Fréchet Queries in Geometric Trees

Abstract

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.

Skeleton-Based Recognition of Shapes in Images via Longest Path Matching

“…This variation is somewhat similar to the notion of a graph isomorphism, but here, our input graphs are geometric graphs rather than arbitrary ones. While fast algorithms for Fréchet distance to a geometric graph have been looked at in some limited settings, such as for trees [12], no one previously has considered the problem where the input path is not given as an embedding into the same frame of reference as the graph G, which adds considerably to the difficulty of the problem.…”

Fast algorithms for approximate Fréchet matching queries in geometric trees

“…We thank an anonymous referee for the conference version [20] of this paper for pointing out reference [17] to us. We also thank the anonymous referees for this journal paper for their constructive remarks that helped improve the presentation.…”

“…Remark: In the preliminary version [20] of this paper, we presented data structures with approximation ratio √ 2(1 + ε) for querying polygonal paths or trees with a query segment. These results hold in R 2 and the space requirements and query times are a logarithmic factor worse than the ones presented in the current paper.…”

“…However, their preprocessing times are faster by a logarithmic factor. The advantage of the results in [20] is that the presentation is completely self-contained. The current version heavily uses the results from Driemel and Har-Peled [17] that were mentioned before.…”

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

“…This model has been used for several generalizations of the Fréchet distance, such as map matching, 21 the mean curve problem, 22 a variant of the Fréchet distance allowing shortcuts, 11 and Fréchet matching queries in trees. 23 Together with introducing the notion of c-packed curves, Driemel et al presented a (1 + ε)-approximation for the continuous Fréchet distance in time O(cn/ε+cn log n), which works in any R d , d ≥ 2. Assuming SETH, the following lower bounds have been shown for c-packed curves: (1) For sufficiently small constant ε > 0 there is no (1+ε)-approximation in time O ((cn) 1−δ ) for any δ > 0.…”