Round-Trip Voronoi Diagrams and Doubling Density in Geographic Networks

TL;DR

This paper introduces new properties and algorithms for round-trip Voronoi diagrams in geographic networks, relating doubling density to Voronoi region complexity, with practical implications for efficient spatial analysis.

Contribution

It establishes novel theoretical bounds connecting doubling density to Voronoi diagram complexity and develops more efficient algorithms for geographic networks with these properties.

Findings

New bounds on the number of Voronoi regions based on doubling density

Algorithms with improved asymptotic efficiency for certain network distributions

Experimental evidence of doubling density properties in real road networks

Abstract

The round-trip distance function on a geographic network (such as a road network, flight network, or utility distribution grid) defines the "distance" from a single vertex to a pair of vertices as the minimum length tour visiting all three vertices and ending at the starting vertex. Given a geographic network and a subset of its vertices called "sites" (for example a road network with a list of grocery stores), a two-site round-trip Voronoi diagram labels each vertex in the network with the pair of sites that minimizes the round-trip distance from that vertex. Alternatively, given a geographic network and two sets of sites of different types (for example grocery stores and coffee shops), a two-color round-trip Voronoi diagram labels each vertex with the pair of sites of different types minimizing the round-trip distance. In this paper, we prove several new properties of two-site and two-color round-trip Voronoi diagrams in a geographic network, including a relationship between the "doubling density" of sites and an upper bound on the number of non-empty Voronoi regions. We show how those lemmas can be used in new algorithms asymptotically more efficient than previous known algorithms when the networks have reasonable distribution properties related to doubling density, and we provide experimental data suggesting that road networks with standard point-of-interest sites have these properties.