License
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://drops.dagstuhl.de/opus/volltexte/2016/6270/
Go to the corresponding LIPIcs Volume Portal


Harsha, Prahladh ; Jain, Rahul ; Radhakrishnan, Jaikumar

Partition Bound Is Quadratically Tight for Product Distributions

pdf-format:
LIPIcs-ICALP-2016-135.pdf (0.6 MB)


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}
}

Keywords: partition bound, product distribution, communication complexity, query complexity
Seminar: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 17.08.2016


DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI