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Communication Complexity and Discrepancy of Halfplanes

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

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ACM Subject Classification
  • Theory of computation → Communication complexity
Keywords
  • Randomized communication complexity
  • Discrepancy theory
  • factorization norm

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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) https://doi.org/10.4230/LIPIcs.SoCG.2024.5

Author Details

Manasseh Ahmed
  • Marianopolis College, Montreal, Canada
Tsun-Ming Cheung
  • School of Computer Science, McGill University, Montreal, Canada
Hamed Hatami
  • School of Computer Science, McGill University, Montreal, Canada
Kusha Sareen
  • School of Computer Science, McGill University, Montreal, Canada

Funding

  • Hatami, Hamed: Supported by an NSERC grant.

Acknowledgements

We wish to thank an anonymous reviewer for pointing out a possible modification in the proof of Proposition 5, which improves the resultant bound.

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