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

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 $d\ge 2$, 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 $n^{\alpha }$ for some $\alpha \in (0,1)$, we can compute their Fréchet 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 $d\ge 2$ and any $\epsilon >0$ 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.