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Geodesic Fréchet Distance Inside a Simple Polygon
[article]

2008
*
arXiv
*
pre-print

We present the first algorithm for the

arXiv:0802.2846v1
fatcat:l2n3oybxu5hplcra4teqzrx5sq
*geodesic**Fréchet**distance*between two*polygonal*curves*A*and B*inside**a**simple*bounding*polygon*P. ... Results are also presented for the*geodesic**Fréchet**distance*in*a**polygonal*domain with obstacles and the*geodesic*Hausdorff*distance*for sets of points or sets of line segments*inside**a**simple**polygon*... ij(ε) b ij(ε) ε*a*ij(ε) b kj(ε) C kj c)*Distance*ÙÖ Ö Ø Ð Ú ÐÙ × Ó Ø Ö Ø ×Ø Ò ÓÖÓÐÐ ÖÝ º¾º Ì Ö ×Ô ÓÒ ÐÐ ÓÙÖ ÓÙÒ Ö × Ó Ö ×Ô ÐÐ Ò ÓÙÒ Ò O(log k) Ø Ñ Ý ÓÑÔÙØ Ò t 1 Ò t 2 ÓÖ ÓÙÒ ÖÝº º¾º Ó × Ö Ø × ÓÒ ÈÖÓ Ð ...##
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Geodesic Fréchet distance inside a simple polygon

2010
*
ACM Transactions on Algorithms
*

We present the first algorithm for the

doi:10.1145/1868237.1868247
fatcat:34bns3us2fbrvnjq5kjb3di4ri
*geodesic**Fréchet**distance*between two*polygonal*curves*A*and B*inside**a**simple**polygon*P . ... Our work is different because we measure all*distances*using*geodesics**inside**a**simple**polygon*P , where*a**geodesic*is*a*path that avoids all obstacles and cannot be shortened by slight perturbations [ ...*Geodesic**Fréchet**Distance*Runtime Theorem 4: The exact*geodesic**Fréchet**distance*between two*polygonal*curves*A*and B*inside**a**simple*bounding*polygon*P can be computed in O(k + N 2 log kN log N ) expected ...##
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Walking your dog in the woods in polynomial time

2008
*
Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08
*

We describe

doi:10.1145/1377676.1377694
dblp:conf/compgeom/ChambersVELLT08
fatcat:b7kuo5qnrrbwrn2n5iz7gtsndu
*a*polynomial-time algorithm to compute the homotopic*Fréchet**distance*between two given*polygonal*curves in the plane minus*a*given set of obstacles, which are either points or*polygons*. ... We propose*a*natural extension of*Fréchet**distance*to more general metric spaces, which requires the leash itself to move continuously over time. ... This continuity requirement is satisfied automatically for curves*inside**a**simple**polygon*, but not in more general environments like convex polyhedra. ...##
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Homotopic Fréchet distance between curves or, walking your dog in the woods in polynomial time

2010
*
Computational geometry
*

We describe

doi:10.1016/j.comgeo.2009.02.008
fatcat:vqtemc2z6nffxd7si62be32s7u
*a*polynomial-time algorithm to compute the homotopic*Fréchet**distance*between two given*polygonal*curves in the plane minus*a*given set of*polygonal*obstacles. ... We propose*a*natural extension of*Fréchet**distance*to more general metric spaces, which requires the leash itself to move continuously over time. ... Acknowledgments This research was initiated during*a*visit to INRIA Lorraine in Nancy, made possible by*a*UIUC-CNRS-INRIA travel grant. ...##
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How to Stay Socially Distant: A Geometric Approach
[article]

2022
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arXiv
*
pre-print

We then consider social

arXiv:2203.04548v1
fatcat:iv42zsg5kbcwxmbrmzininpobu
*distance*width among*polygons*, which measures the minimum*distance*that two agents can maintain while restricted to travel in or on the boundary of the same*polygon*. ... We first study the social*distance*width of two*polygonal*curves in one and two dimensions, providing conditional lower bounds and matching algorithms. ... We then consider social*distance*width among*polygons*, which measures the minimum*distance*that two agents can maintain while restricted to travel in or on the boundary of the same*polygon*. ...##
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Fréchet Distance Problems in Weighted Regions
[article]

2010
*
arXiv
*
pre-print

In both cases, we give algorithms for finding

arXiv:1004.4901v1
fatcat:ygs2tbsk7vfvrkhrcvg4ah5smq
*a*(1+epsilon)-factor approximation of the Fr\'echet*distance*between two*polygonal*curves. ... We also consider the Fr\'echet*distance*between two*polygonal*curves among polyhedral obstacles in R^3 and present*a*(1+epsilon)-factor approximation algorithm. ... They gave the first polynomial-time algorithm for computing the*Fréchet**distance*between*simple**polygons*. In the second category, the*distance*between two points is the*geodesic**distance*. ...##
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Development of a metric and the methods for quantitative estimation of the segmentation of biomedical images

2017
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Eastern-European Journal of Enterprise Technologies
*

Here and hereafter, d denotes the

doi:10.15587/1729-4061.2017.119493
fatcat:viqnkj5srfd6znvrvar7ixufd4
*geodesic**distance**inside*the tree, that is, d(t, t 0 ) indicates the length of the arc*inside*T, which connects t and t 0 . Mapping of root trees. ... In paper [17] , authors show an algorithm for computing the*Fréchet**distance*for surfaces, which are represented by*simple**polygons*. The algorithm has polynomial complexity. ... S k*a*c h k o v ...##
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Applied Similarity Problems Using Frechet Distance
[article]

