An Optimal Randomized Algorithm for Finding the Saddlepoint
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An Optimal Randomized Algorithm for Finding the Saddlepoint

Authors Justin Dallant , Frederik Haagensen , Riko Jacob , László Kozma , Sebastian Wild

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Author Details

Justin Dallant
  • Department of Computer Science, Université libre de Bruxelles, Belgium
Frederik Haagensen
  • Department of Computer Science, IT University of Copenhagen, Denmark
Riko Jacob
  • Department of Computer Science, IT University of Copenhagen, Denmark
László Kozma
  • Institut für Informatik, Freie Universität Berlin, Germany
Sebastian Wild
  • Department of Computer Science, University of Liverpool, UK

Funding

  • Dallant, Justin: Supported by the French Community of Belgium via the funding of a FRIA grant.
  • Haagensen, Frederik: Supported by Independent Research Fund Denmark, grant 0136-00144B, "DISTRUST" project.
  • Kozma, László: Supported by DFG grant KO 6140/1-2.
  • Wild, Sebastian: Supported by EPSRC grant EP/X039447/1.

Acknowledgements

This work was initiated at Dagstuhl Seminar 23211 "Scalable Data Structures".

Cite As Get BibTex

Justin Dallant, Frederik Haagensen, Riko Jacob, László Kozma, and Sebastian Wild. An Optimal Randomized Algorithm for Finding the Saddlepoint. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 44:1-44:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.44

Abstract

A saddlepoint of an n × n matrix is an entry that is the maximum of its row and the minimum of its column. Saddlepoints give the value of a two-player zero-sum game, corresponding to its pure-strategy Nash equilibria; efficiently finding a saddlepoint is thus a natural and fundamental algorithmic task. 
For finding a strict saddlepoint (an entry that is the strict maximum of its row and the strict minimum of its column) an O(n log* n)-time algorithm was recently obtained by Dallant, Haagensen, Jacob, Kozma, and Wild, improving the O(n log n) bounds from 1991 of Bienstock, Chung, Fredman, Schäffer, Shor, Suri and of Byrne and Vaserstein. 
In this paper we present an optimal O(n)-time algorithm for finding a strict saddlepoint based on random sampling. Our algorithm, like earlier approaches, accesses matrix entries only via unit-cost binary comparisons. For finding a (non-strict) saddlepoint, we extend an existing lower bound to randomized algorithms, showing that the trivial O(n²) runtime cannot be improved even with the use of randomness.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • saddlepoint
  • matrix
  • comparison
  • search
  • randomized algorithms

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