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DOI: 10.4230/LIPIcs.ICALP.2017.51

Sensitivity Conjecture and Log-Rank Conjecture for Functions with Small Alternating Numbers

Authors: Chengyu Lin and Shengyu Zhang

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for f(x and y) and f(x xor y) with monotone functions f, where and and xor are bit-wise AND and XOR , respectively. In this paper, we extend these results to functions f which alternate values for a relatively small number of times on any monotone path from 0^n to 1^n. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers.

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Chengyu Lin and Shengyu Zhang. Sensitivity Conjecture and Log-Rank Conjecture for Functions with Small Alternating Numbers. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{lin_et_al:LIPIcs.ICALP.2017.51,
  author =	{Lin, Chengyu and Zhang, Shengyu},
  title =	{{Sensitivity Conjecture and Log-Rank Conjecture for Functions with Small Alternating Numbers}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{51:1--51:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.51},
  URN =		{urn:nbn:de:0030-drops-74045},
  doi =		{10.4230/LIPIcs.ICALP.2017.51},
  annote =	{Keywords: Analysis of Boolean functions, Sensitivity Conjecture, Log-rank Conjecture, Alternating Number}
}

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Sensitivity Conjecture and Log-Rank Conjecture for Functions with Small Alternating Numbers

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