On Optimal Polyline Simplification using the Hausdorff and Fr\'echet Distance

TL;DR

This paper investigates the problem of polygonal line simplification using Hausdorff and Fréchet distances, analyzing existing algorithms, proving NP-hardness results, and providing an exact algorithm for the Fréchet distance case.

Contribution

It offers a comprehensive analysis of line simplification under Hausdorff and Fréchet distances, including complexity results and an exact algorithm for the Fréchet distance.

Findings

Douglas-Peucker and Imai-Iri algorithms are compared to optimal solutions.

Computing optimal simplification with undirected Hausdorff distance is NP-hard.

An $O(kn^5)$ time algorithm for optimal Fréchet distance simplification is provided.

Abstract

We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fr\'echet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given $\varepsilon$ > 0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fr\'echet distance between the input and output polylines is at most $\varepsilon$. We analyze how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible, also in the situation where the algorithms are given a considerably larger error threshold than $\varepsilon$. Furthermore, we show that computing an optimal simplification using the undirected Hausdorff distance is NP-hard. The same holds when using the directed Hausdorff distance from the input to the output polyline, whereas the reverse can be computed in polynomial time. Finally, to compute the optimal simplification from a polygonal line consisting of $n$ vertices under the Fr\'echet distance, we give an $O(kn^5)$ time algorithm that requires $O(kn^2)$ space, where $k$ is the output complexity of the simplification.