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A new approach for the geodesic Voronoi diagram of points in a simple polygon and other restricted polygonal domains
, 1998
"... We introduce a new method for computing the geodesic Voronoi diagram of point sites in a simple polygon and other restricted polygonal domains. Our method combines a sweep of the polygonal domain with the merging step of a usual divideandconquer algorithm. The time complexity is O((n + k) log(n + ..."
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We introduce a new method for computing the geodesic Voronoi diagram of point sites in a simple polygon and other restricted polygonal domains. Our method combines a sweep of the polygonal domain with the merging step of a usual divideandconquer algorithm. The time complexity is O((n + k) log(n + k)) where n is the number of vertices and k is the number of points, improving upon previously known bounds. Space is O(n+ k). Other polygonal domains where our method is applicable include (among others) a polygonal domain of parallel disjoint line segments and a polygonal domain of rectangles in the L 1 metric. Key Words: Geodesic Voronoi diagrams, Shortest paths, Polygon triangulation, Topological plane sweep, Computational Geometry. 1 Introduction The geodesic Voronoi diagram of a set S of k point sites in a polygonal domain P is the partitioning of P into k cells, one to each site, called (geodesic) Voronoi cells. A point x 2 P belongs to the geodesic Voronoi cell of a site s if and o...
Geometric Data Structures and Their Selected Applications
"... Abstract—Finding the shortest path between two positions is a fundamental problem in transportation, routing, and communications applications. In robot motion planning, the robot should pass around the obstacles touching none of them, i.e. the goal is to find a collisionfree path from a starting to ..."
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Abstract—Finding the shortest path between two positions is a fundamental problem in transportation, routing, and communications applications. In robot motion planning, the robot should pass around the obstacles touching none of them, i.e. the goal is to find a collisionfree path from a starting to a target position. This task has many specific formulations depending on the shape of obstacles, allowable directions of movements, knowledge of the scene, etc. Research of path planning has yielded many fundamentally different approaches to its solution, mainly based on various decomposition and roadmap methods. In this paper, we show a possible use of visibility graphs in pointtopoint motion planning in the Euclidean plane and an alternative approach using Voronoi diagrams that decreases the probability of collisions with obstacles. The second application area, investigated here, is focused on problems of finding minimal networks connecting a set of given points in the plane using either only straight connections between pairs of points (minimum spanning tree) or allowing the addition of auxiliary points to the set to obtain shorter spanning networks (minimum Steiner tree). Keywords—motion planning, spanning tree, Steiner tree, Delaunay triangulation, Voronoi diagram. I.
Voronoi Diagrams and their Applications
"... Abstract: The Voronoi diagram is a fundamental structure in computational geometry and arises naturally in various branches of science. This paper surveys basic properties of the Voronoi diagram, algorithms for its construction, and typical applications of the Voronoi diagram and its geometric dual, ..."
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Abstract: The Voronoi diagram is a fundamental structure in computational geometry and arises naturally in various branches of science. This paper surveys basic properties of the Voronoi diagram, algorithms for its construction, and typical applications of the Voronoi diagram and its geometric dual, the Delaunay triangulation.
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"... Abstract—Finding the shortest path between two positions is a fundamental problem in transportation, routing, and communications applications. In robot motion planning, the robot should pass around the obstacles touching none of them, i.e. the goal is to find a collisionfree path from a starting to ..."
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Abstract—Finding the shortest path between two positions is a fundamental problem in transportation, routing, and communications applications. In robot motion planning, the robot should pass around the obstacles touching none of them, i.e. the goal is to find a collisionfree path from a starting to a target position. This task has many specific formulations depending on the shape of obstacles, allowable directions of movements, knowledge of the scene, etc. Research of path planning has yielded many fundamentally different approaches to its solution, mainly based on various decomposition and roadmap methods. In this paper, we show a possible use of visibility graphs in pointtopoint motion planning in the Euclidean plane and an alternative approach using Voronoi diagrams that decreases the probability of collisions with obstacles. The second application area, investigated here, is focused on problems of finding minimal networks connecting a set of given points in the plane using either only straight connections between pairs of points (minimum spanning tree) or allowing the addition of auxiliary points to the set to obtain shorter spanning networks (minimum Steiner tree). Keywords—motion planning, spanning tree, Steiner tree, Delaunay triangulation, Voronoi diagram.
Finding Rectilinear Least Cost Paths in the Presence of Convex Polygonal Congested Regions
, 2004
"... This paper considers the problem of finding the least cost rectilinear distance path in the presence of convex polygonal congested regions. An upper bound for the number of entry/exit points for a congested region is obtained. Based on this key finding, we demonstrate that there are a finite, though ..."
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This paper considers the problem of finding the least cost rectilinear distance path in the presence of convex polygonal congested regions. An upper bound for the number of entry/exit points for a congested region is obtained. Based on this key finding, we demonstrate that there are a finite, though exponential number of potential least cost paths between a specified pair of origindestination points. A “memorybased probing algorithm ” is proposed for the problem and computational experience for various problem instances is reported. Two heuristics are also proposed for larger problem instances.
A New Approach for the Geodesic Voronoi Diagram of Points in a Simple Polygon and Other Restricted
"... Abstract. We introduce a new method for computing the geodesic Voronoi diagram of point sites in a simple polygon and other restricted polygonal domains. Our method combines a sweep of the polygonal domain with the merging step of a usual divideandconquer algorithm. The time complexity is O..nCk / ..."
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Abstract. We introduce a new method for computing the geodesic Voronoi diagram of point sites in a simple polygon and other restricted polygonal domains. Our method combines a sweep of the polygonal domain with the merging step of a usual divideandconquer algorithm. The time complexity is O..nCk / log.nCk//where n is the number of vertices and k is the number of points, improving upon previously known bounds. Space is O.n C k/. Other polygonal domains where our method is applicable include (among others) a polygonal domain of parallel disjoint line segments and a polygonal domain of rectangles in the L1 metric.