Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain

TL;DR

This paper presents efficient algorithms for computing the diameter and radius of rectilinear polygonal domains under link distance, using a novel graph of oriented distances to improve computational complexity.

Contribution

It introduces a new graph-based approach to compute rectilinear link diameter and radius more efficiently than previous methods.

Findings

Algorithms run in $O(n^ ext{omega})$ and $O(n^2 ext{log} n)$ time.

The graph of oriented distances encodes pairwise distances effectively.

Provides faster algorithms for diameter computation in rectilinear polygons.

Abstract

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of $n$ vertices and $h$ holes. We introduce a \emph{graph of oriented distances} to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in $\min \{\,O(n^\omega), O(n^2 + nh \log h + \chi^2)\,\}$ time, where $\omega<2.373$ denotes the matrix multiplication exponent and $\chi\in \Omega(n)\cap O(n^2)$ is the number of edges of the graph of oriented distances. We also provide a faster algorithm for computing the diameter that runs in $O(n^2 \log n)$ time.