Approximating the packedness of polygonal curves

TL;DR

This paper introduces approximation algorithms for determining the minimal c such that a polygonal curve is c-packed, with practical experiments showing the model's usefulness for real-world trajectories.

Contribution

It presents the first algorithms for approximating the packedness of polygonal curves, including a 2-approximation and a faster (6+ε)-approximation for 2D curves.

Findings

The algorithms effectively approximate the packedness of real-world trajectories.

Experiments show c-packedness is a realistic model for many curves.

The first implementation demonstrates practical applicability.

Abstract

In 2012 Driemel et al. \cite{DBLP:journals/dcg/DriemelHW12} introduced the concept of $c$-packed curves as a realistic input model. In the case when $c$ is a constant they gave a near linear time $(1+\varepsilon)$-approximation algorithm for computing the Fr\'echet distance between two $c$-packed polygonal curves. Since then a number of papers have used the model. In this paper we consider the problem of computing the smallest $c$ for which a given polygonal curve in $\mathbb{R}^d$ is $c$-packed. We present two approximation algorithms. The first algorithm is a $2$-approximation algorithm and runs in $O(dn^2 \log n)$ time. In the case $d=2$ we develop a faster algorithm that returns a $(6+\varepsilon)$-approximation and runs in $O((n/\varepsilon^3)^{4/3} polylog (n/\varepsilon)))$ time. We also implemented the first algorithm and computed the approximate packedness-value for 16 sets of real-world trajectories. The experiments indicate that the notion of $c$-packedness is a useful realistic input model for many curves and trajectories.