On the Expected Complexity of Voronoi Diagrams on Terrains

TL;DR

This paper analyzes the expected complexity of geodesic Voronoi diagrams on terrains, showing that under certain probabilistic assumptions, the complexity is linear in the number of terrain features and sites, with implications for understanding their behavior.

Contribution

It proves that the expected complexity of Voronoi diagrams on terrains is linear under relaxed assumptions and provides a lower bound for worst-case scenarios, extending previous worst-case results.

Findings

Expected complexity is O(n + m) under probabilistic assumptions.

A lower bound of Vmega(n m^{2/3}) is established for worst-case terrain complexity.

Expected fatness of a cell in a random planar Voronoi diagram is bounded by a constant.

Abstract

We investigate the combinatorial complexity of geodesic Voronoi diagrams on polyhedral terrains using a probabilistic analysis. Aronov etal [ABT08] prove that, if one makes certain realistic input assumptions on the terrain, this complexity is \Theta(n + m \sqrt n) in the worst case, where n denotes the number of triangles that define the terrain and m denotes the number of Voronoi sites. We prove that under a relaxed set of assumptions the Voronoi diagram has expected complexity O(n+m), given that the sites have a uniform distribution on the domain of the terrain(or the surface of the terrain). Furthermore, we present a worst-case construction of a terrain which implies a lower bound of Vmega(n m2/3) on the expected worst-case complexity if these assumptions on the terrain are dropped. As an additional result, we can show that the expected fatness of a cell in a random planar Voronoi diagram is bounded by a constant.