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

TL;DR

This paper establishes criteria for Euclidean graph classes to admit linear time single-source shortest path algorithms, extending known results from planar graphs by demonstrating sublinear separators in contracted graphs.

Contribution

It introduces criteria ensuring linear time algorithms for Euclidean graphs and proves these classes have sublinear separators in contracted graphs.

Findings

Euclidean graph classes satisfying the criteria admit linear time SSSP algorithms

Contracted graphs of these classes have sublinear separators

Extends planar graph SSSP results to broader Euclidean classes

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.