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Approximating the Packedness of Polygonal Curves

Authors Joachim Gudmundsson, Yuan Sha, Sampson Wong

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Joachim Gudmundsson
  • The University of Sydney, Australia
Yuan Sha
  • The University of Sydney, Australia
Sampson Wong
  • The University of Sydney, Australia

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Joachim Gudmundsson, Yuan Sha, and Sampson Wong. Approximating the Packedness of Polygonal Curves. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ISAAC.2020.9

Abstract

In 2012 Driemel et al. [Anne Driemel et al., 2012] 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+ε)-approximation algorithm for computing the Fréchet 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 ℝ^d is c-packed. We present two approximation algorithms. The first algorithm is a 2-approximation algorithm and runs in O(dn² log n) time. In the case d = 2 we develop a faster algorithm that returns a (6+ε)-approximation and runs in O((n/ε³)^{4/3} polylog (n/ε))) 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Computational geometry
  • trajectories
  • realistic input models

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