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# Partition Bound Is Quadratically Tight for Product Distributions

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## Cite As

Prahladh Harsha, Rahul Jain, and Jaikumar Radhakrishnan. Partition Bound Is Quadratically Tight for Product Distributions. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 135:1-135:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ICALP.2016.135

## Abstract

Let f: {0,1}^n*{0,1}^n -> {0,1} be a 2-party function. For every product distribution mu on {0,1}^n*{0,1}^n, we show that CC^{mu}_{0.49}(f) = O(log(prt_{1/8}(f))*log(log(prt_{1/8}(f)))^2), where CC^{mu}_{epsilon}(f) is the distributional communication complexity of f with error at most epsilon under the distribution mu and prt_{1/8}(f) is the partition bound of f, as defined by Jain and Klauck [Proc. 25th CCC, 2010]. We also prove a similar bound in terms of IC_{1/8}(f), the information complexity of f, namely, CC^{mu}_{0.49}(f) = O((IC_{1/8}(f)*log(IC_{1/8}(f)))^2). The latter bound was recently and independently established by Kol [Proc. 48th STOC, 2016] using a different technique. We show a similar result for query complexity under product distributions. Let g: {0,1}^n -> {0,1} be a function. For every bit-wise product distribution mu on {0,1}^n, we show that QC^{mu}_{0.49}(g) = O((log(qprt_{1/8}(g))*log(log(qprt_{1/8}(g))))^2), where QC^{mu}_{epsilon}(g) is the distributional query complexity of f with error at most epsilon under the distribution mu and qprt_{1/8}(g) is the query partition bound of the function g. Partition bounds were introduced (in both communication complexity and query complexity models) to provide LP-based lower bounds for randomized communication complexity and randomized query complexity. Our results demonstrate that these lower bounds are polynomially tight for product distributions.
##### Keywords
• partition bound
• product distribution
• communication complexity
• query complexity

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## References

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