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

TL;DR

This work tackles single-source shortest paths in weighted unit-disk graphs, achieving an improved time bound of $O(n \log^2 n / \log\log n)$ by speeding up a bottleneck subproblem called offline insertion-only additively-weighted nearest neighbor with a separating line. The core technical advance is a linear-time merging procedure for additively-weighted Voronoi diagrams above a separating line, coupled with a data-structure for IOAWNN-SL that supports fast insertions and queries. Under the algebraic decision tree model, the authors further show that IOAWNN-SL can be solved in $O(n \log n)$ comparisons, thereby matching the $\Omega(n \log n)$ lower bound for the SSSP problem in this model. Together, these contributions yield faster practical algorithms and establish optimal ADC bounds for the problem, bridging gaps in both geometric graph shortest paths and dynamic nearest-neighbor computations.

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 $Ω(n\log n)$ 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 $Ω(n\log n)$ lower bound.

Figures (5)

• Figure 1: Illustrating $\square_a$ (the central highlighted square) and $\boxplus_a$ (the gray area).
• Figure 2: Illustrating an additively-weighted Voronoi diagram. The dashed horizontal line is the $x$-axis $\ell$.
• Figure 3: Illustrating the contour between two sets of points. The dashed horizontal line is the $x$-axis $\ell$.
• Figure 4: Illustrating the additively-weighted Voronoi diagram of four points $\{p_1,p_2,p_3,p_4\}$ for Observation \ref{['obser:example']}.
• Figure 5: The dotted segments are spokes. Our algorithm only uses the portions of these spokes above $\ell$, the dashed line.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof