Let $V$ be a set of $n$ points in the plane. The unit-disk graph $G = (V, E)$ has vertex set $V$ and an edge $e_{uv} \in E$ between vertices $u, v \in V$ if the Euclidean distance between $u$ and $v$ is at most 1. The weight of each edge $e_{uv}$ is the Euclidean distance between $u$ and $v$. Given $V$ and a source point $s\in V$, we consider the problem of computing shortest paths in $G$ from $s$ to all other vertices. The previously best algorithm for this problem runs in $O(n \log^2 n)$ time [Wang and Xue, SoCG'19]. The problem has an $\Omega(n\log n)$ lower bound under the algebraic decision tree model. In this paper, we present an improved algorithm of $O(n \log^2 n / \log \log n)$ time (under the standard real RAM model). Furthermore, we show that the problem can be solved using $O(n\log n)$ comparisons under the algebraic decision tree model, matching the $\Omega(n\log n)$ lower bound.

翻译：令 $V$ 为平面上 $n$ 个点的集合。单位圆盘图 $G = (V, E)$ 的顶点集为 $V$，且对于任意顶点 $u, v \in V$，若 $u$ 与 $v$ 之间的欧几里得距离不超过 1，则存在边 $e_{uv} \in E$。每条边 $e_{uv}$ 的权重为 $u$ 与 $v$ 之间的欧几里得距离。给定 $V$ 及源点 $s\in V$，我们考虑在 $G$ 中计算从 $s$ 到所有其他顶点的最短路径问题。该问题先前的最佳算法时间复杂度为 $O(n \log^2 n)$ [Wang and Xue, SoCG'19]。在代数判定树模型下，该问题存在 $\Omega(n\log n)$ 的下界。本文提出一种改进算法，时间复杂度为 $O(n \log^2 n / \log \log n)$（基于标准实数 RAM 模型）。此外，我们证明在代数判定树模型下，该问题可通过 $O(n\log n)$ 次比较求解，从而匹配了 $\Omega(n\log n)$ 的下界。