Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

Buchin, Kevin; Buchin, Maike; Meulemans, Wouter; Mulzer, Wolfgang

doi:10.1007/s00454-017-9878-7

Cited by 34 publications

(22 citation statements)

References 71 publications

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“…Only quite recently there has finally been progress on this question. First, Buchin et al [12] presented an algorithm with a slightly improved (but still superquadratic) running time. Then, Bringmann [7] proved that no significantly faster algorithm for computing the Fréchet distance between two general polygonal curves exists unless the Strong Exponential Time Hypothesis (SETH) fails.…”

Section: Introductionmentioning

confidence: 99%

SETH Says: Weak Fréchet Distance is Faster, but only if it is Continuous and in One Dimension

Buchin¹,

Ophelders²,

Speckmann³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We show by reduction from the Orthogonal Vectors problem that algorithms with strongly subquadratic running time cannot approximate the Fréchet distance between curves better than a factor 3 unless SETH fails. We show that similar reductions cannot achieve a lower bound with a factor better than 3. Our lower bound holds for the continuous, the discrete, and the weak discrete Fréchet distance even for curves in one dimension. Interestingly, the continuous weak Fréchet distance behaves differently. Our lower bound still holds for curves in two dimensions and higher. However, for curves in one dimension, we provide an exact algorithm to compute the weak Fréchet distance in linear time. ACM Subject Classification I.3.5 Computational Geometry and Object ModelingWeak Fréchet Distance is Faster if it is Continuous and in One Dimension distance in one dimension can also be used as a subroutine for approximating the Fréchet distance for curves in two and higher dimensions [8]. Bringmann's lower bound sparked renewed interest in the computation of the Fréchet distance between one-dimensional curves. Cabello and Korman showed that for two 1D curves that do not overlap, the Fréchet distance can be computed in linear time (personal communication, referenced in [8]). Furthermore, Buchin et al. [15] proved that if one of the curves visits any location at most a constant number of times, then the Fréchet distance can be computed in near linear time. Both results apply only to restricted classes of curves and hence the general case in 1D remained open.Our results. In this paper we settle the general question for one dimension: we give a conditional lower bound for the Fréchet distance between two general polygonal curves in 1D. To do so we reduce (in linear time) from the Orthogonal Vector Problem: given two sets of vectors, is there a pair of orthogonal vectors, one from each set? For vectors of dimension d = ω(log n) no algorithm running in strongly subquadratic time is known. Furthermore, an algorithm with such a running time does not exist in various computational models [24] and would have far-reaching consequences [1]. In particular, the existence of a strongly subquadratic algorithm for the Orthogonal Vector Problem would imply that the Strong Exponential Time Hypothesis fails. Our reduction hence implies that no strongly subquadratic algorithm for approximating the Fréchet distance within a factor less than 3 exists unless SETH fails.Our result also improves upon the previously best known conditional lower bound for curves in 2D by Bringmann and Mulzer [9] (approximation within a factor less than 1.399). Furthermore, we argue that similar reductions, based on a "traditional" encoding of the Orthogonal Vectors Problem, cannot achieve a lower bound better than 3.Section 2 gives various definitions and background. In particular, we recall an asymmetric variant of the Fréchet distance introduced by Alt and Godau [4], the so-called partial Fréchet distance. In Section 3 we succinctly state all our results and in Section 4 we b...

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Section: Introductionmentioning

confidence: 99%

SETH Says: Weak Fréchet Distance is Faster, but only if it is Continuous and in One Dimension

Buchin¹,

Ophelders²,

Speckmann³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In 1992 Alt and Godau [2] were the first to consider the problem and gave an O(n 2 log n) time algorithm for the problem. The only improvement since then is a randomized algorithm with running time O(n 2 (log log n) 2 ) in the word RAM model by Buchin et al [7]. In 2014 Bringmann [4] showed that, conditional on the Strong Exponential Time Hypothesis (SETH), there cannot exist an algorithm with running time O(n 2−ε ) for any ε > 0.…”

Section: Introductionmentioning

confidence: 99%

Translation Invariant Fréchet Distance Queries

Gudmundsson¹,

Renssen²,

Saeidi³

et al. 2021

Preprint

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The Fréchet distance is a popular similarity measure between curves. For some applications, it is desirable to match the curves under translation before computing the Fréchet distance between them. This variant is called the Translation Invariant Fréchet distance, and algorithms to compute it are well studied. The query version, however, is much less well understood.We study Translation Invariant Fréchet distance queries in a restricted setting of horizontal query segments. More specifically, we prepocess a trajectory in O(n 2 log 2 n) time and space, such that for any subtrajectory and any horizontal query segment we can compute their Translation Invariant Fréchet distance in O(polylog n) time. We hope this will be a step towards answering Translation Invariant Fréchet queries between arbitrary trajectories.

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“…Another common approach is to perform a so-called image skeletonization [6,8] instead of contour selection . Computing the Frechet distance between curves, both open and closed, is a relatively well researched problem [1,3,5,7] . Since an image contour is always some sort of a curve, comparing contours with the Frechet metric does not impose much difficulty .…”

Section: Introductionmentioning

confidence: 99%