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.

翻译：令 $S$ 为多边形 $P$（具有 $m$ 个顶点）中的 $n$ 个点集。由 $S$ 诱导的测地线单位圆盘图 $G(S)$ 以 $S$ 为顶点集，当且仅当两点在 $P$ 中的测地距离不超过 $1$ 时，两者之间存在一条边。在加权版本中，每条边的权重等于其端点之间的测地距离；在无权版本中，每条边的权重均为 $1$。给定源点 $s \in S$，我们研究计算从 $s$ 到 $G(S)$ 所有顶点的最短路径问题。据我们所知，该问题此前尚未被研究。一种朴素方法显式构造 $G(S)$，然后应用针对一般图的标准最短路径算法，但由于 $G(S)$ 可能包含 $\Omega(n^2)$ 条边，最坏情况下该算法需二次时间。本文首次提出该问题的次二次时间算法。对于加权情形，当 $P$ 为简单多边形时，我们得到 $O(m + n \log^{2} n \log^{2} m)$ 时间算法。对于无权情形，我们为简单多边形提供 $O(m + n \log n \log^{2} m)$ 时间算法，为带孔多边形提供 $O(\sqrt{n} (n+m)\log(n+m))$ 时间算法。为实现这些结果，我们开发了仅删除操作的测地线单位圆盘范围空查询数据结构，以及用于在简单多边形中构造隐式加法加权测地线沃罗诺伊图的数据结构。此外，我们提出一种动态数据结构，将 Bentley 的对数方法从插入操作扩展至优先队列更新操作（即插入和删除最小值操作）。这些结果可能具有独立研究价值。