Voronoi diagrams
cs.nyu.edu http://newtextiles.media.mit.edu/?p=1906
www.grasshopper3d.com www.sonycsl.co.jp
Motivation
• Solves so-called post office problem
– The goal is to plan a placement of new post
office/supermarket/…
– How many people will find the new supermarket
attractive?
– Let’s consider the following simplified requirements:
• The price of all goods is the same in all supermarkets
• Total cost = cost for the goods + travelling cost to the
supermarket
• Travelling cost to the supermarket = Euclidean distance to
the supermarket x fixed cost per distance unit
• The goal of the customer is to minimize the costs
– Consequence: the customers are using the service of
the nearest supermarket
Motivation
• This model induces the division of the space
to subregions according to the location of the
supermarkets – each subregion contains all
points being closer to the given supermarket
than to any other supermarket
• Such a space division is called Voronoi
diagram
Euclidean distance
• Euclidean distance between two points P =
[px, py] and Q = [qx,qy] is defined as
|PQ| = dist(P,Q) = √(𝑝 𝑥 − 𝑞 𝑥)2
+(𝑝 𝑦 − 𝑞 𝑦)2
VD definition
• Let P = {P1, …, Pn} be a set of n different points
in space, called generating points.
• Voronoi diagram of P is the division to n cells
connected with points Pi in that way that an
arbitrary point Q lies in the cell of Pi only when
|QPi| < |QPj| for all Pj ∈ P, j ≠ i
VD definition
• Lets denote the Voronoi diagram of P as
Vor(P)
• A cell of Vor(P), belonging to point Pi, is
denoted as ƴ(Pi) and we call it a Voronoi cell
of point Pi
VD examples
VD properties
• If all points in P are colinear, Vor(P) consists of
n – 1 parallel lines
VD properties
• If the points are not colinear, Vor(P) is
continuous and its edges are line segments or
half-segments
VD properties
• Voronoi cell ƴ(Pi) is unlimited only when the
point Pi belongs to an edge of the convex hull
of P
VD properties
• If P contains 4 or more vertices lying on one
circle, there is a Voronoi vertex formed by the
intersection of Voronoi edges whose number
corresponds to the number of points on that
circle – we call it a degenerated Voronoi
diagram
Algorithms for VD construction
• Generally, creating VD for n points lies in
O(n log n)
• Algorithms:
– Naïve approach
– Incremental algorithm
– Divide and conquer
– Sweep line (Fortune’s algorithm)
…
Naïve approach
• Each region ƴ(Pi) of Voronoi diagram is
generated as an intersection between
halfplanes h(Pi, Pj), for all j ≠ i.
• The complexity of finding one region = O(n log
n)
• Total complexity = O(n2 log n)
Incremental algorithm
1. For all points P:
1. In the current VD, we localize the corresponding
Voronoi cell containing Pi+1 → ƴ(Pi1)
2. We create the axis of line segment Pi+1Pi1
3. We determine the intersections of this axis of
line segment Pi+1Pi1 with the boundary of ƴ(Pi1)
4. We select one of the intersections which
determines the Voronoi cell with which our
algorithm will continue in the next step → ƴ(Pi2)
Incremental algorithm
5. We create the axis of the line segment Pi+1Pi2 and
its intersections with the boundary of ƴ(Pi2). We
select an intersection not lying on the common
edge of ƴ(Pi1) and ƴ(Pi2) and we continue
6. We repeat step 5, until we reach the second
intersection of the axis of line intersection Pi+1Pi1
with the boundary of ƴ(Pi1)
7. We remove the edges inside
the newly created Voronoi cell
Incremental algorithm
• Complexity O(n2), in special cases even O(n)
http://www.personal.kent.edu/~rmuhamma/Compgeometry/MyCG/Voronoi
/Incremental2/incremental2.htm
Divide and conquer
• The input set is recursively divided to two
subsets until we reach the set of three points
for which we construct the VD easily
• The crucial part is the „backtracking step“,
where the individual solutions have to be
merged to one VD
• Complexity O(n log n)
Divide and conquer
• We sort the input points and divide them
