Crossing Patterns in Nonplanar Road Networks

TL;DR

This paper investigates the crossing graphs of real-world road networks, revealing their sparsity and structural properties, which enable efficient algorithms for geographic information systems.

Contribution

It introduces the study of crossing graph sparsity in road networks and proves that such sparsity leads to desirable algorithmic properties like polynomial expansion.

Findings

Crossing graphs of large road networks are mostly trees or sparse.

Sparse crossing graphs have bounded degeneracy.

Graphs with sparse crossing graphs have polynomial expansion, enabling fast algorithms.

Abstract

We define the crossing graph of a given embedded graph (such as a road network) to be a graph with a vertex for each edge of the embedding, with two crossing graph vertices adjacent when the corresponding two edges of the embedding cross each other. In this paper, we study the sparsity properties of crossing graphs of real-world road networks. We show that, in large road networks (the Urban Road Network Dataset), the crossing graphs have connected components that are primarily trees, and that the remaining non-tree components are typically sparse (technically, that they have bounded degeneracy). We prove theoretically that when an embedded graph has a sparse crossing graph, it has other desirable properties that lead to fast algorithms for shortest paths and other algorithms important in geographic information systems. Notably, these graphs have polynomial expansion, meaning that they and all their subgraphs have small separators.