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DOI: 10.4230/LIPIcs.SoCG.2024.5

Communication Complexity and Discrepancy of Halfplanes

Authors: Manasseh Ahmed, Tsun-Ming Cheung, Hamed Hatami, and Kusha Sareen

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract

We study the discrepancy of the following communication problem. Alice receives a halfplane, and Bob receives a point in the plane, and their goal is to determine whether Bob’s point belongs to Alice’s halfplane. This communication task corresponds to determining whether x₁y₁+y₂ ≥ x₂, where the first player knows (x₁,x₂) and the second player knows (y₁,y₂). Denoting n = m³, we show that when the inputs are chosen from [m] × [m²], the communication discrepancy of the above problem is O(n^{-1/6} log^{3/2} n). On the other hand, through the connections to the notion of hereditary discrepancy by Matoušek, Nikolov, and Tawler (IMRN 2020) and a classical result of Matoušek (Discrete Comput. Geom. 1995), we show that the communication discrepancy of every set of n points and n halfplanes is at least Ω(n^{-1/4} log^{-1} n).

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Manasseh Ahmed, Tsun-Ming Cheung, Hamed Hatami, and Kusha Sareen. Communication Complexity and Discrepancy of Halfplanes. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ahmed_et_al:LIPIcs.SoCG.2024.5,
  author =	{Ahmed, Manasseh and Cheung, Tsun-Ming and Hatami, Hamed and Sareen, Kusha},
  title =	{{Communication Complexity and Discrepancy of Halfplanes}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{5:1--5:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.5},
  URN =		{urn:nbn:de:0030-drops-199504},
  doi =		{10.4230/LIPIcs.SoCG.2024.5},
  annote =	{Keywords: Randomized communication complexity, Discrepancy theory, factorization norm}
}

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Communication Complexity and Discrepancy of Halfplanes

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