tech_report_guanfeng_MEQ - Multiparty Equality Function...

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Unformatted text preview: Multiparty Equality Function Computation in Networks with Point-to-Point Links Guanfeng Liang and Nitin Vaidya Department of Electrical and Computer Engineering, and Coordinated Science Laboratory University of Illinois at Urbana-Champaign gliang2@illinois.edu, nhv@illinois.edu Technical Report October 26, 2010 This research is supported in part by Army Research Oce grant W-911-NF-0710287 and National Science Foundation award 1059540. Any opinions, findings, and conclusions or recommendations ex- pressed here are those of the authors and do not necessarily reect the views of the funding agencies or the U.S. government. 1 Introduction In this report, we study the multiparty communication complexity problem of the multiparty equality function (MEQ): EQ ( x 1 , ,x n ) = { 0 if x 1 = = x n 1 otherwise . (1) The input vector x = ( x 1 , ,x n ) is distributed among n 2 nodes, with x i known to node i , where x i is chosen from the set { 1 , ,M } , for some integer M > 0. 1.1 Communication Complexity The notion of communication complexity (CC) was introduced by Yao in 1979 [2], who investigated the following problem involving two separated parties (Alice and Bob) want to mutually compute a Boolean function that is defined on pairs of inputs. Formally, let f : X Y 7 { , 1 } be a Boolean function. The communication problem for f is the following two-party game: Alice receives x X and Bob receives y Y , and the goal is for them to compute f ( x,y ), collaboratively. Alice and Bob have unlimited computational power and a full description of f , but they do not know each others input. They determine the output value by exchanging messages. The computation ends when either Alice or Bob has enough information to determine f ( x,y ), and sends a special symbol halt to the other party. A protocol P for computing f is an algorithm, according to which Alice and Bob send binary messages to each other. A protocol proceeds in rounds. In every round, the protocol specifies whose turn it is to send a message. Each party in his/her turn sends one bit that may depend on his/her input and the previous messages he/she has received. A correct protocol for f should terminate for every input pair ( x,y ) X Y , when either Alice or Bob knows f ( x,y ). The communication complexity of a protocol P is the number of bits exchanged for the worst case input pair. The communication complexity of a Boolean function f : X Y 7 { , 1 } , is that of the protocols for f with the least complexity. 1.2 Multiparty Communication Complexity There is more than one way to generalize communication complexity to a multiparty setting. The most commonly used model is the number on the forehead model introduced in [1]. Formally, there is some function f : n i =1 X i 7 { , 1 } , and the input is ( x 1 ,x 2 , ,x n ) where each x i X i . The i-th party can see all the x j such that j = i . As in the 2-party case, the....
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This note was uploaded on 02/08/2012 for the course ECE 428 taught by Professor Hu during the Spring '08 term at University of Illinois, Urbana Champaign.

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tech_report_guanfeng_MEQ - Multiparty Equality Function...

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