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.

翻译：Fréchet距离的细粒度复杂度计算是近期大量研究的主题，始于2014年Bringmann基于SETH假设的二次条件性下界。后续研究在1维情形下建立了基本相同的复杂度下界。然而，Bringmann证明在维度$d\geq 2$时紧的不平衡情形仍未被解决。填补这一空白，我们证明在非平衡情形下有可能实现更快的Fréchet距离算法：给定两条1维曲线，复杂度分别为$n$和$n^{\alpha}$（$\alpha \in (0,1)$），可在$O(n^{2\alpha} \log^2 n + n \log n)$时间内计算其Fréchet距离。这排除了Bringmann针对$d \geq 2$且任意$\varepsilon>0$所证$O((nm)^{1-\epsilon})$形式条件性下界的可能性，进而表明与$d=1$情形的严格分离。我们方法的核心是一种存储复杂度为$n$的1维曲线$P$的数据结构，支持以复杂度为$m$的曲线$Q$查询$P$与$Q$的连续Fréchet距离。该数据结构空间复杂度为$\mathcal{O}(n\log n)$，查询时间复杂度为$\mathcal{O}(m^2 \log^2 n)$。我们的证明使用了一个基于访问顺序概念的关键引理，该引理可能具有独立研究价值。我们通过显著简化Driemel、Krivošija和Sohler于2015年提出的聚类算法的正确性证明来展示这一点。