Computing the Fr\'echet Distance When Just One Curve is $c$-Packed: A Simple Almost-Tight Algorithm

We study approximating the continuous Fr\'echet distance of two curves with complexity $n$ and $m$, under the assumption that only one of the two curves is $c$-packed. Driemel, Har{-}Peled and Wenk DCG'12 studied Fr\'echet distance approximations under the assumption that both curves are $c$-packed. In $\mathbb{R}^d$, they prove a $(1+\varepsilon)$-approximation in $\tilde{O}(d \, c\,\frac{n+m}{\varepsilon})$ time. Bringmann and K\"unnemann IJCGA'17 improved this to $\tilde{O}(c\,\frac{n + m }{\sqrt{\varepsilon}})$ time, which they showed is near-tight under SETH. Recently, Gudmundsson, Mai, and Wong ISAAC'24 studied our setting where only one of the curves is $c$-packed. They provide an involved $\tilde{O}( d \cdot (c+\varepsilon^{-1})(cn\varepsilon^{-2} + c^2m\varepsilon^{-7} + \varepsilon^{-2d-1}))$-time algorithm when the $c$-packed curve has $n$ vertices and the arbitrary curve has $m$, where $d$ is the dimension in Euclidean space. In this paper, we show a simple technique to compute a $(1+\varepsilon)$-approximation in $\mathbb{R}^d$ in time $O(d \cdot c\,\frac{n+m}{\varepsilon}\log\frac{n+m}{\varepsilon})$ when one of the curves is $c$-packed. Our approach is not only simpler than previous work, but also significantly improves the dependencies on $c$, $\varepsilon$, and $d$. Moreover, it almost matches the asymptotically tight bound for when both curves are $c$-packed. Our algorithm is robust in the sense that it does not require knowledge of $c$, nor information about which of the two input curves is $c$-packed.