Computing the $L_1$ Geodesic Diameter and Center of a Polygonal Domain

TL;DR

This paper introduces the first algorithms for efficiently computing the $L_1$ geodesic diameter and center of a polygonal domain, significantly outperforming existing Euclidean methods.

Contribution

The paper presents novel algorithms for $L_1$ geodesic diameter and center in polygonal domains, with complexities much lower than Euclidean counterparts, and no prior algorithms existed for these problems.

Findings

Algorithms run in $O(n^2+h^4)$ and $O((n^4+n^2 h^4)\alpha(n))$ time

Algorithms are significantly faster than Euclidean geodesic algorithms

Based on new insights into $L_1$ shortest paths in polygonal domains

Abstract

For a polygonal domain with $h$ holes and a total of $n$ vertices, we present algorithms that compute the $L_1$ geodesic diameter in $O(n^2+h^4)$ time and the $L_1$ geodesic center in $O((n^4+n^2 h^4)\alpha(n))$ time, respectively, where $\alpha(\cdot)$ denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in $O(n^{7.73})$ or $O(n^7(h+\log n))$ time, and compute the geodesic center in $O(n^{11}\log n)$ time. Therefore, our algorithms are significantly faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on $L_1$ shortest paths in polygonal domains.