COMPUTING THE DISCRETE FRÉCHET DISTANCE WITH IMPRECISE INPUT

2012 ◽  
Vol 22 (01) ◽  
pp. 27-44 ◽  
Author(s):  
HEE-KAP AHN ◽  
CHRISTIAN KNAUER ◽  
MARC SCHERFENBERG ◽  
LENA SCHLIPF ◽  
ANTOINE VIGNERON
Keyword(s):  
Time Algorithm ◽  
Constant Factor ◽  
Fréchet Distance ◽  
Planar Case ◽  
Running Time ◽  
Frechet Distance ◽  
Axis Parallel ◽  
Polygonal Curves ◽  
Orthogonal Case ◽  
Improved Algorithm

We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time 2O(d2) m2n2 log 2(mn) the minimum Fréchet distance between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O(mn log 3(mn) + (m2+n2) log (mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L∞ distance, we give an O(dmn log (dmn))-time algorithm. We also give efficient O(dmn)-time algorithms to approximate the maximum Fréchet distance, as well as the minimum and maximum Fréchet distance under translation. These algorithms achieve constant factor approximation ratios in "realistic" settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).

scholarly journals Fast Fréchet Distance Between Curves with Long Edges

2019 ◽  
Vol 29 (02) ◽  
pp. 161-187
Author(s):  
Joachim Gudmundsson ◽  
Majid Mirzanezhad ◽  
Ali Mohades ◽  
Carola Wenk
Keyword(s):  
Data Structure ◽  
Approximation Algorithm ◽  
Special Class ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Fréchet Distance ◽  
Frechet Distance ◽  
Polygonal Curves

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 [Formula: see text] and [Formula: see text] be two polygonal curves in [Formula: see text] with [Formula: see text] and [Formula: see text] 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 [Formula: see text] time, (3) a linear-time [Formula: see text]-approximation algorithm, and (4) a data structure that supports [Formula: see text]-time decision queries, where [Formula: see text] is the number of vertices of the query curve and [Formula: see text] the number of vertices of the preprocessed curve.

scholarly journals Discrete Fréchet Distance under Translation

2021 ◽  
Vol 17 (3) ◽  
pp. 1-42
Author(s):  
Karl Bringmann ◽  
Marvin KüNnemann ◽  
André Nusser
Keyword(s):  
Grid Graph ◽  
Fréchet Distance ◽  
Worst Case ◽  
Grid Graphs ◽  
Exponential Time ◽  
Running Time ◽  
Frechet Distance ◽  
Spatial Domains ◽  
Polygonal Curves ◽  
Important Variant

The discrete Fréchet distance is a popular measure for comparing polygonal curves. An important variant is the discrete Fréchet distance under translation, which enables detection of similar movement patterns in different spatial domains. For polygonal curves of length n in the plane, the fastest known algorithm runs in time Õ( n 5 ) [12]. This is achieved by constructing an arrangement of disks of size Õ( n 4 ), and then traversing its faces while updating reachability in a directed grid graph of size N := Õ( n 5 ), which can be done in time Õ(√ N ) per update [27]. The contribution of this article is two-fold. First, although it is an open problem to solve dynamic reachability in directed grid graphs faster than Õ(√ N ), we improve this part of the algorithm: We observe that an offline variant of dynamic s - t -reachability in directed grid graphs suffices, and we solve this variant in amortized time Õ( N 1/3 ) per update, resulting in an improved running time of Õ( N 4.66 ) for the discrete Fréchet distance under translation. Second, we provide evidence that constructing the arrangement of size Õ( N 4 ) is necessary in the worst case by proving a conditional lower bound of n 4 - o(1) on the running time for the discrete Fréchet distance under translation, assuming the Strong Exponential Time Hypothesis.

scholarly journals A Modification of the Frechet Distance for Nonisomorphic Trees

2021 ◽  
pp. 20-27
Author(s):  
Yevgen V. Vololazskiy ◽  
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Fréchet Distance ◽  
Reference Tree ◽  
Frechet Distance

The paper presents a modification of the Frechet distance for nonisomorphic trees. While the classical Frechet distance between nonisomorphic trees is undefined, a new measure called similarity of a tree to a reference tree is given that is defined for a wider class of trees. A polynomial-time algorithm is given to determine whether one tree’s similarity to another is less than a given number.

