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The Prophet and the Voronoi Diagram

Author Sariel Har-Peled

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Voronoi diagram
  • prophet inequality
  • secretary problem

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Abstract

Consider a stream of n random points (say, from the unit square) arriving one by one, where a player must make an immediate and irreversible decision upon each point’s arrival, whether to pick it. The player must pick exactly one such point, and the payoff is the area of the cell of the picked point, in the final Voronoi diagram of all the points. We show that there is a simple strategy so that with probability ≥ 1 - Õ(1/√n), the player’s payoff is only a constant factor smaller than the optimal choice (i.e., the one made by the prophet). This competitiveness is somewhat surprising, as both the optimal payoff and this strategy’s payoff are larger by a factor of Θ(log n) than the average payoff.

Cite As Get BibTex

Sariel Har-Peled. The Prophet and the Voronoi Diagram. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 41:1-41:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.41

Author Details

Sariel Har-Peled
  • Department of Computer Science, University of Illinois, Urbana, IL, USA

Funding

  • Har-Peled, Sariel: Work on this paper was partially supported by NSF AF award CCF-2317241.

Acknowledgements

The author thanks Farouk Harb for his useful comments on the manuscript.

References

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