Degree of Convexity and Expected Distances in Polygons

TL;DR

This paper introduces algorithms for calculating the Beer-index, visibility pairs, and expected distances in polygons, providing efficient solutions for complex geometric probability and distance computations.

Contribution

It presents new algorithms for computing the Beer-index, visibility pairs, and expected distances in polygons with improved time complexities.

Findings

Beer-index can be computed in O(n^2) time.

Visibility pairs among points can be counted efficiently.

Expected L1-distance can be computed in linear time.

Abstract

We present an algorithm for computing the so-called Beer-index of a polygon $P$ in $O(n^2)$ time, where $n$ is the number of corners. The polygon $P$ may have holes. The Beer-index is the probability that two points chosen independently and uniformly at random in $P$ can see each other. Given a finite set $M$ of $m$ points in a simple polygon $P$, we also show how the number of pairs in $M$ that see each other can be computed in $O(n\log n+m^{4/3}\log^\alpha m\log n)$ time, where $\alpha<1.78$ is a constant. We likewise study the problem of computing the expected geodesic distance between two points chosen independently and uniformly at random in a simple polygon $P$. We show how the expected $L_1$-distance can be computed in optimal $O(n)$ time by a conceptually very simple algorithm. We then describe an algorithm that outputs a closed-form expression for the expected $L_2$-distance in $O(n^2)$ time.