Approximating the Fréchet distance for realistic curves in near linear time

Driemel, Anne; Har-Peled, Sariel; Wenk, Carola

doi:10.1145/1810959.1811019

Cited by 43 publications

(88 citation statements)

References 25 publications

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“…The notion of a tree being c-packed was introduced by Driemel et al [14]. We start by recalling the definition.…”

Section: C-packed Trees and µ-Simplificationsmentioning

confidence: 99%

“…Measuring the similarity between curves has been extensively studied in the last 20 years in computational geometry [1,4,14,20], as well in other areas, such as data mining [16,18], GIScience [6,7,21] and image processing [19].…”

Section: Introductionmentioning

confidence: 99%

“…Until recently, subquadratic time algorithms were only known for restricted cases, such as closed convex curves and κ-bounded curves [5]. In 2010, Driemel et al [14] introduced a new class of realistic curves, the so-called c-packed curves. A polygonal path is c-packed if the total length of the edges inside any ball is bounded by c times the radius of the ball.…”

Section: Introductionmentioning

confidence: 99%

“…Driemel et al showed that a (1 + ε)-approximation of the Fréchet distance between two c-packed curves with a total of n vertices in R d can be computed in O(cn/ε + cn log n) time. The notion of c-packedness has been argued [10,14] to capture many realistic settings for geometric paths and graphs. Chen et al [10] experimentally verified that maps of real world cities are φ-low density for a constant φ 1 .…”

Section: Introductionmentioning

confidence: 99%

“…A geometric graph G is φ-low-density, if for any radius ρ > 0, any ball with radius ρ intersects at most φ edges of G that are longer than ρ. A c-packed curve is φ-low density for φ = 2c; see [14].…”

Section: Introductionmentioning

confidence: 99%

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Fréchet Queries in Geometric Trees

Gudmundsson

Smid

2013

Lecture Notes in Computer Science

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Let T be a tree that is embedded in the plane and let ∆ > 0 be a real number. The aim is to preprocess T into a data structure, such that, for any query polygonal path Q, we can decide if T contains a path P whose Fréchet distance δ F (P, Q) to Q is less than ∆. For any real number ε > 0, we present an efficient data structure that solves an approximate version of this problem, for the case when T is c-packed and each of the edges of T and Q has length Ω(∆): If the data structure returns NO, then there is no such path P . If it returns YES, then δ F (P, Q) ≤ √ 2(1 + ε)∆ if Q is a line segment, and δ F (P, Q) ≤ 3∆ otherwise.

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“…The notion of a tree being c-packed was introduced by Driemel et al [14]. We start by recalling the definition.…”

Section: C-packed Trees and µ-Simplificationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fréchet Queries in Geometric Trees

Gudmundsson

Smid

2013

Lecture Notes in Computer Science

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Probabilistic Embeddings of the Fréchet Distance

Driemel

Krivošija

2018

Approximation and Online Algorithms

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The Fréchet distance is a popular distance measure for curves which naturally lends itself to fundamental computational tasks, such as clustering, nearest-neighbor searching, and spherical range searching in the corresponding metric space. However, its inherent complexity poses considerable computational challenges in practice. To address this problem we study distortion of the probabilistic embedding that results from projecting the curves to a randomly chosen line. Such an embedding could be used in combination with, e.g. locality-sensitive hashing. We show that in the worst case and under reasonable assumptions, the discrete Fréchet distance between two polygonal curves of complexity t in IR d , where d ∈ {2, 3, 4, 5}, degrades by a factor linear in t with constant probability. We show upper and lower bounds on the distortion. We also evaluate our findings empirically on a benchmark data set. The preliminary experimental results stand in stark contrast with our lower bounds. They indicate that highly distorted projections happen very rarely in practice, and only for strongly conditioned input curves. problem has a special structure in this case [14]. Clustering under the Fréchet distance can be done efficiently for 1-dimensional curves [19], but seems to be harder for curves in the plane or higher dimensions. Bringmann and Künnemann used projections to lines to speed up their approximation algorithm for the Fréchet distance [12]. They showed that the distance computation can be done in linear time if the convex hulls of the two curves are disjoint. It is tempting to believe that the curves being restricted to 1-dimensional space makes the problem significantly easier. However, in the general case, there are no algorithms known which are faster for 1-dimensional curves than for curves in higher dimensions. In practice, it is very common to separate the coordinates of trajectories to simplify computational tasks. It seems that in practice the inherent character of a trajectory is often largely preserved when restricted to one of the coordinates of the ambient space. Mathematically, this amounts to projecting the trajectory to a line.This motivates our study of probabilistic embeddings of the Fréchet distance into the space of 1-dimensional curves. Concretely, we study distortion of the probabilistic embedding that results from projecting the curves to a randomly chosen line. Such a random projection could be used in combination with probabilistic data structures, e.g. locality-sensitive hashing [20], but also with the multi-level data structures for Fréchet range searching given by Afshani and Driemel [2]. See below for a more in-depth discussion of these data structures.We show that in the worst case and under certain assumptions, the discrete Fréchet distance between two polygonal curves of complexity t in IR d , where d = {2, 3, 4, 5}, degrades by a factor linear in t with constant probability. In particular, we show upper and lower bounds on the change in distance for the class of c-packed curves. The notion...

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