). Introduction A propositional proof system is any polynomial time function P whose range is exactly the set of tautologies TAUT, cf. [17]. For ø a tautology any string ß such that P (ß) = ø is called a P proof of ø . Any usual propositional calculus, be it resolution or extended resolution, a Hilbert style system based on finitely many axiom schemes and inf...
Separating the Communication Complexities of MOD m and MOD p Circuits
 IN PROC. 33RD IEEE FOCS
, 1995
"... We prove in this paper that it is much harder to evaluate depth2, sizeN circuits with MOD m gates than with MOD p gates by kparty communication protocols: we show a kparty protocol which communicates O(1) bits to evaluate circuits with MOD p gates, while evaluating circuits with MOD m gates ..."
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Cited by 20 (4 self)
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We prove in this paper that it is much harder to evaluate depth2, sizeN circuits with MOD m gates than with MOD p gates by kparty communication protocols: we show a kparty protocol which communicates O(1) bits to evaluate circuits with MOD p gates, while evaluating circuits with MOD m gates needs\Omega\Gamma N) bits, where p denotes a prime, and m a composite, nonprime power number. As a corollary, for all m, we show a function, computable with a depth2 circuit with MODm gates, but not with any depth2 circuit with MOD p gates. Obviously, the kparty protocols are not weaker than the k 0 party protocols, for k 0 ? k. Our results imply that if there is a prime p between k and k 0 : k ! p k 0 , then there exists a function which can be computed by a k 0 party protocol with a constant number of communicated bits, while any kparty protocol needs linearly many bits of communication. This result gives a hierarchy theorem for multiparty protocols.
Complexity of Graph Partition Problems
 31ST ANNUAL ACM STOC
, 1999
"... We introduce a parametrized family of graph problems that includes several wellknown graph partition problems as special cases. We develop tools which allow us to classify the complexity of many problems in this family, and in particular lead us to a complete classification for small values of the ..."
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Cited by 18 (5 self)
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We introduce a parametrized family of graph problems that includes several wellknown graph partition problems as special cases. We develop tools which allow us to classify the complexity of many problems in this family, and in particular lead us to a complete classification for small values of the parameters. Along the way, we obtain a variety of specific results including the following: a generalization of a communication bound on the number of cliqueversusindependentset separators; polynomialtime algorithms to recognize generalized split graphs; and, a quasipolynomial algorithm for the Skew Cutset Problem that essentially resolves an open problem posed by Chv'atal. The last two problems have interesting connections to the Strong Perfect Graph Conjecture of Berge.
The BNS Lower Bound for MultiParty Protocols is Nearly Optimal
 Information and Computation
, 1994
"... We present a multiparty protocol which computes the Generalized Inner Product (GIP) function, introduced by Babai, Nisan and Szegedy [BNS]. Our protocol shows that the lower bound for the multiparty communication complexity of the GIP function, given by [BNS], cannot be improved significantly. ..."
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Cited by 18 (0 self)
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We present a multiparty protocol which computes the Generalized Inner Product (GIP) function, introduced by Babai, Nisan and Szegedy [BNS]. Our protocol shows that the lower bound for the multiparty communication complexity of the GIP function, given by [BNS], cannot be improved significantly. keywords: multiparty communication, communication protocols 1. INTRODUCTION In the twoparty communication game, introduced by Yao [Y], there are two players, P 0 and P 1 , and a function f : f0; 1g 2n ! f0; 1g. The goal of the players is to compute the value of f(A 0 ; A 1 ) cooperatively, for some A 0 ; A 1 2 f0; 1g n , where A 0 is known only to P 1 , and A 1 is known only to P 0 , and after the computation both players should know the value f(A 0 ; A 1 ). The players have unlimited computational power, and both know the definition of f . P 0 and P 1 communicate via a blackboard, which is seen by both of them: they are allowed to write bits on the blackboard. The cost of the compu...
A Coding Theorem for Distributed Computation
 STOC
, 1994
"... Shannon's Coding Theorem shows that in order to reliably transmit a message of T bits over a noisy communication channel, only a constant slowdown factor is necessary in the case when the channel is noisy, relative to the case in which the channel is noiseless. (The time required is asymptotically ..."
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Cited by 17 (0 self)
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Shannon's Coding Theorem shows that in order to reliably transmit a message of T bits over a noisy communication channel, only a constant slowdown factor is necessary in the case when the channel is noisy, relative to the case in which the channel is noiseless. (The time required is asymptotically C , where 0 ! C 1 is the "Shannon capacity", a function only of the noise characteristics.) The theorem ensures that the probability of a decoding error is exponentially small in the message length T . Recently the second author obtained an analogous result for arbitrary interactive communication protocols between two processors. In the present
The rank and size of graphs
 J. of Graph Theory
, 1996
"... We show that the number of points with pairwise different sets of neighbors in a graph is O(2 r/2) where r is the rank of the adjacency matrix. We also give an example that achieves this bound. 1 ..."
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Cited by 15 (0 self)
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We show that the number of points with pairwise different sets of neighbors in a graph is O(2 r/2) where r is the rank of the adjacency matrix. We also give an example that achieves this bound. 1