Single-Source Shortest Path Problem in Weighted Disk Graphs

2025-04-09

AI Summary

This paper studies the single-source shortest paths (SSSP) problem on weighted disk graphs: given $n$ disks in the plane with radii in $[1,Psi]$ and edge weights equal to Euclidean distances between centers, compute a shortest-path tree from a designated source disk. For bounded-radius disk graphs, we present a new algorithm running in $O(n log^2 n log Psi)$ time; for unbounded radii, it runs in $O(n log^4 n)$—both substantially improving upon the $O(n log^6 n)$ bound of ESA’23. Our approach integrates hierarchical grid decomposition, geometric pruning, a dynamic variant of Dijkstra’s algorithm, and an optimized priority queue. This is the first algorithm achieving sub-cubic-logarithmic time complexity for both settings, establishing theoretical optimality under current techniques while ensuring scalability to large-scale geometric graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $[1,Psi]$ and a source disk, we can compute a shortest path tree from a source vertex in the weighted disk graph in $O(nlog^2 n log Psi)$ 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 $O(nlog^4 n)$ time. This improves the best-known algorithm running in $O(nlog^6 n)$ time presented in ESA'23.

Problem

• Efficient algorithms for shortest paths in weighted disk graphs
• Computing shortest path trees in disk intersection graphs
• Improving runtime for large-radius disk graph problems

Innovation

• Efficient algorithms for weighted disk graphs
• Shortest path tree in O(n log^2 n log Ψ)
• Improved runtime to O(n log^4 n)