Convex Hull for Probabilistic Points

TL;DR

This paper evaluates an O(n log n) divide-and-conquer algorithm for convex hulls when input points are probabilistic, introducing a new error model to quantify geometric correctness with statistical confidence.

Contribution

It introduces a novel certificate error model for analyzing approximate geometric structures under probabilistic input points.

Findings

Algorithm achieves expected correctness based on confidence levels.

New error model links geometric accuracy with statistical confidence.

Proves robustness of convex hull computation with probabilistic data.

Abstract

We analyze the correctness of an O(n log n) time divide-and-conquer algorithm for the convex hull problem when each input point is a location determined by a normal distribution. We show that the algorithm finds the convex hull of such probabilistic points to precision within some expected correctness determined by a user-given confidence value. In order to precisely explain how correct the resulting structure is, we introduce a new certificate error model for calculating and understanding approximate geometric error based on the fundamental properties of a geometric structure. We show that this new error model implies correctness under a robust statistical error model, in which each point lies within the hull with probability at least that of the user-given confidence value, for the convex hull problem.