The Prophet and the Voronoi Diagram
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917434725761024 |
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| author | Har-Peled, Sariel |
| author_facet | Har-Peled, Sariel |
| contents | Consider a stream of $n$ random points (say, from the unit square) arriving one by one, where a player has to make an irreversible immediate decision for each arriving point whether to pick it. The player has to pick a single point, and the payoff is the area of the cell of the picked point, in the final Voronoi diagram of \emph{all} the points. We show that there is a simple strategy so that with probability $\geq 1 - \tilde O(1/\sqrt{n})$, 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 this payoff is larger by a factor of $Θ( \log n)$ than the average payoff. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23021 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Prophet and the Voronoi Diagram Har-Peled, Sariel Computational Geometry Consider a stream of $n$ random points (say, from the unit square) arriving one by one, where a player has to make an irreversible immediate decision for each arriving point whether to pick it. The player has to pick a single point, and the payoff is the area of the cell of the picked point, in the final Voronoi diagram of \emph{all} the points. We show that there is a simple strategy so that with probability $\geq 1 - \tilde O(1/\sqrt{n})$, 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 this payoff is larger by a factor of $Θ( \log n)$ than the average payoff. |
| title | The Prophet and the Voronoi Diagram |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2604.23021 |