From information to exact communication

Mark Braverman, Ankit Garg, Denis Pankratov, Omri Weinstein

Research output: Chapter in Book/Report/Conference proceedingConference contribution

51 Scopus citations

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 ± 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/r 2), 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) = ¬ ∨ni =1 AND(xi, yi) with error tending to 0, which turns out to be = CDISJ · n ± 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 rround protocol will be sub-optimal by at least Ω(n/r2) bits of communication. We also obtain the tight bound of 2/ ln 2k ± o(k) on the communication complexity of disjointness of sets of size ≤ k. An asymptotic bound of Θ(k) was previously shown by Håstad and Wigderson.

Original language English (US) STOC 2013 - Proceedings of the 2013 ACM Symposium on Theory of Computing 151-160 10 https://doi.org/10.1145/2488608.2488628 Published - Jul 11 2013 45th Annual ACM Symposium on Theory of Computing, STOC 2013 - Palo Alto, CA, United StatesDuration: Jun 1 2013 → Jun 4 2013

Publication series

Name Proceedings of the Annual ACM Symposium on Theory of Computing 0737-8017

Other

Other 45th Annual ACM Symposium on Theory of Computing, STOC 2013 United States Palo Alto, CA 6/1/13 → 6/4/13

• Software

Keywords

• Communication complexity
• Disjointness
• Information complexity