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arXiv:1508.05189 · Leibniz international proceedings in informatics

Correlation in Hard Distributions in Communication Complexity

Ralph Bottesch, Dmitry Gavinsky, Hartmut Klauck

Published Jan 1, 2015• 1 citation
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Abstract

We study the effect that the amount of correlation in a bipartite distribution has on the communication complexity of a problem under that distribution. We introduce a new family of complexity measures that interpolates between the two previously studied extreme cases: the (standard) randomised communication complexity and the case of distributional complexity under product distributions. We give a tight characterisation of the randomised complexity of Disjointness under distributions with mutual information $k$, showing that it is $Θ(\sqrt{n(k+1)})$ for all $0\leq k\leq n$. This smoothly interpolates between the lower bounds of Babai, Frankl and Simon for the product distribution case ($k=0$), and the bound of Razborov for the randomised case. The upper bounds improve and generalise what was known for product distributions, and imply that any tight bound for Disjointness needs $Ω(n)$ bits of mutual information in the corresponding distribution. We study the same question in the distributional quantum setting, and show a lower bound of $Ω((n(k+1))^{1/4})$, and an upper bound, matching up to a logarithmic factor. We show that there are total Boolean functions $f_d$ on $2n$ inputs that have distributional communication complexity $O(\log n)$ under all distributions of information up to $o(n)$, while the (interactive) distributional complexity maximised over all distributions is $Θ(\log d)$ for $6n\leq d\leq 2^{n/100}$. We show that in the setting of one-way communication under product distributions, the dependence of communication cost on the allowed error $ε$ is multiplicative in $\log(1/ε)$ -- the previous upper bounds had the dependence of more than $1/ε$.

DOI
10.4230/lipics.approx-random.2015.544

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We show two new direct product results in two different models of communication complexity. Our first result is in the one-way public-coin model. Let f ⊆ X × Y × Z be a relation and e > 0 be a constant. Let R1,pube(f) represent the communication complexity of f, with worst-case error e in this model. We show that if for computing fk (k independent copies of f) in this model, o(k c R1, pub1/3(f)) communication is used, then the success is exponentially small in k. We show a new tight characterization of communication complexity in this model which strengthens the tight characterization shown in Jain et al. [2008]. We use this new characterization to show our direct product result and this characterization may also be of independent interest.Our second direct product result is in the model of two-way public-coin communication complexity. We show a direct product result for all relations in this model in terms of a new complexity measure that we define. Our new measure is a generalization to nonproduct distributions, of the two-way product subdistribution bound of Jain et al. [2008]. Our direct product result therefore generalizes to nonproduct distributions, their direct product result in terms of the two-way product subdistribution bound. As an application of our new direct product result, we reproduce (via completely different arguments) strong direct product result for the set-disjointness problem which was previously shown by Klauck [2010]. We show this by proving that our new complexity measure gives a tight lower bound of Ω(n) for the set-disjointness problem on n-bit inputs (this strengthens the linear lower bound on the rectangle/corruption bound for set-disjointness shown by Razborov [1992]). In addition, we show that many previously known direct product results in this model are uniformly implied and often strengthened by our result.
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We study an extension of the standard two-party communication model in which Alice and Bob hold probability distributions ppp and qqq over domains XXX and YYY, respectively. Their goal is to estimate Ex∼p, y∼q[f(x,y)] \mathbb{E}_{x \sim p,\, y \sim q}[f(x, y)] Ex∼p,y∼q​[f(x,y)] to within additive error ε\varepsilonε for a bounded function fff, known to both parties. We refer to this as the distributed estimation problem. Special cases of this problem arise in a variety of areas including sketching, databases and learning. Our goal is to understand how the required communication scales with the communication complexity of fff and the error parameter ε\varepsilonε. The random sampling approach -- estimating the mean by averaging fff over O(1/ε2)O(1/\varepsilon^2)O(1/ε2) random samples -- requires O(R(f)/ε2)O(R(f)/\varepsilon^2)O(R(f)/ε2) total communication, where R(f)R(f)R(f) is the randomized communication complexity of fff. We design a new debiasing protocol which improves the dependence on 1/ε1/\varepsilon1/ε to be linear instead of quadratic. Additionally we show better upper bounds for several special classes of functions, including the Equality and Greater-than functions. We introduce lower bound techniques based on spectral methods and discrepancy, and show the optimality of many of our protocols: the debiasing protocol is tight for general functions, and that our protocols for the equality and greater-than functions are also optimal. Furthermore, we show that among full-rank Boolean functions, Equality is essentially the easiest.