scholarly journals Fréchet Similarity of Closed Polygonal Curves

2016 ◽  
Vol 26 (01) ◽  
pp. 53-66 ◽  
Author(s):  
M. I. Schlesinger ◽  
E. V. Vodolazskiy ◽  
V. M. Yakovenko
Keyword(s):  
Hausdorff Metric ◽  
Fréchet Distance ◽  
Frechet Distance ◽  
Number Formula ◽  
Polygonal Curves

The article analyzes similarity of closed polygonal curves with respect to the Fréchet metric, which is stronger than the well-known Hausdorff metric and therefore is more appropriate in some applications. An algorithm is described that determines whether the Fréchet distance between two closed polygonal curves with [Formula: see text] and [Formula: see text] vertices is less than a given number [Formula: see text]. The algorithm takes [Formula: see text] time whereas the previously known algorithms take [Formula: see text] time.

COMPUTING THE FRÉCHET DISTANCE BETWEEN TWO POLYGONAL CURVES

1995 ◽  
Vol 05 (01n02) ◽  
pp. 75-91 ◽  
Author(s):  
HELMUT ALT ◽  
MICHAEL GODAU
Keyword(s):  
Distance Function ◽  
Fréchet Distance ◽  
Closed Curves ◽  
Frechet Distance ◽  
Polygonal Curves ◽  
Arbitrary Dimensions ◽  
Polygonal Chains ◽  
Distance Measuring

As a measure for the resemblance of curves in arbitrary dimensions we consider the so-called Fréchet-distance, which is compatible with parametrizations of the curves. For polygonal chains P and Q consisting of p and q edges an algorithm of runtime O(pq log(pq)) measuring the Fréchet-distance between P and Q is developed. Then some important variants are considered, namely the Fréchet-distance for closed curves, the nonmonotone Fréchet-distance and a distance function derived from the Fréchet-distance measuring whether P resembles some part of the curve Q.

ON PIERCING SETS OF AXIS-PARALLEL RECTANGLES AND RINGS

1999 ◽  
Vol 09 (03) ◽  
pp. 219-233 ◽  
Author(s):  
MICHAEL SEGAL
Keyword(s):  
Dimensional Space ◽  
Time Algorithm ◽  
Efficient Algorithms ◽  
Center Problem ◽  
Running Time ◽  
Higher Dimensional ◽  
Axis Parallel ◽  
The Given ◽  
Existing Result

We consider the p-piercing problem for axis-parallel rectangles. We are given a collection of axis-parallel rectangles in the plane and wish to determine whether there exists a set of p points whose union intersects all the given rectangles. We present efficient algorithms for finding a piercing set (i.e, a set of p points as above) for values of p=1,2,3,4,5. The result for 4 and 5-piercing improves an existing result of O(n  log 3 n) and O(n  log 4 n) to O(n  log  n) time. The result for 5-piercing can be applied find an O(n  log 2 n) time algorithm for planar rectilinear 5-center problem, in which we are given a set S of n points in the pane, and wish to find 5 axis-parallel congruent squares of smallest possible size whose union covers S. We improve the existing algorithm for general (but fixed) p to O(np-4 log  n) running time, and we also extend our algorithms to higher dimensional space. We also consider the problem of piercing a set of rectangular rings.

scholarly journals FRÉCHET DISTANCE PROBLEMS IN WEIGHTED REGIONS

2010 ◽  
Vol 02 (02) ◽  
pp. 161-179 ◽  
Author(s):  
YAM KI CHEUNG ◽  
OVIDIU DAESCU
Keyword(s):  
Approximation Algorithm ◽  
Shortest Path ◽  
Line Segment ◽  
Fréchet Distance ◽  
Weighted Regions ◽  
Frechet Distance ◽  
Polygonal Curves ◽  
Polyhedral Obstacles ◽  
Distance Problem ◽  
Planar Subdivisions

We discuss two versions of the Fréchet distance problem in weighted planar subdivisions. In the first one, the distance between two points is the weighted length of the line segment joining the points. In the second one, the distance between two points is the length of the shortest path between the points. In both cases, we give algorithms for finding a (1 + ∊)-factor approximation of the Fréchet distance between two polygonal curves. We also consider the Fréchet distance between two polygonal curves among polyhedral obstacles in [Formula: see text] (1/∞ weighted region problem) and present a (1 + ∊)-factor approximation algorithm.

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