Fréchet Edit Distance

TL;DR

This work introduces the Fréchet edit distance, defining minimal edits to a curve σ so that the Fréchet distance to a fixed π is within a threshold $\delta$, across continuous and discrete, strong and weak variants. It develops a unifying DAG-complex framework and product-space reasoning to model all admissible edits, enabling polynomial-time algorithms for the strong continuous and strong discrete variants (with deletions, insertions, or both) and establishing clear hardness boundaries for the weak variants via 3SAT-based reductions. The results include concrete runtimes such as $O(mn^3)$ for strong continuous deletion-only, $O(nm^5)$ for plane insertion-only, and $O(m^2 + mn)$ for discrete deletion-only, among others, illustrating a sharp divide between tractable and intractable cases. The paper connects to shortcut Fréchet distances and minimum-vertex-curves, showing how DAG-embedding and canonical subcurves yield efficient solutions and providing a framework for future extensions, including substitutions and broader edit-operation sets. Practically, these findings offer robust, edit-distance-based similarity measures for noisy, densely sampled curves (e.g., GPS traces) with formal performance guarantees.

Abstract

We define and investigate the Fréchet edit distance problem. Given two polygonal curves $π$ and $σ$ and a threshhold value $δ>0$, we seek the minimum number of edits to $σ$ such that the Fréchet distance between the edited $σ$ and $π$ is at most $δ$. For the edit operations we consider three cases, namely, deletion of vertices, insertion of vertices, or both. For this basic problem we consider a number of variants. Specifically, we provide polynomial time algorithms for both discrete and continuous Fréchet edit distance variants, as well as hardness results for weak Fréchet edit distance variants.

Figures (9)

• Figure 2: Three free spaces diagrams with free spaces represented by circles, and edits permitted on $\sigma$ only. The values along the axes are the curve coordinates in $\mathbb{R}^1$.
• Figure 3: Opposing vertical gaps, where bridging one with a row deletion creates a horizontal gap at the other.
• Figure 4: Abstract free space structure. This example is satisfied by setting $\mathcal{X}_2, \lnot \mathcal{X}_5 =\textsf{True}\xspace$.
• Figure 5: Basic clause gadget, consisting of 9 (highlighted) diagonals made by pairs of $L$'s and $L^R$'s which have been glued together such that the free-space has 3 paths.
• Figure 6: Free space example for $\mathsf{Ded}_{\EuScript{DF}}^\mathrm{w} \!\left({\pi, \sigma}\right)$ reduction. Observe that deletion of row 54 closes the vertical gap for $\mathcal{X}_5$, but creates a horizontal gap for $\lnot\mathcal{X}_5$, i.e. setting $\mathcal{X}_5$ to True sets $\lnot \mathcal{X}_5$ to False (and vice versa for deleting 56). The containment gadget restricts deletion to the variable layer only.

Theorems & Definitions (24)

• Theorem 2
• Theorem 3
• Corollary 4
• Definition 5
• Theorem 7: ghms-apsmlp-93
• Lemma 10
• Definition 11
• Corollary 12
• Remark 13
• Theorem 14