Fortune’s algorithm is a sweep line algorithm for generating a Voronoi diagram from a set of points in a plane using O(n log n) time and O(n) space. A sweepline algorithm for Voronoi Diagrams. 1. A sweepline algorithm for Voronoi Diagrams Steven Fortune Algorithmica, By: Himanshi. Computing Voronoi Diagrams: There are a number of algorithms for computing Voronoi Behind the sweep line you have constructed the Voronoi diagram.
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As the sweep line progresses, the vertices of the beach line, at which two parabolas cross, trace out the edges of the Voronoi diagram.
Views Read Edit View history. These events include the addition doagrams another parabola to the beach line when the sweep line crosses another input point and the removal of a curve from the beach line when the sweep line becomes tangent to a circle through some three input points whose parabolas form consecutive segments of the beach line. A sweepline algorithm for Voronoi diagrams.
As Fortune describes in ref. The algorithm maintains as data structures a binary search tree describing the combinatorial structure of the beach line, and a priority queue listing potential future events that could change the beach line structure. Weighted sites may be used to control the algroithm of the Voronoi cells when using Voronoi diagrams to construct treemaps.
Fortune’s algorithm – Wikipedia
Computing the Voronoi Diagram: The beach line is not a straight line, but a complicated, piecewise curve to the left of the sweep line, composed of pieces of parabolas ; it divides the portion of the plane within which the Voronoi diagram can be known, regardless of what other points might be right of the sweep line, from the rest of the plane.
The algorithm maintains both a sweep line and a beach linewhich both move through the plane as the algorithm progresses.
This page was last edited on 27 Decemberat As there are O n events to process each being associated with some feature of the Voronoi diagram and O log n time to process an event each consisting of a constant number of binary search tree and priority queue operations the total time is O n log n.
Pseudocode description of the algorithm. The sweep line is a straight line, which we may by convention assume to be vertical and moving left to right across the plane. From Wikipedia, the free encyclopedia. The algorithm itself then consists of repeatedly removing the next event from the priority queue, finding the changes the event causes in the beach line, and updating the data structures. For each point left of the sweep line, one can define a parabola of points equidistant from that point and from the sweep line; the beach line is the boundary of the union of these parabolas.
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At any time during the algorithm, the input points left of the sweep line will have been incorporated into the Voronoi diagram, while the points right of the sweep line will not have been considered yet. Proceedings of the second annual symposium on Computational geometry. Mathematically, this means each parabola is formed by using the sweep line as the directrix and the input point as the focus.
Each such event may be prioritized by the x -coordinate of the sweep line at the point the event occurs. Articles with example pseudocode. Retrieved from ” https: