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tech_report_guanfeng_MEQ - Multiparty Equality Function...

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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 [email protected], [email protected] Technical Report October 26, 2010 This research is supported in part by Army Research Office 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 reflect the views of the funding agencies or the U.S. government.
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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→ { 0 , 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 other’s 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→ { 0 , 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 : Π n i =1 X i 7→ { 0 , 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 n players have an agreed-upon protocol for communication, and all this communication is posted on a “public blackboard”. At the
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