2013
*
arXiv
*
pre-print

We also study this problem in the setting where the

arXiv:1307.6628v1
fatcat:ayxp6ajgnzga7pcnz4urfgrugy
*polygonal*curves are*inside**a**simple**polygon*. ... As the last part of this thesis, given*a*point set S and*a**polygonal*curve P in R^d, we study the problem of finding*a**polygonal*curve Q through S, which has*a*minimum*Frechet**distance*to P. ...*Geodesic**Fréchet**Distance*In [28] , Cook and Wenk described an algorithm for the*geodesic**Fréchet**distance*between two*polygonal*curves P and Q*inside**a**simple**polygon*K. ...##
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Locally correct Fréchet matchings

2018
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Computational geometry
*

The

doi:10.1016/j.comgeo.2018.09.002
fatcat:ppwzxzef6zdlzh6fxyifd6p6yu
*Fréchet**distance*is*a*metric to compare two curves, which is based on monotonous matchings between these curves. We call*a*matching that results in the*Fréchet**distance**a**Fréchet*matching. ... We prove that at least one such*a*matching exists for two*polygonal*curves and give an algorithm to compute it. ... The*Fréchet**distance*is the maximal*distance*between two matched points minimized over all monotonous matchings of the curves. We call*a*matching resulting in the*Fréchet**distance**a**Fréchet*matching. ...##
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How to walk your dog in the mountains with no magic leash

2012
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Proceedings of the 2012 symposuim on Computational Geometry - SoCG '12
*

We describe

doi:10.1145/2261250.2261269
dblp:conf/compgeom/Har-PeledNSS12
fatcat:hkoomihgwvbkjhjg2iduhacc3a
*a*O(log n)-approximation algorithm for computing the homotopic*Frechét**distance*between two*polygonal*curves that lie on the boundary of*a*triangulated topological disk. ... Surprisingly, it is not even known if computing either the homotopic*Frechét**distance*, or the minimum height of*a*homotopy, is in NP. ... [EGH + 02] considered the*Frechét**distance**inside**a**simple**polygon*as*a*way to facilitate sweeping it efficiently. ...##
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How to Walk Your Dog in the Mountains with No Magic Leash

2015
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Discrete & Computational Geometry
*

We describe

doi:10.1007/s00454-015-9737-3
fatcat:s7y6pxj2i5h3rp777deqjbbaxq
*a*O(log n)-approximation algorithm for computing the homotopic*Frechét**distance*between two*polygonal*curves that lie on the boundary of*a*triangulated topological disk. ... Surprisingly, it is not even known if computing either the homotopic*Frechét**distance*, or the minimum height of*a*homotopy, is in NP. ... [EGH + 02] considered the*Frechét**distance**inside**a**simple**polygon*as*a*way to facilitate sweeping it efficiently. ...##
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How to Walk Your Dog in the Mountains with No Magic Leash
[article]

2015
*
arXiv
*
pre-print

We describe

arXiv:1401.7042v2
fatcat:remyhych2nb4ldfgrmumfl2gde
*a*O( n )-approximation algorithm for computing the homotopic*distance*between two*polygonal*curves that lie on the boundary of*a*triangulated topological disk. ... Surprisingly, it is not even known if computing either the homotopic*distance*, or the minimum height of*a*homotopy, is in NP. ... [EGH + 02] considered the*Fréchet**distance**inside**a**simple**polygon*as*a*way to facilitate sweeping it efficiently. ...##
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Computing Optimal Homotopies over a Spiked Plane with Polygonal Boundary

2017
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European Symposium on Algorithms
*

The algorithm combines structural properties of homotopies arising from the geometry with methodology for computing

doi:10.4230/lipics.esa.2017.23
dblp:conf/esa/BurtonCKMOS17
fatcat:tltyqyrvafeqrlitqfqim4fzru
*Fréchet**distances*. ... In this paper, we examine this problem in*a*geometric setting, where we consider the boundary of*a**polygonal*domain with spikes, point obstacles that can be crossed at an additive cost. ... The former is*a*variant where the leashes must stay*inside**a**polygonal*boundary and remain*geodesic*, but no obstacles are present*inside*the*polygon*[13] . ...##
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Page 3736 of Mathematical Reviews Vol. , Issue 98F
[page]

1998
*
Mathematical Reviews
*

*A*

*simple*piecewise linear Markov transformation on an interval is considered. ... i.e. there exists

*a*positive constant € such that distinct orbits eventually separate by

*a*

*distance*of at least ¢. ...

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Simple Curve Embedding
[article]

2013
*
arXiv
*
pre-print

, as well as under weak

arXiv:1303.0821v1
fatcat:kkrd3xnf4zdi3konbfqdntc3by
*Frechet**distance*if S is*a*terrain. ... Given*a*curve f and*a*surface S, how hard is it to find*a**simple*curve f' in S that is the most similar to f? ... Acknowledgements The authors thank Sariel Har-Peled, Anne Driemel, and Benjamin Raichel for initial discussion of the*simple*curve embedding problem. ...
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