An Improved Algorithm for Shortest Paths in Weighted Unit-Disk Graphs

TL;DR

This paper introduces a faster algorithm for computing shortest paths in weighted unit-disk graphs, improving the time complexity and matching theoretical lower bounds, which benefits geometric network analysis.

Contribution

The paper presents an improved algorithm with $O(n rac{ ext{log}^2 n}{ ext{log} ext{log} n})$ time and demonstrates it can be solved with $O(n ext{log} n)$ comparisons, matching lower bounds.

Findings

Achieved $O(n rac{ ext{log}^2 n}{ ext{log} ext{log} n})$ time complexity.

Established $O(n ext{log} n)$ comparison bound matching lower bounds.

Improved the efficiency of shortest path computation in weighted unit-disk graphs.

Abstract

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.