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DOI: 10.4230/LIPIcs.FSTTCS.2016.15

On the Sensitivity Conjecture for Disjunctive Normal Forms

Authors: Karthik C. S. and Sébastien Tavenas

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Abstract

The sensitivity conjecture of Nisan and Szegedy [CC'94] asks whether for any Boolean function f, the maximum sensitivity s(f), is polynomially related to its block sensitivity bs(f), and hence to other major complexity measures. Despite major advances in the analysis of Boolean functions over the last decade, the problem remains widely open. In this paper, we consider a restriction on the class of Boolean functions through a model of computation (DNF), and refer to the functions adhering to this restriction as admitting the Normalized Block property. We prove that for any function f admitting the Normalized Block property, bs(f) <= 4 * s(f)^2. We note that (almost) all the functions mentioned in literature that achieve a quadratic separation between sensitivity and block sensitivity admit the Normalized Block property. Recently, Gopalan et al. [ITCS'16] showed that every Boolean function f is uniquely specified by its values on a Hamming ball of radius at most 2 * s(f). We extend this result and also construct examples of Boolean functions which provide the matching lower bounds.

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Karthik C. S. and Sébastien Tavenas. On the Sensitivity Conjecture for Disjunctive Normal Forms. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{s._et_al:LIPIcs.FSTTCS.2016.15,
  author =	{S., Karthik C. and Tavenas, S\'{e}bastien},
  title =	{{On the Sensitivity Conjecture for Disjunctive Normal Forms}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{15:1--15:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.15},
  URN =		{urn:nbn:de:0030-drops-68504},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.15},
  annote =	{Keywords: Boolean function, Sensitivity, Block sensitivity, DNF}
}

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DOI: 10.4230/LIPIcs.SOCG.2015.642

Building Efficient and Compact Data Structures for Simplicial Complexes

Authors: Jean-Daniel Boissonnat, Karthik C. S., and Sébastien Tavenas

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract

The Simplex Tree (ST) is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows efficient implementation of a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the Simplex Tree while retaining its functionalities. In addition, we propose two new data structures called Maximal Simplex Tree (MxST) and Simplex Array List (SAL). We analyze the compressed Simplex Tree, the Maximal Simplex Tree, and the Simplex Array List under various settings.

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Jean-Daniel Boissonnat, Karthik C. S., and Sébastien Tavenas. Building Efficient and Compact Data Structures for Simplicial Complexes. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 642-657, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{boissonnat_et_al:LIPIcs.SOCG.2015.642,
  author =	{Boissonnat, Jean-Daniel and S., Karthik C. and Tavenas, S\'{e}bastien},
  title =	{{Building Efficient and Compact Data Structures for Simplicial Complexes}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{642--657},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.642},
  URN =		{urn:nbn:de:0030-drops-50981},
  doi =		{10.4230/LIPIcs.SOCG.2015.642},
  annote =	{Keywords: Simplicial complex, Compact data structures, Automaton, NP-hard}
}

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Documents authored by S., Karthik C.

On the Sensitivity Conjecture for Disjunctive Normal Forms

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Building Efficient and Compact Data Structures for Simplicial Complexes

Abstract

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