Global Curve Simplification

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 $P^{\prime }$ with a smaller number of vertices such that $P^{\prime }$ 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 $P^{\prime }$. We consider six different curve distance measures: three variants of the \emph{Hausdorff} distance and three variants of the \emph{Fréchet} distance. And we study different restrictions on the choice of vertices for $P^{\prime }$. 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.

33 pages, 16 figures