Going Off-road: Transversal Complexity in Road Networks

TL;DR

This paper analyzes the average transversal complexity of road networks modeled as geometric graphs, demonstrating an O(√n) crossing bound and efficient query algorithms supported by empirical evidence.

Contribution

It introduces a multiscale-dispersed graph model for road networks and proves an O(√n) average crossing bound, with practical algorithms and empirical validation.

Findings

Average crossings per random line are O(√n).

Point location and ray-shooting queries run in O(√n log n) expected time.

Empirical data confirms theoretical bounds and efficiency.

Abstract

A geometric graph is a graph embedded in the plane with vertices at points and edges drawn as curves (which are usually straight line segments) between those points. The average transversal complexity of a geometric graph is the number of edges of that graph that are crossed by random line or line segment. In this paper, we study the average transversal complexity of road networks. By viewing road networks as multiscale-dispersed graphs, we show that a random line will cross the edges of such a graph O(sqrt(n)) times on average. In addition, we provide by empirical evidence from experiments on the road networks of the fifty states of United States and the District of Columbia that this bound holds in practice and has a small constant factor. Combining this result with data structuring techniques from computational geometry, allows us to show that we can then do point location and ray-shooting navigational queries with respect to road networks in O(sqrt(n) log n) expected time. Finally, we provide empirical justification for this claim as well.