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Computer Science, Computational Geometry, cs.CG,Computer Science, Data Structures and Algorithms, cs.DS
Abstract:
The Fr\'echet distance is a well-studied and very popular measure of
similarity of two curves. The best known algorithms have quadratic time
complexity, which has recently been shown to be optimal assuming the Strong
Exponential Time Hypothesis (SETH) [Bringmann FOCS'14].
To overcome the worst-case quadratic time barrier, restricted classes of
curves have been studied that attempt to capture realistic input curves. The
most popular such class are c-packed curves, for which the Fr\'echet distance
has a $(1+\epsilon)$-approximation in time $\tilde{O}(c n /\epsilon)$ [Driemel
et al. DCG'12]. In dimension $d \ge 5$ this cannot be improved to
$O((cn/\sqrt{\epsilon})^{1-\delta})$ for any $\delta > 0$ unless SETH fails
[Bringmann FOCS'14].
In this paper, exploiting properties that prevent stronger lower bounds, we
present an improved algorithm with runtime $\tilde{O}(cn/\sqrt{\epsilon})$.
This is optimal in high dimensions apart from lower order factors unless SETH
fails. Our main new ingredients are as follows: For filling the classical
free-space diagram we project short subcurves onto a line, which yields
one-dimensional separated curves with roughly the same pairwise distances
between vertices. Then we tackle this special case in near-linear time by
carefully extending a greedy algorithm for the Fr\'echet distance of
one-dimensional separated curves.
