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Voronoi Diagrams Fortunes Algorithm and Applications Kevin Wittmer Mentor: Rob Maschal DRP Summer 2016 Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 1 / 15 Introduction What is Computational Geometry? The

  1. Voronoi Diagrams Fortune’s Algorithm and Applications Kevin Wittmer Mentor: Rob Maschal DRP Summer 2016 Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 1 / 15

  2. Introduction What is Computational Geometry? The systematic study of algorithms and data structures to solve geometric problems Focus on exact algorithms that are asymptotically fast Data sets can be incredibly large Input sizes grow exponentially as the number of dimensions increases Gain new insights from reformulating a problem in geometric terms Querying a database: determine set of points contained in n -dimensional cube Many other application areas: robotics, computer graphics... Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 2 / 15

  3. Introduction A Geometric Problem Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 3 / 15

  4. Introduction A Geometric Problem Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 4 / 15

  5. Introduction Definitions Sites: Distinct central places or points of interest Denote the set of n sites by P := { p 1 , p 2 , . . . , p n } Voronoi diagram: The subdivision of the plane into n cells such that a point q lies in the cell corresponding to p i iff dist ( q , p i ) < dist ( q , p j ) for each p j ∈ P with j � = i Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 5 / 15

  6. Introduction A Na¨ ıve Algorithm The set of all points equidistant from two sites p i and p j is the perpendicular bisector of the line segment connecting p i and p j Each Voronoi cell is the intersection of n − 1 half-planes induced by the perpendicular bisectors between p i and all other sites in P Unfortunately, computing the diagram this way runs in O ( n 2 log n ) time... Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 6 / 15

  7. Fortune’s Algorithm Plane Sweep Paradigm Imagine sweeping a horizontal line from top to bottom over the plane Everything above the sweep line has been computed, while everything below has not Maintain information about the intersection of our structure with the sweep line Event points: Locations where this information changes Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 7 / 15

  8. Fortune’s Algorithm Beach Line The Voronoi diagram above the sweep line is affected by event points below the line We know the nearest site of a point q if q lies at least as close to a site as it does to the sweep line The set of points equidistant from a point and a line defines a parabola Beach line: The sequence of parabolic arcs closest to the sweep line Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 8 / 15

  9. Fortune’s Algorithm Break Points The break points of the beach line trace out the edges of the Voronoi diagram as the sweep line moves Source: Wikipedia Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 9 / 15

  10. Fortune’s Algorithm Event Points Site event: Correspond to adding a new edge to the diagram These are our predetermined events Circle event: Correspond to adding a new vertex to the diagram Dynamically added as we process site events Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 10 / 15

  11. Fortune’s Algorithm Fortune’s Algorithm Algorithm VoronoiDiagram ( P ) Input . A set P of point sites in the plane. Output . The Voronoi diagram given inside a bounding box. 1. Initialize an event queue Q with all site events. 2. while Q is not empty 3. do Remove the event with the largest y -coordinate from Q . 4. if the event is a site event 5. then HandleSiteEvent ( p i ) 6. else HandleCircleEvent ( γ ) 7. Compute a bounding box that contains all vertices of the Voronoi diagram in its interior Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 11 / 15

  12. Fortune’s Algorithm Handing Site Events A new parabola is added to the beach line New break points begin to trace out the same new edge Check the new triple of consecutive arcs for any potential circle events Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 12 / 15

  13. Fortune’s Algorithm Handling Circle Events Store a circle event as the lowest point on an empty circle containing 3 or more sites When the sweep line reaches a circle event, a parabola disappears from the beach line The center of this circle is added as a vertex of the Voronoi diagram Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 13 / 15

  14. Fortune’s Algorithm Conclusion Theorem The Voronoi diagram of n point sites can be computed in O ( n log n ) time using Fortune’s algorithm Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 14 / 15

  15. Fortune’s Algorithm References de Berg, Cheong, van Kreveld, Overmars Computational Geometry: Algorithms and Applications Springer, 2008. David M. Mount CMSC754 Computational Geometry Lecture Notes p. 67 - 74 Kevin Wittmer, Mentor: Rob Maschal Voronoi Diagrams DRP Summer 2016 15 / 15

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