Near-Linear Time Approximation Algorithms for Curve Simplification

Abstract

We consider the problem of approximating a polygonal curve P under a given error criterion by another polygonal curve P' whose vertices are a subset of the vertices of P. The goal is to minimize the number of vertices of P' while ensuring that the error between P' and P is below a certain threshold. We consider two different error measures: Hausdorff and Frechet. For both error criteria, we present near-linear time approximation algorithms that, given a parameter ź > 0, compute a simplified polygonal curve P' whose error is less than ź and size at most the size of an optimal simplified polygonal curve with error ź/2. We consider monotone curves in ź2 in the case of the Hausdorff error measure under the uniform distance metric and arbitrary curves in any dimension for the Frechet error measure under Lp metrics. We present experimental results demonstrating that our algorithms are simple and fast, and produce close to optimal simplifications in practice.