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An Optimal Algorithm for Shortest Paths in Unweighted Disk Graphs

Bruce W. Brewer, Haitao Wang

TL;DR

This work solves the unweighted single-source shortest paths problem in disk graphs G(S) in the plane in optimal time. It leverages a static additively-weighted Voronoi diagram and its dual DT(S) to perform a BFS-like expansion from the source, testing reachability via d_{S_{i-1}}(v) ≤ r_v and updating frontier through DT adjacencies. The main contribution is an O(n log n) algorithm, matching the Ω(n log n) lower bound and offering a simpler, self-contained description; it also handles the containment case and extends to L_infty and L1 metrics with the same asymptotic efficiency. The approach reinforces the utility of Voronoi-based structures in geometric graph problems and yields practical, near-linear-time SSSP for disk graphs.

Abstract

Given in the plane a set $S$ of $n$ points and a set of disks centered at these points, the disk graph $G(S)$ induced by these disks has vertex set $S$ and an edge between two vertices if their disks intersect. Note that the disks may have different radii. We consider the problem of computing shortest paths from a source point $s\in S$ to all vertices in $G(S)$ where the length of a path in $G(S)$ is defined as the number of edges in the path. The previously best algorithm solves the problem in $O(n\log^2 n)$ time. A lower bound of $Ω(n\log n)$ is also known for this problem under the algebraic decision tree model. In this paper, we present an $O(n\log n)$ time algorithm, which matches the lower bound and thus is optimal. Another virtue of our algorithm is that it is quite simple.

An Optimal Algorithm for Shortest Paths in Unweighted Disk Graphs

TL;DR

This work solves the unweighted single-source shortest paths problem in disk graphs G(S) in the plane in optimal time. It leverages a static additively-weighted Voronoi diagram and its dual DT(S) to perform a BFS-like expansion from the source, testing reachability via d_{S_{i-1}}(v) ≤ r_v and updating frontier through DT adjacencies. The main contribution is an O(n log n) algorithm, matching the Ω(n log n) lower bound and offering a simpler, self-contained description; it also handles the containment case and extends to L_infty and L1 metrics with the same asymptotic efficiency. The approach reinforces the utility of Voronoi-based structures in geometric graph problems and yields practical, near-linear-time SSSP for disk graphs.

Abstract

Given in the plane a set of points and a set of disks centered at these points, the disk graph induced by these disks has vertex set and an edge between two vertices if their disks intersect. Note that the disks may have different radii. We consider the problem of computing shortest paths from a source point to all vertices in where the length of a path in is defined as the number of edges in the path. The previously best algorithm solves the problem in time. A lower bound of is also known for this problem under the algebraic decision tree model. In this paper, we present an time algorithm, which matches the lower bound and thus is optimal. Another virtue of our algorithm is that it is quite simple.

Paper Structure

This paper contains 9 sections, 2 theorems, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work.
  3. Preliminaries
  4. The algorithm
  5. Algorithm description
  6. Algorithm implementation and time analysis
  7. Algorithm correctness
  8. The containment case.
  9. Extension to the $\boldsymbol{L_{\infty}}$ (resp., $\boldsymbol{L_1}$) metric.

Key Result

Lemma 3

For any $i > 0$ and any site $v \in S_i$, there exist a site $u \in S_{i - 1}$ and a $u$-$v$ path in $\mathcal{DT}(S)$ such that all internal vertices of the path are in $S_i$.

Theorems & Definitions (2)

  • Lemma 3
  • Theorem 4