A Subquadratic $n^ε$-approximation for the Continuous Fréchet Distance

The Fréchet distance is a commonly used similarity measure between curves. It is known how to compute the continuous Fréchet distance between two polylines with $m$ and $n$ vertices in $\mathbb{R}^d$ in $O(mn (\log \log n)^2)$ time; doing so in strongly subquadratic time is a longstanding open problem. Recent conditional lower bounds suggest that it is unlikely that a strongly subquadratic algorithm exists. Moreover, it is unlikely that we can approximate the Fréchet distance to within a factor $3$ in strongly subquadratic time, even if $d=1$. The best current results establish a tradeoff between approximation quality and running time. Specifically, Colombe and Fox (SoCG, 2021) give an $O(α)$-approximate algorithm that runs in $O((n^3 / α^2) \log n)$ time for any $α\in [\sqrt{n}, n]$, assuming $m = n$. In this paper, we improve this result with an $O(α)$-approximate algorithm that runs in $O((n + mn / α) \log^3 n)$ time for any $α\in [1, n]$, assuming $m \leq n$ and constant dimension $d$.

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