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Geodesic Fréchet distance inside a simple polygon
 Proceedings of the 25th International Symposium on Theoretical Aspects of Computer Science (STACS
, 2008
"... Abstract. We unveil an alluring alternative to parametric search that applies to both the nongeodesic and geodesic Fréchet optimization problems. This randomized approach is based on a variant of redblue intersections and is appealing due to its elegance and practical efficiency when compared to p ..."
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Abstract. We unveil an alluring alternative to parametric search that applies to both the nongeodesic and geodesic Fréchet optimization problems. This randomized approach is based on a variant of redblue intersections and is appealing due to its elegance and practical efficiency when compared to parametric search. We present the first algorithm for the geodesic Fréchet distance between two polygonal curves A and B inside a simple bounding polygon P. The geodesic Fréchet decision problem is solved almost as fast as its nongeodesic sibling and requires O(N 2 log k) time and O(k + N) space after O(k) preprocessing, where N is the larger of the complexities of A and B and k is the complexity of P. The geodesic Fréchet optimization problem is solved by a randomized approach in O(k +N 2 log kN log N) expected time and O(k +N 2) space. This runtime is only a logarithmic factor larger than the standard nongeodesic Fréchet algorithm [4]. 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 P. 1.
Walking Your Dog in the Woods in Polynomial Time
"... Given two input curves, the Fréchet distance, sometimes called the dogleash distance, between them is defined as the minimum length of a leash required to connect a dog and its owner as they walk without backtracking along their respective curves from one endpoint to the other. Fréchet distance is ..."
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Cited by 3 (0 self)
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Given two input curves, the Fréchet distance, sometimes called the dogleash distance, between them is defined as the minimum length of a leash required to connect a dog and its owner as they walk without backtracking along their respective curves from one endpoint to the other. Fréchet distance is used as a measure of similarity of the two curves in many different applications [1, 2]. When the two curves are embedded in a general metric space, the distance between two points on the curves (the length of the shortest leash joining them) is not the Euclidean distance but a geodesic distance. For instance, this is the case if the space containing the two curves has obstacle regions which the leash cannot penetrate [3] or if the two curves lie on a terrain
HOMOTOPIC FRÉCHET DISTANCE BETWEEN CURVES OR, WALKING YOUR DOG IN THE WOODS IN POLYNOMIAL TIME
, 2008
"... The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requi ..."
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Cited by 3 (0 self)
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The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles (“trees”). We describe a polynomialtime algorithm to compute the homotopic Fréchet distance between two given polygonal curves in the plane minus a given set of polygonal obstacles.
GEODESIC FRÉCHET DISTANCE WITH POLYGONAL OBSTACLES
"... We present the first algorithm to compute the geodesic Fréchet distance between two polygonal curves in a plane with polygonal obstacles, where distances between points are measured as the length of a shortest path between them. Using shortest path structures that we call dynamic and static spotligh ..."
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We present the first algorithm to compute the geodesic Fréchet distance between two polygonal curves in a plane with polygonal obstacles, where distances between points are measured as the length of a shortest path between them. Using shortest path structures that we call dynamic and static spotlights, we efficiently construct and propagate reachability information through the free space diagram to solve the Fréchet distance. We also show how to construct a novel shortest path map from a line segment source (instead of from a point source). This shortest path map supports geodesic distance queries from any point s ∈ ab to any point t ∈ cd in optimal logarithmic time and permits the shortest path to be reported in outputsensitive fashion.
Shortest Path Problems on a Polyhedral Surface
, 2009
"... 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. ..."
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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.
Morphing Polylines: A Step Towards Continuous Generalization ⋆
"... We study the problem of morphing between two polylines that represent linear geographical features like roads or rivers generalized at two different scales. This problem occurs frequently during continuous zooming in interactive maps. Situations in which generalization operators like typification an ..."
