Improved Approximation for Fréchet Distance on c-packed Curves Matching Conditional Lower Bounds

The Fréchet 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 cpacked curves, for which the Fréchet distance has a (1 + ε)-approximation in time Õ(cn/ε) [Driemel et al. DCG’12]. In dimension d > 5 this cannot be improved to O((cn/ √ ε )1−δ) for any δ > 0 unless SETH fails [Bringmann FOCS’14]. In this paper, exploiting properties that prevent stronger lower bounds, we present an improved algorithm with runtime Õ(cn/ √ ε ). 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 Fréchet distance of one-dimensional separated curves. ∗Max Planck Institute for Informatics, Campus E1 4, 66123 Saarbrücken, Germany; [email protected]. Karl Bringmann is a recipient of the Google Europe Fellowship in Randomized Algorithms, and this research is supported in part by this Google Fellowship. †Max Planck Institute for Informatics, Campus E1 4, 66123 Saarbrücken, Germany; [email protected]. Saarbrücken Graduate School of Computer Science, Germany 1 ar X iv :1 40 8. 13 40 v1 [ cs .C G ] 6 A ug 2 01 4