The Frechet distance is a well-studied and 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 Frechet distance has a (1 + ε)-approximation in time O(cn/ε + cn log n) [Driemel et al. DCG'12]. In dimension d ≥ 5 this cannot be improved to O((cn/{the square root of}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 time complexity O(cn log~2(1/ε)/{the square root of}ε + cn log n). This improves upon the algorithm by Driemel et al. for any ε 1/log n, and matches the conditional lower bound (up to lower order factors of the form n~(o(1))).
展开▼
