An Optimal Algorithm for Shortest Paths in Unweighted Disk Graphs

TL;DR

This paper introduces an optimal $O(n \,\log n)$ algorithm for shortest path computation in unweighted disk graphs, matching the known lower bound and improving upon previous solutions.

Contribution

The paper presents a simple, optimal algorithm for shortest paths in unweighted disk graphs, matching the lower bound and improving efficiency over prior methods.

Findings

Achieves $O(n \log n)$ time complexity, matching the lower bound.

Simplifies the algorithm for shortest path computation in disk graphs.

Provides a practical solution with optimal theoretical performance.

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 $\Omega(n\log n)$ 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.