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We study the problem of morphing between two polylines that represent linear geographical features like roads or rivers generalized at two different scales. This problem occurs frequently during continuous zooming in interactive maps. Situations in which generalization operators like typification and simplification replace, for example, a series of consecutive bends by fewer bends are not always handled well by traditional morphing algorithms. We attempt to cope with such cases by modeling the problem as an optimal correspondence problem between characteristic parts of each polyline. A dynamic programming algorithm is presented that solves the matching problem in O(nm) time, where n and m are the respective numbers of characteristic parts of the two polylines. In a case study we demonstrate that the algorithm yields good results when being applied to data from mountain roads, a river and a region boundary at various scales. Key words: continuous generalization, morphing, dynamic programming, line simplification
WALKING YOUR DOG IN THE WOODS IN POLYNOMIAL TIME 1
, 2008
"... Abstract. The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, wh ..."
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Abstract. The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles (“trees”). We describe a polynomialtime 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.
Geodesic Fréchet Distance Inside a Simple Polygon ∗
"... Abstract — We present the first algorithm for the geodesic Fréchet distance between two polygonal curves A and B inside a simple polygon P. If A and B have total complexity N and P has complexity k, then the algorithm runs in O(k+N 2 log kN log N) expected time and O(k+N 2) space. This runtime is qu ..."
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Abstract — We present the first algorithm for the geodesic Fréchet distance between two polygonal curves A and B inside a simple polygon P. If A and B have total complexity N and P has complexity k, then the algorithm runs in O(k+N 2 log kN log N) expected time and O(k+N 2) space. This runtime is quite good as it is only a logarithmic factor larger than the nongeodesic Fréchet algorithm [2]. We also unveil an alluring alternative to parametric search that applies to both the nongeodesic and geodesic Fréchet distance algorithms. This randomized approach is based on a variant of redblue intersections and is appealing due to its elegance and practical efficiency when compared to parametric search.
Parametric Search Package: Tutorial
, 2005
"... This document is the tutorial of the CGAL extension package for parametric search. It gives an overview of the parametricsearch framework, and explains how to use it by describing implementations of several applications of parametric search. The full documentation of all classes and functions that ..."
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This document is the tutorial of the CGAL extension package for parametric search. It gives an overview of the parametricsearch framework, and explains how to use it by describing implementations of several applications of parametric search. The full documentation of all classes and functions that are meant to be called from usercode can be found in the reference manual. 1 1
COORDINATING TEAM MOVEMENTS IN DANGEROUS TERRITORY
"... Suppose a twomember team is required to navigate through dangerous territory (a plane with polygonal obstacles) along two preselected polygonal paths. The team could consist of police officers, soldiers, or even robots on the surface of Mars. Team members are allowed to coordinate their movements, ..."
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Suppose a twomember team is required to navigate through dangerous territory (a plane with polygonal obstacles) along two preselected polygonal paths. The team could consist of police officers, soldiers, or even robots on the surface of Mars. Team members are allowed to coordinate their movements, and it is imperative to keep the team close together during the mission so that if an emergency occurs, one team member can quickly run to the other member and offer assistance. We show how the team can navigate through a plane with polygonal obstacles such that the worstcase separation between the two team members is minimized during the mission. As part of this safety exercise, we develop geodesic and minlink algorithms for shortest path maps and the Fréchet distance [2] in a plane with obstacles. We also utilize a correspondence between shortest path maps and the free space diagram to help compute the Fréchet distance. We present the first algorithms for three types of Fréchet distance: (1) geodesic Fréchet distance in a polygonal domain (a plane with polygonal obstacles), (2) minlink Fréchet distance in a simple polygon, and (3) minlink Fréchet distance in a polygonal domain. A novel subquadratic approximation algorithm for the minlink (and traditional) Fréchet distance is also given. We also develop shortest path maps including (4) the first minlink shortest path map in a polygonal domain that supports queries from a continuous set of source points and (5) a geodesic shortest path map between two line segments in a polygonal domain. The latter structure is related to work by Chiang and Mitchell [8] but uses a new approach based on shortest path structures that we call dynamic and static spotlights.