Shortest Paths in Geodesic Unit-Disk Graphs

TL;DR

This paper introduces the first subquadratic algorithms for shortest path computations in geodesic unit-disk graphs within polygons, significantly improving efficiency over naive methods by leveraging novel data structures and geometric techniques.

Contribution

It presents the first subquadratic algorithms for shortest paths in geodesic unit-disk graphs in polygons, utilizing new data structures for range emptiness and Voronoi diagrams.

Findings

Weighted case algorithm runs in O(m + n log^2 n log^2 m) time.

Unweighted simple polygon algorithm runs in O(m + n log n log^2 m) time.

Unweighted polygon with holes algorithm runs in O(√n (n+m) log(n+m)) time.

Abstract

Let $S$ be a set of $n$ points in a polygon $P$ with $m$ vertices. The geodesic unit-disk graph $G(S)$ induced by $S$ has vertex set $S$ and contains an edge between two vertices whenever their geodesic distance in $P$ is at most one. In the weighted version, each edge is assigned weight equal to the geodesic distance between its endpoints; in the unweighted version, every edge has weight $1$. Given a source point $s \in S$, we study the problem of computing shortest paths from $s$ to all vertices of $G(S)$. To the best of our knowledge, this problem has not been investigated previously. A naive approach constructs $G(S)$ explicitly and then applies a standard shortest path algorithm for general graphs, but this requires quadratic time in the worst case, since $G(S)$ may contain $\Omega(n^2)$ edges. In this paper, we give the first subquadratic-time algorithms for this problem. For the weighted case, when $P$ is a simple polygon, we obtain an $O(m + n \log^{2} n \log^{2} m)$-time algorithm. For the unweighted case, we provide an $O(m + n \log n \log^{2} m)$-time algorithm for simple polygons, and an $O(\sqrt{n} (n+m)\log(n+m))$-time algorithm for polygons with holes. To achieve these results, we develop a data structure for deletion-only geodesic unit-disk range emptiness queries, as well as a data structure for constructing implicit additively weighted geodesic Voronoi diagrams in simple polygons. In addition, we propose a dynamic data structure that extends Bentley's logarithmic method from insertions to priority-queue updates, namely insertion and delete-min operations. These results may be of independent interest.