Fréchet Distance for Curves, Revisited

Aronov, Boris; Har-Peled, Sariel; Knauer, Christian; Wang, Yusu; Wenk, Carola

doi:10.1007/11841036_8

Cited by 104 publications

(100 citation statements)

References 22 publications

(22 reference statements)

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“…In such cases, Hausdorff-based algorithms can be used to approximate the Fréchet one. On the other hand, [7] considers κ-bounded and backbone curves, and introduces efficient approximation and pseudo-output-sensitive exact algorithms that calculate the discrete-Fréchet distance. A general discussion on various distance-measures for point sets and their computations is presented in [16].…”

Section: Related Workmentioning

confidence: 99%

Dynamics-aware similarity of moving objects trajectories

Trajcevski¹,

Scheuermann²,

Tamassia

et al. 2007

Proceedings of the 15th Annual ACM International Symposium on Advances in Geographic Information Systems

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This work addresses the problem of obtaining the degree of similarity between trajectories of moving objects. Typically, a Moving Objects Database (MOD) contains sequences of (location,time) points describing the motion of individual objects, however, they also have an implicit information about the velocity, which is an important attribute describing the dynamics of a particular object. Our main goal is to extend the MOD functionalities with the capability of reasoning about how similar are the trajectories of objects that, possibly, move along geographically different routes. Towards this, we use a distance function which balances the lack of temporal-awareness of the Hausdorff distance with the generality (and complexity of calculation) of the Fréchet distance. Based on the observation that, as a firstapproximation in practice, the individual segments of trajectories are assumed to have constant speed, we provide efficient algorithms for: (1) optimal matching between trajectories; and (2) approximate matching between trajectories, both under translations and rotations, where the approximate algorithm guarantees a bounded error-quality with respect to the optimal one.

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Section: Related Workmentioning

confidence: 99%

Dynamics-aware similarity of moving objects trajectories

Trajcevski¹,

Scheuermann²,

Tamassia

et al. 2007

Proceedings of the 15th Annual ACM International Symposium on Advances in Geographic Information Systems

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show abstract

“…They argued that the Fréchet distance is better suited as a similarity measure, and they described an O(n 2 log n) time algorithm to compute it on a real RAM or pointer machine. 1 Since Alt and Godau's seminal paper, there has been a wealth of research in various directions, such as extensions to higher dimensions [7,23,26,28,33,46], approximation algorithms [9,10,37], the geodesic and the homotopic Fréchet distance [29,34,38,48], and much more [2,22,25,35,36,51,54,55]. Most known approximation algorithms make further assumptions on the curves, and only an O(n 2 )-time approximation algorithm is known for arbitrary polygonal curves [24].…”

Section: Introductionmentioning

confidence: 99%

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

Buchin

Meulemans

et al. 2017

Discrete Comput Geom

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Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One popular measure is the Fréchet distance. Since it was proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original O(n 2 log n) algorithm by Alt and Godau for computing the Fréchet distance remains the state of the art (here, n denotes the number of edges on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUM-hard. In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fréchet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fréchet distance between two polygonal curves in time O(n 2 √ log n(log log n) 3/2 ) on a pointer machine and in time O(n 2 (log log n) 2 ) on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth O(n 2−ε ), for some ε > 0. We believe that this reveals an intriguing new aspect of this well-studied problem. Finally, we show how to obtain the first subquadratic algorithm for computing the weak Fréchet distance on a word RAM.

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“…The Fréchet distance measures the similarity of continuous shapes [5,8,11,26,39] by calculating a distance-based value that represents the similarity of the shapes. Although the traditional Fréchet distance operates in a space that is free of obstacles, recent works have realized the potential of the Fréchet distance in domains with obstacles.…”

Section: Introductionmentioning

confidence: 99%

Shortest Path Problems on a Polyhedral Surface

Cook

Wenk

2012

Algorithmica

Self Cite

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We develop algorithms to compute edge sequences, Voronoi diagrams, shortest path maps, the Fréchet distance, and the diameter for a polyhedral surface. Distances on the surface are measured either by the length of a Euclidean shortest path or by link distance.Key words: Polyhedral Surface, Voronoi Diagram, Shortest Path Map, Fréchet distance, Diameter, Link Distance, Euclidean Shortest Path IntroductionTwo questions are invariably encountered when dealing with shortest path problems. The first question is how to represent the combinatorial structure of a shortest path. In the plane with polygonal obstacles, a shortest path can only turn at obstacle vertices, so a shortest path can be combinatorially described as a sequence of obstacle vertices [28]. On a polyhedral surface, a shortest path need not turn at vertices [35], so a path is often described combinatorially by an edge sequence that represents the sequence of edges encountered by the path [1]. A benefit of representing shortest paths by edge sequences is that a series of unfolding rotations can be used to reduce the problem of computing a shortest path on a polyhedral surface into a two-dimensional problem. This process is described more fully in section 3.The second commonly encountered shortest path question is how to compute shortest paths in a problem space with M vertices. The following preprocessing schemes compute combinatorial representations of all possible shortest paths. In a $ This work has been supported by the National Science Foundation grant NSF CAREER CCF-0643597.$$ A previous version of this work has been presented in [21]. February 21, 2009 simple polygon, Guibas et al. [28] give an optimal Θ(M ) preprocessing scheme that permits a shortest path to be computed between any two points in O(log M ) time. In the plane with polygonal obstacles, Chiang and Mitchell [16] support shortest path queries between any two points after O(M 11 ) preprocessing. On a convex polyhedral surface, Mount [37] shows that Θ(M 4 ) combinatorially distinct shortest path edge sequences exist, and Schevon and O'Rourke [40] show that only Θ(M 3 ) of these edge sequences are maximal (i.e., they cannot be extended at either end without creating a suboptimal path). Agarwal et al. Preprint submitted to Elsevier[1] use these properties to compute the Θ(M 4 ) shortest path edge sequences in O(M 6 2 α(M ) log M ) time and the diameter in O(M 8 log M ) time, where α(M ) is the inverse Ackermann function. The diameter is the largest shortest path distance between any two points on the surface. Despite recent efforts by Chandru et al. [13] to improve the runtimes of Agarwal et al.[1], these runtimes have not improved since 1997. Our main result improves the edge sequence and diameter algorithms of [1] by a linear factor. We achieve this improvement by combining the star unfolding technique of [1] with the kinetic Voronoi diagram structure of Albers et al. [3]. A kinetic Voronoi diagram allows its defining point sites to move. A popular alternative to precomputing...

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Fréchet Distance for Curves, Revisited

Cited by 104 publications

References 22 publications

Dynamics-aware similarity of moving objects trajectories

Dynamics-aware similarity of moving objects trajectories

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

Shortest Path Problems on a Polyhedral Surface

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