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arXiv · 2604.23021

The Prophet and the Voronoi Diagram

Sariel Har-Peled
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Abstract

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 $\Theta( \log n)$ than the average payoff.

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Sariel Har-Peled. 2026-04-24. The Prophet and the Voronoi Diagram. https://arxiv.org/abs/2604.23021

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