Farthest-point Voronoi diagrams in the presence of rectangular obstacles

TL;DR

This paper introduces the first optimal algorithm for constructing the geodesic L1 farthest-point Voronoi diagram among rectangular obstacles, enabling efficient farthest-neighbor queries in obstacle-rich environments.

Contribution

It presents a novel optimal algorithm for Voronoi diagram construction with obstacles, improving computational efficiency and query performance.

Findings

Algorithm runs in O(nm + n log n + m log m) time

Supports fast farthest-neighbor queries in obstacle environments

First optimal solution for this problem setting

Abstract

We present an algorithm to compute the geodesic $L_1$ farthest-point Voronoi diagram of $m$ point sites in the presence of $n$ rectangular obstacles in the plane. It takes $O(nm+n \log n + m\log m)$ construction time using $O(nm)$ space. This is the first optimal algorithm for constructing the farthest-point Voronoi diagram in the presence of obstacles. We can construct a data structure in the same construction time and space that answers a farthest-neighbor query in $O(\log(n+m))$ time.