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Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of t ..."
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Cited by 149 (13 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
Voronoi Diagrams
 Handbook of Computational Geometry
"... Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such t ..."
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Cited by 143 (20 self)
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Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.
Approximating Weighted Shortest Paths on Polyhedral Surfaces
 In 6th Annual Video Review of Computational Geometry, Proc. 13th ACM Symp. Computational Geometry
, 1996
"... Consider a simple polyhedron P, possibly nonconvex, composed of n triangular regions (faces), each assigned a positive weight indicating the cost of travel in that region. We present and experimentally study several algorithms to compute an approximate weighted geodesic shortest path, ß 0 (s; t) ..."
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Cited by 47 (5 self)
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Consider a simple polyhedron P, possibly nonconvex, composed of n triangular regions (faces), each assigned a positive weight indicating the cost of travel in that region. We present and experimentally study several algorithms to compute an approximate weighted geodesic shortest path, ß 0 (s; t), between two points s and t on the surface of P. Our algorithms are simple, practical, less prone to numerical problems, adaptable to a wide spectrum of weight functions, and use only elementary data structures. An additional feature of our algorithms is that execution time and space utilization can be traded off for accuracy; likewise, a sequence of approximate shortest paths for a given pair of points can be computed with increasing accuracy (and execution time) if desired. Dynamic changes to the polyhedron (removal, insertions of vertices or faces) are easily handled. The key step in these algorithms is the construction of a graph by introducing Steiner points on the edges of the given p...
Using interpolation to improve path planning: The Field D* algorithm
 Journal of Field Robotics
, 2006
"... ..."
A New Algorithm for Computing Shortest Paths in Weighted Planar Subdivisions (Extended Abstract)
 In Proc. 13th Annu. ACM Sympos. Comput. Geom
, 1997
"... ) Cristian S. Mata Joseph S. B. Mitchell y Abstract We present a practical new algorithm for the problem of computing lowcost paths in a weighted planar subdivision or on a weighted polyhedral surface. The algorithm is based on constructing a relatively sparse graph, a "pathnet", that links se ..."
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Cited by 33 (3 self)
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) Cristian S. Mata Joseph S. B. Mitchell y Abstract We present a practical new algorithm for the problem of computing lowcost paths in a weighted planar subdivision or on a weighted polyhedral surface. The algorithm is based on constructing a relatively sparse graph, a "pathnet", that links selected pairs of subdivision vertices with locally optimal paths. The pathnet can be searched for paths that are provably close to optimal and approach optimal, as one varies the parameter that controls the sparsity of the pathnet. We analyze our algorithm both analytically and experimentally. We report on the results of a set of experiments comparing the new algorithm with other standard methods. 1 Introduction For a given weight function, F : ! 2 ! !, the weighted length of an st path ß in the plane is the path integral, R ß F (x; y)doe, of the weight function along the path ß, linking the start s to the goal t. The weighted region metric associated with F defines the distance dF (...
Approximating Shortest Paths on Weighted Polyhedral Surfaces
"... Shortest path problems are among the... In this paper we propose several simple and practical algorithms (schemes) to compute an approximated weighted shortest path Π'(s, t) points s and t on the surface of a polyhedron P. ..."
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Cited by 29 (6 self)
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Shortest path problems are among the... In this paper we propose several simple and practical algorithms (schemes) to compute an approximated weighted shortest path Π'(s, t) points s and t on the surface of a polyhedron P.
Rectilinear Paths among Rectilinear Obstacles
 Discrete Applied Mathematics
, 1996
"... Given a set of obstacles and two distinguished points in the plane the problem of finding a collision free path subject to a certain optimization function is a fundamental problem that arises in many fields, such as motion planning in robotics, wire routing in VLSI and logistics in operations resear ..."
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Cited by 23 (3 self)
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Given a set of obstacles and two distinguished points in the plane the problem of finding a collision free path subject to a certain optimization function is a fundamental problem that arises in many fields, such as motion planning in robotics, wire routing in VLSI and logistics in operations research. In this survey we emphasize its applications to VLSI design and limit ourselves to the rectilinear domain in which the goal path to be computed and the underlying obstacles are all rectilinearly oriented, i.e., the segments are either horizontal or vertical. We consider different routing environments, and various optimization criteria pertaining to VLSI design, and provide a survey of results that have been developed in the past, present current results and give open problems for future research. 1 Introduction Given a set of obstacles and two distinguished points in the plane, the problem of finding a collision free path subject to a certain optimization function is a fundamental probl...
Applications of Computational Geometry to Geographic Information Systems
"... Contents 1 Introduction 2 2 Map Data Modeling 4 2.1 TwoDimensional Spatial Entities and Relationships . . . . . . . . . . . . . . . . . . . . . 4 2.2 Raster and Vector Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Subdivisions as Cell Complexes . . . . . . . ..."
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Cited by 21 (1 self)
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Contents 1 Introduction 2 2 Map Data Modeling 4 2.1 TwoDimensional Spatial Entities and Relationships . . . . . . . . . . . . . . . . . . . . . 4 2.2 Raster and Vector Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Subdivisions as Cell Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Topological Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Multiresolution Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Map data processing 8 3.1 Spatial Queries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Map Overlay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Geometric Problems in Map Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Map Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . .
Theta*: AnyAngle Path Planning on Grids
"... Grids with blocked and unblocked cells are often used to represent terrain in computer games and robotics. However, paths formed by grid edges can be suboptimal and unrealistic looking, since the possible headings are artificially constrained. We present Theta*, a variant of A*, that propagates inf ..."
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Cited by 19 (2 self)
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Grids with blocked and unblocked cells are often used to represent terrain in computer games and robotics. However, paths formed by grid edges can be suboptimal and unrealistic looking, since the possible headings are artificially constrained. We present Theta*, a variant of A*, that propagates information along grid edges without constraining the paths to grid edges. Theta * is simple, fast and finds short and realistic looking paths. We compare Theta * against both Field D*, the only other variant of A * that propagates information along grid edges without constraining the paths to grid edges, and A * with postsmoothed paths. Although neither path planning method is guaranteed to find shortest paths, we show experimentally that Theta * finds shorter and more realistic looking paths than either of these existing techniques.
An Efficient Approximation Algorithm for Weighted Region Shortest Path Problem
, 2000
"... In this paper we present an approximation algorithm for solving the following optimal motion planning problem: Given a planar space composed of triangular regions, each of which is associated with a positive unit weight, and two points s and t, find a path from s to t with the minimum weight, where ..."
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Cited by 17 (5 self)
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In this paper we present an approximation algorithm for solving the following optimal motion planning problem: Given a planar space composed of triangular regions, each of which is associated with a positive unit weight, and two points s and t, find a path from s to t with the minimum weight, where the weight of a path is defined to be the weighted sum of the lengths of the subpaths within each region.