Omri Weinstein

From information to exact communication

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 IC^ext(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 <i>set disjointness</i> function Disj_n(X,Y) = - v_i=1ⁿ AND(x_i,y_i) with error tending to 0, which turns out to be = C_DISJ⋅ n pm o(n), where C_DISJ≅ 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.

Direct Products in Communication Complexity

We give exponentially small upper bounds on the success probability for computing the direct product of any function over any distribution using a communication protocol. Let suc(μ, f, C) denote the maximum success probability of a 2-party communication protocol for computing the boolean function f(x, y) with C bits of communication, when the inputs (x, y) are drawn from the distribution μ. Let μⁿ be the product distribution on n inputs and fⁿ denote the function that computes n copies of f on these inputs. We prove that if T log^3/2 T ≪ (C - 1)√n and suc(μ, f, C) <; 2/3, then suc(μⁿ, fⁿ, T) ≤ exp(-Ω(n)). When μ is a product distribution, we prove a nearly optimal result: as long as T log² T ≪ Cn, we must have suc(μⁿ, fⁿ, T) ≤ exp(-Ω(n)).

A Discrepancy Lower Bound for Information Complexity

This paper provides the first general technique for proving information lower bounds on two-party unbounded-rounds communication problems. We show that the discrepancy lower bound, which applies to randomized communication complexity, also applies to information complexity. More precisely, if the discrepancy of a two-party function f with respect to a distribution

Information Lower Bounds via Self-reducibility

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The Simultaneous Communication of Disjointness with Applications to Data Streams

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Information Complexity and the Quest for Interactive Compression

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