lect07 - 6.841 Advanced Complexity Theory Feb 7 2005...

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Unformatted text preview: 6.841 Advanced Complexity Theory Feb 7, 2005 Lecture 7: Communication Complexity and Lower Bounds Lecturer: Madhu Sudan Scribe: David Chen This lecture gives an introduction to Communication Complexity. We go over the properties and examples of Communication Complexity, Karchmer-Wigderson games, a lower bound related to PARITY, and log-rank lower bounds. 1 Introduction to Communication Complexity Communication complexity was introduced by Yao in 1979, and involves the following interac- tion between Alice and Bob. Alice knows x ∈ { , 1 } n , and Bob knows y ∈ { , 1 } n . Alice and Bob are allowed to send each other one bit at a time. The goal is to have both of them compute some z ∈ { , 1 } m such that z = f ( x,y ). Note that the more general version of this interaction requires them to output z such that ( x,y,z ) ∈ R for some predetermined relation R ⊆ { , 1 } n × { , 1 } n × { , 1 } m . Given a relation R , then, we ask: how many bits are needed for Alice and Bob to output the same z that satisfies the relation? To address the problem, we protocols . A protocol Π specifies the following: • Given a history of sent bits b 1 ,...,b i , whether the interaction should stop (and Alice and Bob computes their output with the bits sent) • How to calculate z . The protocol should specify functions f A and f B such that, upon stop- ping, Alice and Bob can calculate z = f A ( b 1 ,...,b i ,x ) = f B ( b 1 ,...,b i ,y ) • Who sends the next bit • How to calculate the next bit sent. That is, if Alice is to send bit b i +1 , the protocol should specify a g such that b i +1 = g ( b 1 ,...,b i ,x ). A protocol Π solves a relation R if it halts and outputs a z such that ( x,y,z ) ∈ R . 2 Communication Complexity We define the Communication complexity of a protocol to be CC (Π) = the number of bits transmitted with Π in the worst case . We consider R that are well-defined in the sense that for every pair ( x,y ), there exists some z such that ( x,y,z ) ∈ R . Then, the communication complexity of a relation is CC ( R ) = min Π solving R { CC (Π) } Using this model, we ignore all computation that Alice and Bob do on their own, and focus on the number of bits sent between the two. Trivially, an upper bound to this is 2 n . Alice and Bob can both send each other the entirety of their strings, and each can calculate f ( x,y ). The question)....
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This note was uploaded on 04/02/2010 for the course CS 6.841 taught by Professor Madhusudan during the Spring '09 term at MIT.

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lect07 - 6.841 Advanced Complexity Theory Feb 7 2005...

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