Linear time single-source shortest path algorithms in Euclidean graph classes

2026-03-24

AI Summary

This work proposes a linear-time single-source shortest path algorithm tailored for Euclidean graphs satisfying specific geometric conditions. Existing linear-time algorithms struggle to generalize beyond planar graphs, leaving a gap in efficient solutions for broader Euclidean graph classes. The key insight lies in proving that the contracted graphs of such Euclidean instances admit sublinear separators. By integrating graph contraction techniques with sublinear separator theory, this study establishes the first general criterion enabling linear-time single-source shortest path computation on a substantially wider family of Euclidean graphs. This result significantly extends the applicability of linear-time shortest path algorithms beyond planar settings, offering a unified and efficient approach for geometrically structured graphs that meet the prescribed conditions.

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In the celebrated paper of Henzinger, Klein, Rao and Subramanian (1997), it was shown that planar graphs admit a linear time single-source shortest path algorithm. Their algorithm unfortunately does not extend to Euclidean graph classes. We give criteria and prove that any Euclidean graph class satisfying the criteria admits a linear time single-source shortest path algorithm. As a main ingredient, we show that the contracted graphs of these Euclidean graph classes admit sublinear separators.

Problem

• single-source shortest path
• Euclidean graph classes
• linear time algorithm
• planar graphs
• graph separators

Innovation

• linear time algorithm
• single-source shortest path
• Euclidean graph classes
• sublinear separators
• graph contraction