The Prophet and the Voronoi Diagram

2026-04-24

AI Summary

This study addresses the online selection of a point from a sequentially arriving random point stream, aiming to maximize the area of its cell in the final Voronoi diagram relative to the optimal solution achievable by a prophet with full foresight. The authors propose a simple online strategy that makes an irrevocable selection decision upon each point’s arrival. This strategy achieves, for the first time, a constant-factor competitive ratio, guaranteeing— with probability at least $1 - \tilde{O}(1/\sqrt{n})$—a cell area within a constant factor of the prophet’s optimum. Notably, the expected performance significantly outperforms the average baseline by a factor of $\Theta(\log n)$. By integrating probabilistic analysis, online algorithms, and computational geometry, this work demonstrates that strong approximation guarantees are attainable in online geometric decision-making.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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

Problem

• prophet inequality
• Voronoi diagram
• online selection
• stochastic geometry
• competitive analysis

Innovation

• online selection
• Voronoi diagram
• prophet inequality
• competitive ratio
• stochastic geometry