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arXiv · 1809.10269

Global Curve Simplification

Mees van de Kerkhof·Irina Kostitsyna·Maarten Löffler·Majid Mirzanezhad·Carola Wenk
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Abstract

Due to its many applications, \emph{curve simplification} is a long-studied problem in computational geometry and adjacent disciplines, such as graphics, geographical information science, etc. Given a polygonal curve $P$ with $n$ vertices, the goal is to find another polygonal curve $P'$ with a smaller number of vertices such that $P'$ is sufficiently similar to $P$. Quality guarantees of a simplification are usually given in a \emph{local} sense, bounding the distance between a shortcut and its corresponding section of the curve. In this work, we aim to provide a systematic overview of curve simplification problems under \emph{global} distance measures that bound the distance between $P$ and $P'$. We consider six different curve distance measures: three variants of the \emph{Hausdorff} distance and three variants of the \emph{Fr\'echet} distance. And we study different restrictions on the choice of vertices for $P'$. We provide polynomial-time algorithms for some variants of the global curve simplification problem and show NP-hardness for other variants. Through this systematic study we observe, for the first time, some surprising patterns, and suggest directions for future research in this important area.

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Mees van de Kerkhof, Irina Kostitsyna, Maarten Löffler, Majid Mirzanezhad, Carola Wenk. 2018-09-26. Global Curve Simplification. https://arxiv.org/abs/1809.10269

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