Area and Perimeter of the Convex Hull of Stochastic Points

TL;DR

This paper investigates the probability distribution of the area and perimeter of the convex hull of random subsets of points, providing complexity results and approximation algorithms for these probabilistic measures.

Contribution

It establishes ext{ extbackslash#}P-hardness for computing these probabilities and introduces efficient approximation algorithms with probabilistic guarantees.

Findings

ext{ extbackslash#}P-hard to compute exact probabilities

Provides polynomial-time approximation algorithms

Offers Monte Carlo methods with probabilistic error bounds

Abstract

Given a set $P$ of $n$ points in the plane, we study the computation of the probability distribution function of both the area and perimeter of the convex hull of a random subset $S$ of $P$. The random subset $S$ is formed by drawing each point $p$ of $P$ independently with a given rational probability $\pi_p$. For both measures of the convex hull, we show that it is \#P-hard to compute the probability that the measure is at least a given bound $w$. For $\varepsilon\in(0,1)$, we provide an algorithm that runs in $O(n^{6}/\varepsilon)$ time and returns a value that is between the probability that the area is at least $w$, and the probability that the area is at least $(1-\varepsilon)w$. For the perimeter, we show a similar algorithm running in $O(n^{6}/\varepsilon)$ time. Finally, given $\varepsilon,\delta\in(0,1)$ and for any measure, we show an $O(n\log n+ (n/\varepsilon^2)\log(1/\delta))$-time Monte Carlo algorithm that returns a value that, with probability of success at least $1-\delta$, differs at most $\varepsilon$ from the probability that the measure is at least $w$.