An $O(nlog n)$ Algorithm for Single-Source Shortest Paths in Disk Graphs

2025-06-09

AI Summary

This paper studies the single-source shortest paths (SSSP) problem on disk graphs and fat-triangle intersection graphs. To address the high time complexity of classical geometric SSSP algorithms, we propose a novel framework combining geometric divide-and-conquer with plane-sweep techniques, integrated with an enhanced Dijkstra’s algorithm and an efficient dynamic priority queue. Our approach achieves the first exact $O(n log n)$-time SSSP algorithm for disk graphs—matching the theoretical lower bound—and extends to fat-triangle intersection graphs with an $O(n log^2 n)$ solution. All algorithms explicitly exploit underlying geometric structure, ensuring both strong theoretical guarantees and practical implementability. The results represent the first optimal or near-optimal exact SSSP algorithms for these fundamental geometric graph classes.

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
• Social Networks and Social Media: Social mining and social search on the Web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We prove that the single-source shortest-path problem on disk graphs can be solved in $O(nlog n)$ time, and that it can be solved on intersection graphs of fat triangles in $O(nlog^2 n)$ time.

Problem

• Efficient algorithm for shortest paths in disk graphs
• Solves single-source shortest-path problem in O(n log n)
• Extends solution to fat triangle intersection graphs

Innovation

• O(n log n) algorithm for disk graphs
• Solves shortest-path problem efficiently
• Extends to fat triangle intersection graphs