An Improved Algorithm for Shortest Paths in Weighted Unit-Disk Graphs
<|reference_start|>An Improved Algorithm for Shortest Paths in Weighted Unit-Disk Graphs: 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.<|reference_end|>
@article{brewer2024an,
    title={An Improved Algorithm for Shortest Paths in Weighted Unit-Disk Graphs},
    author={Bruce W. Brewer and Haitao Wang},
    journal={arXiv preprint arXiv:2407.03176},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.03176},
    primaryClass={cs.CG cs.DS}
}