Shortest Secure Path in a Voronoi Diagram

TL;DR

This paper presents an efficient algorithm for computing the shortest secure path in a Voronoi diagram, incorporating site insertion and combining continuous and discrete Dijkstra methods.

Contribution

It introduces an $O(n \,\log\, n)$ algorithm that uses dynamic weighted Voronoi diagrams and site insertion to find shorter secure paths.

Findings

Algorithm runs in $O(n \log n)$ time.

Inserting new sites can significantly shorten paths.

Implementation using CGAL demonstrates practical efficiency.

Abstract

We investigate the problem of computing the shortest secure path in a Voronoi diagram. Here, a path is secure if it is a sequence of touching Voronoi cells, where each Voronoi cell in the path has a uniform cost of being secured. Importantly, we allow inserting new sites, which in some cases leads to significantly shorter paths. We present an $O(n \log n)$ time algorithm for solving this problem in the plane, which uses a dynamic additive weighted Voronoi diagram to compute this path. The algorithm is an interesting combination of the continuous and discrete Dijkstra algorithms. We also implemented the algorithm using CGAL.