A faster algorithm for the Fréchet distance in 1D for the imbalanced case

TL;DR

This work resolves an open question about the imbalanced Fréchet distance in one dimension by presenting an exact, subquadratic algorithm for 1D curves of complexities $n$ and $m=n^{\alpha}$ (with $\alpha\in(0,1)$), achieving $O(n^{2\alpha}\log^2 n + n\log n)$ time. Central to the approach is a Fréchet distance oracle for 1D curves: after preprocessing a curve $P$ in $O(n\log n)$ time and space, queries against a curve $Q$ of complexity $m$ can be answered in $O(m^2\log^2 n)$ time, enabling binary-search over a set of critical values to compute the exact distance. The methodology rests on signatures, extended signatures, and coupled $\delta$-visiting orders, combined with orthogonal range successor data structures to efficiently navigate the decision problem. The results yield a clean separation between 1D and higher-dimensional Fréchet distance complexity, improve exact computation in the imbalanced 1D regime, and support subcurve queries, broadening practical applicability for geometric data analysis. Overall, the paper advances fine-grained complexity understanding and provides a concrete, scalable tool for exact 1D Fréchet distance computations.

Abstract

The fine-grained complexity of computing the Fréchet distance has been a topic of much recent work, starting with the quadratic SETH-based conditional lower bound by Bringmann from 2014. Subsequent work established largely the same complexity lower bounds for the Fréchet distance in 1D. However, the imbalanced case, which was shown by Bringmann to be tight in dimensions $d\geq 2$, was still left open. Filling in this gap, we show that a faster algorithm for the Fréchet distance in the imbalanced case is possible: Given two 1-dimensional curves of complexity $n$ and $n^α$ for some $α\in (0,1)$, we can compute their Fréchet distance in $O(n^{2α} \log^2 n + n \log n)$ time. This rules out a conditional lower bound of the form $O((nm)^{1-ε})$ that Bringmann showed for $d \geq 2$ and any $\varepsilon>0$ in turn showing a strict separation with the setting $d=1$. At the heart of our approach lies a data structure that stores a 1-dimensional curve $P$ of complexity $n$, and supports queries with a curve $Q$ of complexity~$m$ for the continuous Fréchet distance between $P$ and $Q$. The data structure has size in $\mathcal{O}(n\log n)$ and uses query time in $\mathcal{O}(m^2 \log^2 n)$. Our proof uses a key lemma that is based on the concept of visiting orders and may be of independent interest. We demonstrate this by substantially simplifying the correctness proof of a clustering algorithm by Driemel, Krivošija and Sohler from 2015.

Figures (7)

• Figure 1: In this paper, the vertices of the time series are drawn as vertical segments for clarity. The $\delta$-signature vertices are marked with a red disk and $((i_1, j_1), (i_2, v_2),(i_3, j_2), (i_4, w_4),$$(i_5, w_5), (v_3, j_3), (v_4, j_4), (i_6, j_5), (i_7, j_6))$ is a coupled $\delta$-visiting order.
• Figure 2: The red disks are the $\delta$-signature vertices. The indices $1\leq 2\leq 5\leq 6\leq 7$ of $Q$ are a $\delta$-visiting order of the $\delta$-signature vertices of $P$ on $Q$, resp. $1\leq 2\leq 5\leq 6\leq 7\leq 8\leq 9$ of the $\delta$-signature vertices of $Q$ on $P$. Those are the only existing $\delta$-visiting orders for the $\delta$-signatures. Therefore, there does not exist a coupled $\delta$-visiting order of $P$ and $Q$, since $(3, 5)$ and $(5, 3)$ cross.
• Figure 3: Visualization of parts of the proof of \ref{['t:coupled visiting order']}.
• Figure 4: $P(1), P(3), P(4), P(7)$ and $P(8)$ are the $\delta$-signature vertices of $P$. If $\widehat{Q}$ is obtained by removing $Q(2)$ from $Q$, then $d_F(P, \widehat{Q})> \delta$. Therefore, we define the (enlarged) signature-range $r_1=[P(1)-3\delta, P(1)+3\delta]$. If $\widehat{Q}$ is obtained from $Q$ by removing $Q(4)$, then it still holds that $d_F(P, \widehat{Q})\leq \delta$.
• Figure 5: For the orange cell $[i_k, i_{k+1}]\times [j_l, j_{l+1}]$, we find the smallest index $v_{k, l-1}$ such that $(v_{k, l-1}, P(v_{k, l-1}))\in [v_{k, l}, i_{k+1}]\times B(Q(j_{l+1}),\delta)$ and the smallest index $w_{k+1, l}$ such that $(w_{k+1, l}, Q(w_{k+1, l}))\in [j_l, j_{l+1}]\times B(P(i_{k+1}), \delta)$. The thick lines mark the query intervals for the indices of the exit points. To compute $i_2, j_2, \widetilde{v}$ and $\widetilde{w}$, \ref{['l: minimum first signature']} is used.

Theorems & Definitions (18)

• Lemma 2: Generalization of Lemma 37 in BDNP21
• Definition 3: $\delta$-signature
• Lemma 4: Driemel, Krivošija and Sohler DKS16
• Theorem 5: Driemel, Krivošija and Sohler DKS16
• Definition 6: coupled $\delta$-visiting order
• Theorem 7: A04CG86
• Lemma 8
• Lemma 9: Key Lemma
• Corollary 11: Theorem 3.7 in DKS16
• Definition 12