Transforming Dogs on the Line: On the Fréchet Distance Under Translation or Scaling in 1D

TL;DR

This work establishes 1D algorithms for the continuous Fréchet distance under translation and scaling, achieving $\mathcal{O}(n^{8/3} \log^3 n)$ time—matching discrete-case bounds and surpassing prior $\tilde{\mathcal{O}}(n^4)$ guarantees. A novel 1D framework based on $\delta$-signatures, extended signatures, and deadlocks reduces the continuous problems to discrete-like reachability problems via a modified free-space matrix and offline dynamic grid reachability. The approach systematically handles degeneracies with symbolic perturbation and leverages parametric search to obtain optimization results, providing tight 1D bounds and a blueprint for applying offline dynamic data structures to continuous geometric problems. Overall, the results close the gap between discrete and continuous Fréchet distance under translations or scaling in 1D and introduce techniques that may extend to related 1D transformation-invariant distance problems with similar structure.

Abstract

The Fréchet distance is a computational mainstay for comparing polygonal curves. The Fréchet distance under translation, which is a translation invariant version, considers the similarity of two curves independent of their location in space. It is defined as the minimum Fréchet distance that arises from allowing arbitrary translations of the input curves. This problem and numerous variants of the Fréchet distance under some transformations have been studied, with more work concentrating on the discrete Fréchet distance, leaving a significant gap between the discrete and continuous versions of the Fréchet distance under transformations. Our contribution is twofold: First, we present an algorithm for the Fréchet distance under translation on 1-dimensional curves of complexity n with a running time of $\mathcal{O}(n^{8/3} log^3 n)$. To achieve this, we develop a novel framework for the problem for 1-dimensional curves, which also applies to other scenarios and leads to our second contribution. We present an algorithm with the same running time of $\mathcal{O}(n^{8/3} \log^3 n)$ for the Fréchet distance under scaling for 1-dimensional curves. For both algorithms we match the running times of the discrete case and improve the previously best known bounds of $\tilde{\mathcal{O}}(n^4)$. Our algorithms rely on technical insights but are conceptually simple, essentially reducing the continuous problem to the discrete case across different length scales.

Figures (8)

• Figure 1: Throughout this paper, vertices of time series are drawn as vertical segments for clarity. The red vertices of the time series $P$ are its $\delta$-signature vertices. After linearly interpolating those red vertices, we get the extended $\delta$-signature of $P$.
• Figure 2: The left (resp. right) time series is $\delta$-monotone increasing (resp. decreasing).
• Figure 3: Visualization of a coupled $\delta$-visiting order. Edges are drawn between vertices $P(i)$ and $Q(j)$, when one is a $\delta$-signature vertex and $|P(i)-Q(j)|\leq \delta$. Then, a coupled $\delta$-visiting order consists of a subset of the drawn edges, shown as an orange bipartite graph, where no two edges cross and it contains one incident edge for every $\delta$-signature vertex.
• Figure 4: Minimal $\mathrm{pre}(P)$-matcher on $\mathrm{pre}(Q)$, $w_\mathrm{pre}$, and minimal $\mathrm{suf}(P)$-matcher on $\mathrm{suf}(Q)$, $w_\mathrm{suf}$.
• Figure 5: Example of a Modified Free-Space Matrix $M_{\delta}$. The colored columns (resp. rows) correspond to the $\delta$-signature vertices of $P$ (resp. $Q$). The white entries are all $1$ by \ref{['def:mod_free_space_matrix']} a), the red entries are defined by b), the purple entries by c) and d) and e), and the yellow entries by f). The traversal drawn in blue uses only 1-entries of $M_\delta$. Hence, $M_\delta(n, m)$ is reachable.

Theorems & Definitions (36)

• Theorem 1
• Theorem 2
• Definition 3
• Definition 4: extended $\delta$-signature
• Definition 5: coupled $\delta$-visiting order
• Lemma 5: Lemma 9 of BD24
• Definition 6: Offline Dynamic Grid Reachability
• Theorem 7: Theorem 3.1 of DBLP:journals/talg/BringmannKN21
• Definition 8: prefix and suffix
• Definition 9: minimal matcher