Fast Fréchet Distance Between Curves with Long Edges

Abstract

Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let P and Q be two polygonal curves in R d with n and m vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in O((n + m) log(n + m)) time, (3) a linear-time √ d-approximation algorithm, and (4) a data structure that supports O(m log 2 n)time decision queries, where m is the number of vertices of the query curve and n the number of vertices of the preprocessed curve. 23:2Fast Fréchet Distance Between Curves with Long Edges O(n 2 /φ + n log n)-time algorithm to compute a φ-approximation of the discrete Fréchet distance for any integer 1 ≤ φ ≤ n. Therefore, an n -approximation, for any > 0, can be computed in (strongly) subquadratic time. For the continuous Fréchet distance, there are also a few subquadratic algorithms known for restricted classes of curves such as κ-bounded, backbone and c-packed curves. Alt et al. [3] considered κ-bounded curves and they gave an O(n log n) time algorithm to (κ + 1)-approximate the Fréchet distance. A curve P is κbounded if for any two points x, y ∈ P , the union of the balls with radii r centered at x and y contains the whole P [x, y] where r is equal to (κ/2) times the Euclidean distance between x and y. For any > 0, Aronov et al. [4] provided a near-linear time (1 + )-approximation algorithm for the discrete Fréchet distance for so-called backbone curves that have essentially constant edge length and require a minimum distance between non-consecutive vertices. For c-packed curves a (1 + )-approximation can be computed in O(cn/ + cn log n) time [11]. A curve is c-packed if for any ball B, the length of the portion of P contained in B is at most c times the diameter of B.

Adaptive Computation of the Discrete Fréchet Distance

“…The results described in this work are by far only preliminary. Among the various questions that those preliminary results raise, we discuss here the relation to the long edged sequences recently described by Gudmundsson et al [8]; a potential parameterized conditional lower bound matching our parameterized upper bound on the computational complexity of the Discrete Fréchet Distance; (the not so) similar results on the Orthogonal Vector decision problem; and the possibility of a theory of reductions between parameterized versions of polynomial problems without clear (parameterized or not) computational complexity lower bounds.…”

“…Relation to Long Edged Sequences: In 2018, Gudmundsson et al [8] described an algorithm deciding if the Fréchet distance is equal to a given value f in time linear in the size of the input curves when each edge is longer than the Fréchet Distance between those two curves. Algorithm 3 is more general than Gudmundsson et al's algorithm [8], but it also performs in linear time on long-edged instances: the traversal corresponding to the Fréchet Distance of such an instance is along the diagonal, implying a certificate width of 1. See Figures 1, 2 and 3 for the Euclidean matrix, Fréchet Matrix and Dynamic Program Matrix of a random instance formed of 5 points, each edge of length 100 with a Fréchet Distance of 13.45 (see Appendix A.2 for the Python code used to generate long edged instances).…”

“…Acknowledgements: The author would like to thank Travis Gagie for interesting discussions and for pointing out Gudmundsson et al's work [8]. Funding: Jérémy Barbay is partially funded by the project Fondecyt Regular no.…”

Translation Invariant Fréchet Distance Queries

“…Querying the standard Fréchet distance between a given trajectory and a query trajectory has been studied [8,9,10,12,13], but due to the difficult nature of the query problem, data structures only exist for answering a restricted class of queries. There are two results which are most relevant.…”

“…In the special case when k = 1, the approximation ratio can be improved to (1 + ε) with no increase in preprocessing or query time with respect to n. New ideas are required for exact Fréchet distance queries on arbitrary query trajectories. Other query versions for the standard Fréchet distance have also been considered [8,12,13].…”