The Geodesic Diameter of Polygonal Domains

TL;DR

This paper introduces the first algorithms to compute the geodesic diameter of polygonal domains with holes, extending previous simple polygon results, and reveals that interior points can determine the diameter with multiple shortest paths.

Contribution

It provides the first algorithms for calculating the geodesic diameter in polygonal domains with holes, with worst-case time complexities of O(n^7.73) and O(n^7 (log n + h)).

Findings

Interior points can determine the geodesic diameter.

At least five distinct shortest paths exist between such points.

New algorithms extend simple polygon results to complex domains.

Abstract

This paper studies the geodesic diameter of polygonal domains having h holes and n corners. For simple polygons (i.e., h = 0), the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time, as known by Hershberger and Suri. For general polygonal domains with h >= 1, however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present the first algorithms that compute the geodesic diameter of a given polygonal domain in worst-case time O(n^7.73) or O(n^7 (log n + h)). The main difficulty unlike the simple polygon case relies on the following observation revealed in this paper: two interior points can determine the geodesic diameter and in that case there exist at least five distinct shortest paths between the two.