The Prophet and the Voronoi Diagram

TL;DR

This paper studies an online selection problem where a player chooses a point from a stream to maximize the Voronoi cell area, achieving near-optimal payoff with a simple strategy.

Contribution

It introduces a simple strategy for online point selection that guarantees a constant-factor approximation to the prophet's optimal payoff.

Findings

The strategy achieves probability ≥ 1 - O(1/√n) of near-optimal payoff.

The player's payoff exceeds the average by a factor of Θ(log n).

The approach is surprisingly effective given the problem's complexity.

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.