vertically to two subsets R and B of
approximately the same size
Divide and conquer
• We calculate recursively Vor(R) and Vor(B)
Divide and conquer
• We determine so called separating chain
Divide and conquer
• We remove the part of Vor(R) lying on the
right side from the separating chain and the
par of Vor(B) lying on the left side from the
separating chain
Divide and conquer
• Defining the separating chain:
– First, we find two convex hulls …
Divide and conquer
… and the upper and lower tangent and their
perpendicular half-lines
Divide and conquer
• We start from one of these half-lines and
continue with the following procedure, until we
reach the second half-line:
– Always when there starts an edge e ∈ b(R, B) for
which e ⊂ bij, pi ∈ R, pj ∈ B:
• Search for the intersection of edge e with VorR(pi)
• Search for the intersection of edge e with VorB(pj)
• Select one of these intersections
• Determine pk corresponding to a new starting region
• Replace pi or pj (according to the selected point) by new pk
• Repeat this step with the new edge
Sweep line (Fortune’s algorithm)
• Algorithm uses so-called „sweep line“ and
„beach line“, both of them traversing the
space containing the input points
• The sweep line can be horizontal or vertical,
heading from top to bottom or vice versa
• Invariant of the algorithm = for the input
points already traversed by the sweep line we
have already a correct VD constructed, the
rest of the points was not processed yet
Sweep line (Fortune’s algorithm)
• „Beach line“ is not in fact a line but a curve
above the sweep line, consisting of parts of
parabolas
• A set of all points being closer to some of the
points above the sweep line than to the
sweep line itself is delineated by parabolic
arcs – their connection forms the beach line
Sweep line (Fortune’s algorithm)
Sweep line (Fortune’s algorithm)
• The intersection of arcs lying on the beach line
lie on the edges of the VD. With moving the
sweep line, these intersections create the
edges of VD Vor(P)
• The algorithm contains the following two
operations:
Sweep line (Fortune’s algorithm)
• Site event – a new generating point emerges
on the beach line, we have to add it to the VD
structure
• Circle event – when one of the parabolic arcs
is terminated
Site event
• This event generates a new parabolic arc on the
beach line and its intersection with the current
beach line starts to create a new VD edge
Site event
• Beach line consists of maximally 2n – 1
parabolic arcs, because each generating point
creates one parabola and divides maximally
one existing parabolic arc to two parts
Circle event
• When some of the parabolic arcs is
terminated
• This happens when three parabolas generated
by points Pi, Pj, Pk all intersect in point Q –
then this point Q forms the new Voronoi
vertex
Circle event
Sweep line (Fortune’s algorithm)
• More information, details for implementation:
– http://blog.ivank.net/fortunes-algorithm-and-
implementation.html
Weighted Voronoi diagrams
• One of possible generalizations of VD, when
each generating point is assigned to a weight.
This weight influences the size and shape of
the VD cell.
• Let’s assign weight wi ∈ R to point Pi. Then we
define the corresponding metrics as
distWVD(P, Q) = dist(P, Q) – wi
where dist can be an arbitrary metrics
Weighted Voronoi diagrams
• When increasing the weight of a given point
the corresponding VD cell is increasing which
correspond to the given metric
• When the dist metric is the Euclidean
distance, then distWVD(P, Pi) can be interpreted
as the distance of point P from a circle with
center in Pi and radius wi
• Voronoi edges are in this case parts of
hyperbolas
Weighted Voronoi diagrams
www.cgal.org
www.sciencedirect.com
d.hatena.ne.jp
Assignment
• Use the already constructed Delaunay
triangulation for the construction of Voronoi
diagram
• Visualize it
fr.wikipedia.org