[Paper Review] On Optimal Polyline Simplification Using the Hausdorff and Fréchet Distance

Marc van Kreveld, Maarten Löffler+1 more|arXiv (Cornell University)|Jan 1, 2018

Geographic Information Systems Studies5 references4 citations

TL;DR

This paper investigates optimal polyline simplification using the Hausdorff and Fréchet distances, proving that computing the optimal simplification under the undirected Hausdorff distance is NP-hard, while providing an O(kn⁵)-time, O(kn²)-space dynamic programming algorithm for optimal Fréchet-based simplification. It further shows that classic algorithms like Douglas-Peucker and Imai-Iri are not optimal under these measures, and their performance degrades significantly even with enlarged error thresholds.

ABSTRACT

We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fréchet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given epsilon>0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fréchet distance between the input and output polylines is at most epsilon. 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 epsilon. 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 Fréchet distance, we give an O(kn^5) time algorithm that requires O(kn^2) space, where k is the output complexity of the simplification.

Motivation & Objective

• To analyze the theoretical performance of well-known polyline simplification algorithms—Douglas-Peucker and Imai-Iri—under the Hausdorff and Fréchet distance measures.
• To determine whether these algorithms produce optimal simplifications when using these distance metrics.
• To investigate the computational complexity of computing an optimal simplification under the Hausdorff and Fréchet distances.
• To develop an efficient algorithm for computing the optimal simplification under the Fréchet distance.

Proposed method

• Formalizing the problem as finding the minimum-size subsequence of input vertices such that the Hausdorff or Fréchet distance between input and output polylines is at most ε.
• Using dynamic programming to maintain reachable points along the polyline, where reachability is defined by Fréchet distance constraints.
• Introducing a key insight: for each edge, only the farthest reachable point needs to be tracked to ensure correctness.
• Proving that the number of reachable intervals per edge is bounded by a linear function, enabling efficient computation.
• Designing a dynamic programming table ρ(k, i, ˆt) to store reachability of the farthest point on edge i for a k-link simplification.
• Using geometric properties of the Fréchet distance to reduce the search space and ensure correctness of the recurrence.

Experimental results

Research questions

• RQ1 How does the performance of the Douglas-Peucker and Imai-Iri algorithms compare to the optimal simplification under the Hausdorff distance?
• RQ2 Is computing an optimal simplification under the undirected Hausdorff distance NP-hard?
• RQ3 Can the optimal simplification under the Fréchet distance be computed in polynomial time, and if so, how efficiently?
• RQ4 What is the impact of increasing the error threshold ε on the vertex count of the output from standard algorithms compared to the optimal solution?

Key findings

• Computing an optimal simplification using the undirected Hausdorff distance is NP-hard.
• The directed Hausdorff distance from the output to the input polyline can be computed in polynomial time, but the reverse and undirected versions are NP-hard.
• An optimal simplification under the Fréchet distance can be computed in O(kn⁵) time and O(kn²) space using dynamic programming.
• The Douglas-Peucker and Imai-Iri algorithms can require Ω(n) vertices even when ε is enlarged by any constant factor, while the optimal simplification uses only a constant number of vertices.
• For certain polylines, the Imai-Iri algorithm requires a constant factor more vertices than the optimal solution even when ε is increased by a small constant.
• The optimal Fréchet simplification can be computed efficiently via dynamic programming by tracking only the farthest reachable point per edge, ensuring correctness and efficiency.