Voronoi Diagram of Polygonal Chains under the Discrete Fr\'echet Distance

TL;DR

This paper introduces the first study of Voronoi diagrams for polygonal chains under the discrete Fréchet distance in 2D and 3D, establishing fundamental properties and bounds for these diagrams.

Contribution

It presents the first analysis of Voronoi diagrams of polygonal chains under the discrete Fréchet distance, including bounds and fundamental properties.

Findings

Established upper bounds for the Voronoi diagram complexity.

Established lower bounds for the Voronoi diagram complexity.

Analyzed properties of Voronoi diagrams in 2D and 3D.

Abstract

Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the famous Fr\`{e}chet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in $d$-dimension ($d=2,3$) under the discrete Fr\`{e}chet distance. Given $n$ polygonal chains ${\cal C}$ in $d$-dimension ($d=2,3$), each with at most $k$ vertices, we prove fundamental properties of such a Voronoi diagram {\em VD}$_F({\cal C})$ by presenting the first known upper and lower bounds for {\em VD}$_F({\cal C})$.