UNC Chapel Hill M. C. Lin Computing Voronoi Diagram For each site p i, compute the common inter- section of the half-planes h(p i, p j ) for i  j, using linear programming: O ( n 2 log n ) The worst case time bound is O ( n log n ) => Many, but we’ll describe a algorithm based on plane sweep (Fortune’s algorithm)

UNC Chapel Hill M. C. Lin Plane-Sweep Algorithm Maintaining the status of the sweep line l : the part of Vor(P) above l depends not only on the sites above l but also sites below l Solution: Maintaining the information about the part of Voronoi diagram of the sites above l that cannot be changed by the sites below l. We’ll call this closed half-space l +. Observation: the locus of points that are closer to some (any) site p i  l + than to l is bounded by a parabola (parabolic arcs or the beach line).

UNC Chapel Hill M. C. Lin Beach Line The beach line is the function that -- for each x- coordinate-- passes through the lowest point of all parabolas. The breakpoints between different parabolic arcs forming the beach line lie on the edges of the Voronoi diagram; they exactly trace out the Voronoi diagram as l moves from top to bottom. The combinatorial structure of a beach line changes when a new parabolic arc appears on it, and when a parabolic arc shrinks to a point and disappears.

UNC Chapel Hill M. C. Lin Site Events The only way in which a new arc can appear on the beach line is through a site event, where a new site is encountered by l. At a site event, 2 new breakpoints appear, which start tracing out edges. In fact, the they coincide first, then move in opposite directions to trace out the same edge. The growing edge, initially not connected to the rest of the diagram, will eventually become connected to the rest. Each site encountered gives rise to 1 new arc and splits at most one existing one into two.

UNC Chapel Hill M. C. Lin Circle Events The only way an existing arc can disappear from the beach line is through a circle event, where the sweep line reaches the lowest point of a circle through 3 sites defining 3 consecutive arcs on the beach line. Every Voronoi vertex is defined by means of a circle event.

UNC Chapel Hill M. C. Lin Data Structures Store Voronoi diagrams using doubly-connected edge list. Bound the scene using a bounding box large enough to contain all vertices. Beach line is represented by a binary search tree T as the status structure. Its leaves correspond to the arcs of the beach line. A breakpoint is stored at an internal node by an ordered pair of sites. In T, we also store pointers to other 2 data structures. Every internal node has pointer to a half-edge in the Voronoi diagram being traced out by the breakpoint it represents. The event Q is implemented as a priority queue, where the priority is the y -coordinate.

UNC Chapel Hill M. C. Lin VoronoiDiagram(P) Input: A set P := {p 1, p 2, …, p n } of point sites in the plane. Output: The Voronoi diagram Vor(P) given inside a bounding box in a doubly-connected edge list structure. 1. Initialize the event queue Q 2. while Q is not empty 3. do Consider the event with the largest y -coordinate in Q 4. if the event is a site event, occurring at site p i 5. then HandleSiteEvent( p i ) 6. else HandleCircleEvent( p l ), where p l is the lowest point of the circle causing the event 7. Remove the event from Q

UNC Chapel Hill M. C. Lin VoronoiDiagram(P) 8. The internal nodes still present in T correspond to the half-infinite edges of the Voronoi diagram. Compute a bounding box that contains all vertices of the Voronoi diagram in its interior, and attach the half-infinite edges to the bounding box by updating the doubly-connected edge list appropriately. 9. Traverse the half-edges of the doubly-connected edge list to add the cell records and the pointers to and from them.

UNC Chapel Hill M. C. Lin HandleSiteEvent(p i ) 1. Search in T for the arc  vertically above p i and delete all circle events involving  from Q. 2. Replace the leaf of T that represents  with a subtree having three leaves. The middle leaf stores the new site p i and the other two leaves store the site p j that was originally stored with . Store the tuples and representing the new breakpoints at the two new internal nodes. Perform rebalancing operations on T if needed. 3. Create new records in the Voronoi diagram structure for the two half-edges separating V(p i ) and V(p j ), which will be traced out by the two new breakpoints. 4. Check the triples of consecutive arcs involving one of the three new arcs. Insert the corresponding circle event only if circle intersects sweep line & the event isn’t present yet in Q.

UNC Chapel Hill M. C. Lin HandleCircleEvent(p i ) 1. Search in T for the arc  vertically above p l that is about to disappear, and delete all circle events that involve  from Q 2. Delete the leave that represents  from T. Update the tuples representing the breakpoints at the internal nodes. Perform rebalancing operations on T if needed. 3. Add the center of the circle causing the event as a vertex record in the Voronoi diagram structure and create two half- edge records corresponding to the new breakpoint of the Voronoi diagram. Set the pointers between them appropriately. 4. Check the new triples of consective arcs that arise because of the disappearance of . Insert the corresponding circle event into Q only if the circle intersects the sweep line & circle event isn’t present yet in Q.

UNC Chapel Hill M. C. Lin Algorithm Analysis Degeneracies can be handled –2 or more events on a common horizontal line –Event points coincide, e.g. 4 or more co-circular sites –A site coincides with an event The Voronoi diagram of a set of n point sites in the plane can be computed with a sweep line algorithm in O(n log n) time using O(n) storage.