Computing Voronoi Diagram

1
Computing Voronoi Diagram

• For each site pi, compute the common
  inter-section of the half-planes h(pi , pj) for
  i? j, using linear programming O(n2 log n)
• The worst case time bound is O(n log n)
• gt Many, but well describe a algorithm based on
  plane sweep (Fortunes algorithm)

2
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. Well
  call this closed half-space l.
• Observation the locus of points that are closer
  to some (any) site pi? l than to l is bounded
  by a parabola (parabolic arcs or the beach line).

3
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.

4
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.

5
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.

6
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 ltpi, pjgt. 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.

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VoronoiDiagram(P)

• Input A set P p1, p2, , pn 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 pi
• 5. then HandleSiteEvent(pi)
• 6. else HandleCircleEvent(pl),
  where pl is the lowest
• point of the circle
  causing the event
• 7. Remove the event from Q

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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.

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HandleSiteEvent(pi)

• 1. Search in T for the arc ? vertically above pi
  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 pi and the other two leaves
  store the site pj that was originally stored
  with ?. Store the tuples lt pj , pi gt and lt pi ,
  pj gt 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(pi)
  and V(pj) , 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 isnt present
  yet in Q.

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HandleCircleEvent(pi)

• 1. Search in T for the arc ? vertically above
  pl 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
  isnt present yet in Q.

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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.