Shortest Paths in Geodesic Unit-Disk Graphs

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 $Ω(n^2)$ 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.

Figures (26)

• Figure 1: Example of a geodesic unit disk centered at $s$ (the black solid curve is the boundary of the disk). A unit-length $s$-$t$ path (the dashed path) is also shown.
• Figure 2: Illustrating a the funnel $F_s(\overline{tr})$ (the light blue region).
• Figure 3: The three solid segments represent edges in $G(S)$. The two dotted curves represent two subpaths of $\pi(s,b)$.
• Figure 4: Illustrations of $\Xi(S)$. The region bounded by $\Xi(S)$ and $d$ is $\mathcal{D}_R(S)$.
• Figure 5: Illustrations of $p \prec_R p$.

Theorems & Definitions (88)

• Lemma 1
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• Theorem 1
• Theorem 2
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• Theorem 4
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• Lemma 2
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