Voronoi Diagrams
Properties
R. Inkulu
http://www.iitg.ac.in/rinkulu/
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Problem Description
Preprocess a set S of n points, known as sites, to find a closest site to a query
point q.
Assume all the sites are in general position, and n ≥ 3.
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Running Example
(a)
(c)
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(b)
(d)
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Running Example (cont)
(a)
(b)
(c)
(d)
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Voronoi diagram: Definition
• Given a set S of n sites, partition R2 such that each region of that
partition corresponds to loci of points that are closer to a unique site in S.
Boundaries that define such a partition together are termed as the Voronoi
diagram of S, denoted with VD(S).
• Vertices and edges of Voronoi diagram are termed as the Voronoi vertices
and Voronoi edges respectively. The partition that corresp. to site si is
termed as the Voronoi polygon of si , and is denoted with VP(si ).
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Voronoi polygons are convex
• VP(si ) =
T
1≤j≤n,i6=j h(si , sj )
• Every Voronoi polygon is convex.
Proof by induction.
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Structure of a Voronoi edge
• The Voronoi edges are either line segments or half-lines.
Proof by contradiction.
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Voronoi diagram is connected
• Voronoi diagram is connected.
Proof by contradiction: a strip bounded with parallel lines must exist.
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Degree of Voronoi vertex
• Exactly three Voronoi edges incident to each Voronoi vertex.
Proof by contradiction (suppose degree > 3): violates general position
assumption of sites.
Proof by contradiction (suppose degree < 3): separator must be a full-line
with no scope for vertex.
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Characterizing a Voronoi vertex
p
• A point p is a Voronoi vertex iff largest empty circle centered at p has
exactly three sites on its boundary.
⇒: Use the degree of Voronoi vertex.
⇐: Voronoi vertex must be on the boundary of three Voronoi polygons.
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Characterizing a Voronoi edge
p
• The bisector b between two sites si , sj defines a Voronoi edge iff there is
a point p on b such that the circle centered at p passing through si and sj
does not contain any other site (including on its boundary).
⇒: direct proof.
⇐: such a p must be on the boundary of a Voronoi polygon but it cannot
be a Voronoi vertex.
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Characterizing a Voronoi edge (cont)
b(s i , s j)
sj
Voronoi edge
v
sk
u
si
• Every nearest neighbor of site si defines a Voronoi edge of V(si ).
Assuming sj is closer to si leaves with a contradition.
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Relation between Voronoi diagram and Convex Hull
x
s1
p
s2
si
s3
• Voronoi polygon V(si ) is unbounded iff si is a point on the boundary of
CH(S).
⇒: proof by contrapositive.
⇐: proof by contrapositive.
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Relation between Voronoi diagram and Delaunay
triangulation
Dual: for every black edge, create a corresp. magneta edge.
• Straight-line dual of the Voronoi diagram is the Delaunay triangulation
of S.
dual of any Voronoi vertex does forms a unique triangle.
interiors of no such two triangles intersect.
every point of CH(S) bleongs to a triangle.
• Voronoi edge not necessarily intersects with its corresp. Delaunay edge.
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Number of Voronoi vertices and edges
• The number of Voronoi vertices are at most 2n − 5; and, the number of
Voronoi edges are at most 3n − 6.
apply Euler’s formula together with Handshaking Lemma.
• Average number of edges defining a Voronoi polygon is at most six.
• The number of triangles in the Delaunay triangulation are at most 2n − 5
and the number of edges are at most 3n − 6.
• Average number of edges incident to a site in Delaunay triangulation is
O(1).
hence, Delaunay Theorem makes the (Delaunay triangulation)
construction process efficient in practice.
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