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Single-Source Shortest Path Problem in Weighted Disk Graphs

Shinwoo An, Eunjin Oh, Jie Xue

TL;DR

This work advances SSSP computations on edge-weighted disk graphs by delivering two subquadratic algorithms. For disk graphs with bounded radius ratio $\Psi$, the authors achieve $O(n \log^2 n \log \Psi)$ time using a Dijkstra-like scheme augmented with an Update subroutine and a lazy-update mechanism to bound irregular-edge costs. For arbitrary radius ratios, they extend the approach with a compressed quadtree and heavy-path decomposition to obtain $O(n \log^4 n)$ time, integrating a two-way lazy update strategy and canonical-path techniques. The results improve the previous $O(n \log^6 n)$ bound and provide near-linear preprocessing and subquadratic query performance, with practical impact for implicit geometric networks and related broadcast/communication models. The methods combine hierarchical grids, additively weighted Voronoi structures, and intricate geometric lemmas to control updates across irregular and regular edges.

Abstract

In this paper, we present efficient algorithms for the single-source shortest path problem in weighted disk graphs. A disk graph is the intersection graph of a family of disks in the plane. Here, the weight of an edge is defined as the Euclidean distance between the centers of the disks corresponding to the endpoints of the edge. Given a family of $n$ disks in the plane whose radii lie in $[1,Ψ]$ and a source disk, we can compute a shortest path tree from a source vertex in the weighted disk graph in $O(n\log^2 n \log Ψ)$ time. Moreover, in the case that the radii of disks are arbitrarily large, we can compute a shortest path tree from a source vertex in the weighted disk graph in $O(n\log^4 n)$ time. This improves the best-known algorithm running in $O(n\log^6 n)$ time presented in ESA'23.

Single-Source Shortest Path Problem in Weighted Disk Graphs

TL;DR

This work advances SSSP computations on edge-weighted disk graphs by delivering two subquadratic algorithms. For disk graphs with bounded radius ratio , the authors achieve time using a Dijkstra-like scheme augmented with an Update subroutine and a lazy-update mechanism to bound irregular-edge costs. For arbitrary radius ratios, they extend the approach with a compressed quadtree and heavy-path decomposition to obtain time, integrating a two-way lazy update strategy and canonical-path techniques. The results improve the previous bound and provide near-linear preprocessing and subquadratic query performance, with practical impact for implicit geometric networks and related broadcast/communication models. The methods combine hierarchical grids, additively weighted Voronoi structures, and intricate geometric lemmas to control updates across irregular and regular edges.

Abstract

In this paper, we present efficient algorithms for the single-source shortest path problem in weighted disk graphs. A disk graph is the intersection graph of a family of disks in the plane. Here, the weight of an edge is defined as the Euclidean distance between the centers of the disks corresponding to the endpoints of the edge. Given a family of disks in the plane whose radii lie in and a source disk, we can compute a shortest path tree from a source vertex in the weighted disk graph in time. Moreover, in the case that the radii of disks are arbitrarily large, we can compute a shortest path tree from a source vertex in the weighted disk graph in time. This improves the best-known algorithm running in time presented in ESA'23.

Paper Structure

This paper contains 37 sections, 29 theorems, 24 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related work.
  3. Our results.
  4. Model of computation.
  5. Preliminaries.
  6. SSSP on Disk Graphs of Bounded Radius Ratio Lg
  7. Sketch of Our Algorithm
  8. Regular edge and Irregular edge.
  9. Basic strategy of wang2020near.
  10. Main obstacles and lazy update scheme.
  11. Algorithm
  12. Case 1: $k=\textnormal{dist}(v)$ for a vertex $v \in R$.
  13. Case 2: $k=\textnormal{alarm}(c)$ for a grid cell $c \in \Gamma$.
  14. Correctness
  15. Time Complexity
  16. ...and 22 more sections

Key Result

Lemma 1

Let $uv$ be a regular edge. Then $c_u\in \boxplus_{c_v}$. Symmetrically, $c_v\in \boxplus_{c_u}$.

Figures (4)

  • Figure 1: (a) If $r_u>r_v$ and $|uv|<r_u-r_v$, all neighbors of $v$ are also neighbors of $u$. (b) If $r_v>r_u$ and $|uv|<r_v-r_u$, the predecessor $w$ of $u$ is neighbor of $v$, and the shortest $s$-$v$ path should contain $wv$.
  • Figure 2: Illustrating some points in the proof of Lemma \ref{['lem:correct1']}-(2). (a) An edge $xu$ is not an edge of $G$. (b) If $w$ is a small neighbor of $u$. $\text{alarm}(c_v)$ rings before the round of $k=\text{dist}(v)$.
  • Figure 3: (a) Compressed quadtree $\mathcal{Q}$ and canonical path $\pi$. The lowest cell $c$ and topmost cell $c'$ of $\pi$. (b) Illustration of $\lambda=(c,C)$. (c) Classification of $P_\text{post}(\lambda)$ and $P_\text{large}(\lambda)$. A disk of $P_\text{post}(\lambda)$(green disk) contains the disks of $P_\text{small}(\lambda)$(red disk), while a disk of $P_\text{large}(\lambda)$(blue disk) may not.
  • Figure 4: Illustration of some points in the proof of Lemma \ref{['lem:arbi-correct-main']}. (a) In (3), $|yp|$ is small since $\angle you$ is small. Then $|xy|+|yu|$ is not much larger than $|wu|$. The blue path is shorter than the shortest $s$-$u$ path(red path). (b) In (4), $|xw|$ is small. The alarm rings after $w$ has been processed.

Theorems & Definitions (50)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Corollary 1
  • ...and 40 more