Tight Fréchet bounds for $λ$-low density curves
The Fréchet distance is a well-studied similarity measure between curves. We computing the Fréchet distance between $λ$-low-density curves, the most general of realistic curve assumptions, where every ball of radius $r$ intersects at most $λ$ edges of length at least $r$. Previous algorithms either assumed constant $λ$ or had no tight dependence on $λ$. For two $n$-vertex $λ$-low-density curves in $\mathbb{R}^d$, we give a $(1+\varepsilon)$-approximation algorithm for the continuous and discrete Fréchet distance running in $ \tilde{O}\!\left(\frac{λ^{2/d}n^{2-2/d}}{\varepsilon^2}\right) $ time. Our key insight is a tight property of simplifying $λ$-low density curves: the simplification of any $n$-vertex $λ$-low-density curve is $O(λ^{1/d}n^{1-1/d})$-low-density. We show this is tight, and this provides the structural property under simplification that was previously known for $c$-packed curves. We provide matching lower bounds for $n$ and $λ$: assuming the Orthogonal Vectors Hypothesis, for every $δ>0$, we rule out algorithms with running time $O\!\left( \left( \frac{λ^{2/d}n^{2-2/d}} {\varepsilon^{2-4/d}} \right)^{1-δ} \right). $ We extend our techniques to the map matching problem, where we also give tight bounds.