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From information to exact communication

Published: 01 June 2013 Publication History

Abstract

We develop a new local characterization of the zero-error information complexity function for two-party communication problems, and use it to compute the exact internal and external information complexity of the 2-bit AND function: IC(AND,0) = C≅ 1.4923 bits, and ICext(AND,0) = log2 3 ≅ 1.5839 bits. This leads to a tight (upper and lower bound) characterization of the communication complexity of the set intersection problem on subsets of {1,...,n} (the player are required to compute the intersection of their sets), whose randomized communication complexity tends to C⋅ n pm o(n) as the error tends to zero.
The information-optimal protocol we present has an infinite number of rounds. We show this is necessary by proving that the rate of convergence of the r-round information cost of AND to IC(AND,0)=C behaves like Θ(1/r2), i.e. that the r-round information complexity of AND is C+Θ(1/r2).
We leverage the tight analysis obtained for the information complexity of AND to calculate and prove the exact communication complexity of the set disjointness function Disjn(X,Y) = - vi=1n AND(xi,yi) with error tending to 0, which turns out to be = CDISJ⋅ n pm o(n), where CDISJ≅ 0.4827. Our rate of convergence results imply that an asymptotically optimal protocol for set disjointness will have to use ω(1) rounds of communication, since every r-round protocol will be sub-optimal by at least Ω(n/r2) bits of communication.
We also obtain the tight bound of 2/ln2 k pm o(k) on the communication complexity of disjointness of sets of size ≤ k. An asymptotic bound of Θ(k) was previously shown by Hastad and Wigderson.

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cover image ACM Conferences
STOC '13: Proceedings of the forty-fifth annual ACM symposium on Theory of Computing
June 2013
998 pages
ISBN:9781450320290
DOI:10.1145/2488608
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Published: 01 June 2013

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Author Tags

  1. communication complexity
  2. disjointness
  3. information complexity

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June 1 - 4, 2013
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