Faster Fréchet Distance under Transformations

TL;DR

This work advances the computation of the continuous Fréchet distance between polygonal curves under transformations by transforming the problem into a transformation-space arrangement and a sequence of dynamic reachability problems on refined graph structures. For translations in the plane, it achieves a running time of $\tilde{\mathcal{O}}(n^{7 + \frac{1}{3}})$, substantially improving the previous $\tilde{\mathcal{O}}(n^{8})$ bound, and generalizes to rationally parameterized transformation classes with $k$ degrees of freedom in $\tilde{\mathcal{O}}(n^{3k + \frac{4}{3}} \log^2 n)$ time. The key approach combines a transformation-space arrangement (à la Wenk) with an offline dynamic grid-reachability structure to efficiently propagate changes caused by transformation events. This yields a unified framework that extends the Fréchet distance under transformations to broader transformation families, with concrete time bounds for rotations, scalings, and affine transforms in various dimensions.

Abstract

We study the problem of computing the Fréchet distance between two polygonal curves under transformations. First, we consider translations in the Euclidean plane. Given two curves $π$ and $σ$ of total complexity $n$ and a threshold $δ\geq 0$, we present an $\tilde{\mathcal{O}}(n^{7 + \frac{1}{3}})$ time algorithm to determine whether there exists a translation $t \in \mathbb{R}^2$ such that the Fréchet distance between $π$ and $σ+ t$ is at most $δ$. This improves on the previous best result, which is an $\mathcal{O}(n^8)$ time algorithm. We then generalize this result to any class of rationally parameterized transformations, which includes translation, rotation, scaling, and arbitrary affine transformations. For a class $\mathcal T$ of rationally parametrized transformations with $k$ degrees of freedom, we show that one can determine whether there is a transformation $τ\in \mathcal T$ such that the Fréchet distance between $π$ and $τ(σ)$ is at most $δ$ in $\tilde{\mathcal{O}}(n^{3k+\frac{4}{3}})$ time.

Figures (3)

• Figure 1: From the refined freespace diagram to the refined freespace graph. On the left freespace diagram (FSD), the corner points, critical points, and propagated critical points are marked by squares, blue circles, and red circles, respectively. On the right freespace diagram graph (FSG), the corner, boundary, and interior vertices are marked by squares, circles, and crosses, respectively.
• Figure 2: In the top row of figures, as the segment $\sigma_1 \sigma_2$ translates, the critical points $p$ and $q$ moves down and up, respectively. As $l(p)$ and $l(q)$ (colored in blue) move, they overlap and then separate. In the bottom row, as $\sigma_1 \sigma_2$ translates, a new critical point $p$ appears, and then a new critical point $p'$ appears.
• Figure 3: A visual illustration of the placeholder lines and their relative positions among the grid lines and the freespace diagram boundaries. The row (resp. column) placeholder points and lines are colored in red (resp. green). The grid lines are colored in blue.

Theorems & Definitions (16)

• Lemma 0
• Definition 1
• Definition 2
• Definition 3
• Lemma 4
• Lemma 4
• Lemma 5
• Lemma 5
• Lemma 6
• Lemma 7