A faster algorithm for the Fr\'echet distance in 1D for the imbalanced case

TL;DR

This paper introduces a faster algorithm for computing the Fréchet distance in 1D for the imbalanced case, using a novel data structure and visiting order concepts, improving efficiency over previous bounds.

Contribution

The paper presents the first sub-quadratic algorithm for the imbalanced 1D Fréchet distance, with a new data structure and a key lemma based on visiting orders.

Findings

Achieves $O(n^{2eta} ext{log}^2 n + n ext{log} n)$ time complexity for curves of complexities $n$ and $n^{eta}$.

Provides a data structure with $O(n ext{log} n)$ size and $O(m^2 ext{log}^2 n)$ query time.

Simplifies the proof of a clustering algorithm using the key lemma.

Abstract

The fine-grained complexity of computing the Fr\'echet 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 Fr\'echet 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 Fr\'echet distance in the imbalanced case is possible: Given two 1-dimensional curves of complexity $n$ and $n^{\alpha}$ for some $\alpha \in (0,1)$, we can compute their Fr\'echet distance in $O(n^{2\alpha} \log^2 n + n \log n)$ time. This rules out a conditional lower bound of the form $O((nm)^{1-\epsilon})$ 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 Fr\'echet 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, Krivo\v{s}ija and Sohler from 2015.