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DOI: 10.4230/LIPIcs.ITCS.2021.1

The Entropy of Lies: Playing Twenty Questions with a Liar

Authors: Yuval Dagan, Yuval Filmus, Daniel Kane, and Shay Moran

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract

"Twenty questions" is a guessing game played by two players: Bob thinks of an integer between 1 and n, and Alice’s goal is to recover it using a minimal number of Yes/No questions. Shannon’s entropy has a natural interpretation in this context. It characterizes the average number of questions used by an optimal strategy in the distributional variant of the game: let μ be a distribution over [n], then the average number of questions used by an optimal strategy that recovers x∼ μ is between H(μ) and H(μ)+1. We consider an extension of this game where at most k questions can be answered falsely. We extend the classical result by showing that an optimal strategy uses roughly H(μ) + k H_2(μ) questions, where H_2(μ) = ∑_x μ(x)log log 1/μ(x). This also generalizes a result by Rivest et al. (1980) for the uniform distribution. Moreover, we design near optimal strategies that only use comparison queries of the form "x ≤ c?" for c ∈ [n]. The usage of comparison queries lends itself naturally to the context of sorting, where we derive sorting algorithms in the presence of adversarial noise.

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Yuval Dagan, Yuval Filmus, Daniel Kane, and Shay Moran. The Entropy of Lies: Playing Twenty Questions with a Liar. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dagan_et_al:LIPIcs.ITCS.2021.1,
  author =	{Dagan, Yuval and Filmus, Yuval and Kane, Daniel and Moran, Shay},
  title =	{{The Entropy of Lies: Playing Twenty Questions with a Liar}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{1:1--1:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.1},
  URN =		{urn:nbn:de:0030-drops-135400},
  doi =		{10.4230/LIPIcs.ITCS.2021.1},
  annote =	{Keywords: entropy, twenty questions, algorithms, sorting}
}

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DOI: 10.4230/LIPIcs.CCC.2017.16

Trading Information Complexity for Error

Authors: Yuval Dagan, Yuval Filmus, Hamed Hatami, and Yaqiao Li

Published in: LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

Abstract

We consider the standard two-party communication model. The central problem studied in this article is how much can one save in information complexity by allowing a certain error. * For arbitrary functions, we obtain lower bounds and upper bounds indicating a gain that is of order Omega(h(epsilon)) and O(h(sqrt{epsilon})). Here h denotes the binary entropy function. * We analyze the case of the two-bit AND function in detail to show that for this function the gain is Theta(h(epsilon)). This answers a question of Braverman et al. [Braverman, STOC 2013]. * We obtain sharp bounds for the set disjointness function of order n. For the case of the distributional error, we introduce a new protocol that achieves a gain of Theta(sqrt{h(epsilon)}) provided that n is sufficiently large. We apply these results to answer another of question of Braverman et al. regarding the randomized communication complexity of the set disjointness function. * Answering a question of Braverman [Braverman, STOC 2012], we apply our analysis of the set disjointness function to establish a gap between the two different notions of the prior-free information cost. In light of [Braverman, STOC 2012], this implies that amortized randomized communication complexity is not necessarily equal to the amortized distributional communication complexity with respect to the hardest distribution. As a consequence, we show that the epsilon-error randomized communication complexity of the set disjointness function of order n is n[C_{DISJ} - Theta(h(epsilon))] + o(n), where C_{DISJ} ~ 0.4827$ is the constant found by Braverman et al. [Braverman, STOC 2012].

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Yuval Dagan, Yuval Filmus, Hamed Hatami, and Yaqiao Li. Trading Information Complexity for Error. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 16:1-16:59, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dagan_et_al:LIPIcs.CCC.2017.16,
  author =	{Dagan, Yuval and Filmus, Yuval and Hatami, Hamed and Li, Yaqiao},
  title =	{{Trading Information Complexity for Error}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{16:1--16:59},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{O'Donnell, Ryan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.16},
  URN =		{urn:nbn:de:0030-drops-75179},
  doi =		{10.4230/LIPIcs.CCC.2017.16},
  annote =	{Keywords: communication complexity, information complexity}
}

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The Entropy of Lies: Playing Twenty Questions with a Liar

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Trading Information Complexity for Error

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