CMPUT 498 Voronoi Diagrams

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CMPUT 498Voronoi Diagrams

• Lecturer Sherif Ghali
• Department of Computing Science
• University of Alberta

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Trading Areas

• Assumptions
• Total cost cost of item transportation cost
• Transportation cost k Euclidean distance
• Consumers pursue cheapest option

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Definitions

• p and q are two sites
• P set of sites
• Vor(P) Voronoi diagram of P
• V(pi) Voronoi cell of site pi
• In Vor(p,q)
• h(p,q) halfplane that contains p
• h(q,p) halfplane that contains q

q
p
h(q,p)
h(p,q)
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A Voronoi cell is the intersection of halfplanes
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Characterisation of Vor(P)

• What is P?
• collinear set of points
• Vor(P) n-1 parallel lines
• otherwise
• Edges of Vor(P) are either segments or halflines
• Vor(P) is connected

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Upper bound on complexity of Vor(P)

• Argument complexity of one cell is linear in n
  (where P n) ? complexity of Vor(P) is
  quadratic in n
• Is this tight?

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Tight upper bound on complexity of Vor(P)

• n Number of sites
• ne Number of Voronoi edges ? 2n-5
• nv Number of Voronoi vertices ? 3n-6

• Argument
• Eulers formula

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?

• Every edge has degree 2
• Every vertex has degree at least 3
• ?

?
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Computing the Voronoi diagram

• for each site
• complexity O(n log n) / Voronoi cell
• O(n2 log n) / Voronoi diagram
• The sweep algorithm (Fortunes) takes O(n log n)
  time,
• which is optimal

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Plane sweep?

• Apply the plane sweep paradigm
• Maintain the Voronoi regions above the sweep line
  for the sites above the line
• But that doesnt work
• A not-yet-crossed site induces a cell to appear
  above the line
• A Voronoi vertex is crossed by the sweep line
  before the corresponding site is reached
• Solution
• Find an answer to
• What is the shape of the portion above the sweep
  line for which we know for sure the Voronoi
  regions (for a given location of the sweep line)

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Plane sweep?
we know the Voronoi cells here
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sites already crossed
we cant yet know anything about the Voronoi
cells here
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sites yet to be crossed
we know nothing about the Voronoi cells here
Question What is the shape of and ?
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2
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Plane sweep?
we couldnt know about this vertex (yet)
Question What is the shape of the region we know
about?
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Plane sweep?

• We have to take care of the worst-case scenario
• a site could be encountered anywhere
• what is the shape of
• and ?

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The beach line

• Observations
• The beach line is x-monotone
• The beach line breakpoints lie on edges of Vor(P)

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Event 1 - Site event
Beach line crosses a site
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Event 2 - Circle event
Existing arc shrinks to a point and disappears
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Detecting circle events

• Observations
• Circle events arise from consecutive triples of
  arcs
• Not each consecutive triple of arcs results in
  circle event
• A consecutive triple of arcs that results in a
  circle event does not necessarily actually result
  in one (false alarm)

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Recall DCEL
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Data structures for sweeping

• Representing the Voronoi diagram
• Winged-edge data structure D
• a.k.a. Doubly connected edge list
• Representing the beach line
• Balanced binary tree T
• Internal nodes represent beach
• line breakpoints
• Representing the events
• Priority queue Q

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Algorithm
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Reference
Chap. 7