Lower Bounds for the Complexity of the Voronoi Diagram of Polygonal Curves under the Discrete Frechet Distance

TL;DR

This paper establishes lower bounds on the complexity of Voronoi diagrams for polygonal curves under the discrete Frechet distance, revealing how complexity scales with the number of curves, vertices, and dimension.

Contribution

It provides the first known lower bounds for the combinatorial complexity of Voronoi diagrams of polygonal curves under the discrete Frechet distance, depending on dimension and vertices.

Findings

Complexity is Omega(n^{dk}) for d=1,2.

Complexity is Omega(n^{d(k-1)+2}) for d>2.

Results show the complexity grows polynomially with number of curves and vertices.

Abstract

We give lower bounds for the combinatorial complexity of the Voronoi diagram of polygonal curves under the discrete Frechet distance. We show that the Voronoi diagram of n curves in R^d with k vertices each, has complexity Omega(n^{dk}) for dimension d=1,2 and Omega(n^{d(k-1)+2}) for d>2.