License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2016.135
URN: urn:nbn:de:0030-drops-62708
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6270/
Harsha, Prahladh ;
Jain, Rahul ;
Radhakrishnan, Jaikumar
Partition Bound Is Quadratically Tight for Product Distributions
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.
BibTeX - Entry
@InProceedings{harsha_et_al:LIPIcs:2016:6270,
author = {Prahladh Harsha and Rahul Jain and Jaikumar Radhakrishnan},
title = {{Partition Bound Is Quadratically Tight for Product Distributions}},
booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
pages = {135:1--135:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-013-2},
ISSN = {1868-8969},
year = {2016},
volume = {55},
editor = {Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6270},
URN = {urn:nbn:de:0030-drops-62708},
doi = {10.4230/LIPIcs.ICALP.2016.135},
annote = {Keywords: partition bound, product distribution, communication complexity, query complexity}
}
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Keywords: |
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partition bound, product distribution, communication complexity, query complexity |
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Collection: |
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43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016) |
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Issue Date: |
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2016 |
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Date of publication: |
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23.08.2016 |
