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

Abstract

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 c-packed curves, for which the Fréchet 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/ √ 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)/ √ ε + cn log n). This improves upon the algorithm by Driemel et al. for any ε 1/ log n. Moreover, our algorithm's dependence on c, n and ε 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.

Translation Invariant Fréchet Distance Queries

“…Their algorithm runs in O(cn/ε + cn log n) time for curves in R d . Bringmann and Künnemann [3] improved the running time of the algorithm to O( cn √ ε log 2 (1/ε)+cn log n) for curves in R d . Assuming the Strong Exponential Time Hypothesis, Bringmann [2] showed that (i) for sufficiently small constants ε > 0 there is no (1 + ε)-approximation in time O((cn) 1−δ ) for any δ > 0, and (ii) in any dimension d ≥ 5 there is no (1 + ε)-approximation in time O((cn/ √ ε) 1−δ ) for any δ > 0.…”

“…Existing algorithms [3,12] for computing the Fréchet distance between c-packed curves require that both curves are c-packed. An open problem is whether the Fréchet distance can be approximated efficiently when only one curve is c-packed.…”

A Subquadratic nε-approximation for the Continuous Fréchet Distance

“…These families of curves again include κ-bounded curves, but also c-packed curves. An improved algorithm for c-packed curves, which matches conditional lower bounds, was later given by Bringmann and Künnemann [5]. Gudmundsson et al [16] give a √ d-approximate algorithm for the case where the curves have sufficiently long edges (relative to their Fréchet distance) that runs in linear time.…”

Approximating the packedness of polygonal curves

“…A curve π is c-packed if for any ball B, the length of the portion of π contained in B is at most c times the radius of B. In their paper they considered the problem of computing the Fréchet distance between two c-packed curves and presented a (1 + ε)-approximation algorithm with running time O( cn ε + cn log n), which was later improved to O( cn √ ε log 2 (1/ε) + cn log n) by Bringmann and Künnemann [7]. Other models for realistic curves have also been studied.…”