1 Lecture 8: Voronoi Diagram Computational Geometry Prof. Dr. Th. Ottmann Voronoi Diagrams...

1Lecture 8:Voronoi Diagram

Computational GeometryProf. Dr. Th. Ottmann

Voronoi Diagrams

• Definition

• Characteristics

• Size and Storage

• Construction

• Use



The Voronoi Diagram



Voronoi Regions

Eucledian distance :

dist(p,q) := ²² yyxx qpqp

Let P :={ p1, p2, ...,pn } be a set of n distinct points in a plane.We define the voronoi diagram of P as the subdivision of the planeinto n cells, with the property that a point q lies in the cell correspon-ding to a site pi iff dist(q, pi ) < dist(q, pj ) for each pj P with j i.

We denote the Voronoi diagram of P by Vor(P).The cell that corresponds to a site pi is denotd by V(pi ), called thevoronoi cell of pi.



Example

V(pi) = 1 j n, j i h(pi, pj)

q4

q1

q3

q2

p



Computing the Voronoi DiagramInput: A set of points (sites) Output: A partitioning of the plane into regions of equal nearest neighbors



Animations of the Voronoi diagram

Java Applet zur Animation von Voronoi Diagrammen,Entwickelt von R. Klein u.a., FU Hagen

pi

V(pi)



Characteristics of Voronoi Diagrams(1) Voronoi regions (cells) are bounded by line segments.

Special case :

Collinear points

Theorem : Let P be a set of n points (sites) in the plane. If all the sites are collinear, then Vor(P) consist of n-1 parallel lines and n cells. Otherwise, Vor(P) is a connected graph and its edges are either line segments or half-lines.

e

pi pj

pk

h(pi,pj)

h(pj,pk)

If pi, pj are not collinear with pk, thenh(pi, pj ) and h(pj, pk ) can not be parallel!



Vor(P) is Connected

If Vor(P) is not connected then there would be a Voronoicell V(Pi ) splitting the plane into two halfes. Because Voronoi cells are convex, V(Pi ) would consist of a strip bounded by two parallel full lines, but we know that edges of Voronoi diagram cannot be full lines, hence a contradiction.

Claim: Vor(P) is connected

Proof by contradiction:



Other Characteristics (Assumption: No 4 points are on the circle)

(2) Each vertex (corner) of VD(P) has degree 3

(3) The circle through the three points defining a Vertex of the Voronoi diagram does not contain any further point



(4) Each nearest neighbor of one point defines an edge of the Voronoi region of the point.

(5) The Voronoi region of a point is unbounded iff the point lies exactly on the convex hull of the point set.



Size and Storage

Size of the Voronoi Diagram:

V(p) can have O(n) vertices!



Size of the Voronoi DiagramTheorem: The number of vertices in the Voronoi diagram of a set of n points in the plane is at most 2n-5 and the number of edges is at most 3n-6.Proof: 1. Connect all Half-lines with fictitious point 2. Apply Euler`s formula: v – e + f = 2

For VD(P) + : v = number of vertices of VD(P) + 1e = number of edges of VD(P) f = number of sites of VD(P) = n

Each edge in VD(P) + has exactly two vertices and each vertex of VD(P) + has at least a degree of 3:

sum of the degrees of all vertices of Vor(P) +

= 2 · ( # edges of VD(P) )

3 · ( # vertices of VD(P) + 1 )



Number of vertices of VD(P) = vp

Number of edges of VD(P) = ep

We can apply: (vp + 1) – ep + n = 2

2 ep 3 (vp + 1)

2 ep 3 ( 2 + ep - n)

= 6 + 3ep – 3n

3n – 6 ep

Proof(Continued)



Example



Storage of Voronoi-Diagrams

1

1

3

4

5

6

2

2 3

4

5

• •

•

•

Three Records:

vertex {CoordinatesIncident edge

}; face {

OuterComponentInnerComponents

}; halfedge {

OriginTwinIncidentFaceNextPrev

};

e.g. :

Vertices 1 = {(1,2) | 12}Sites 1 = {15 | [] }Edges 54 = { 4 | 45 | 1 | 43 | 15 }



Computing the Voronoi DiagramInput: A set of points (sites) Output: A partitioning of the plane into regions of equal

nearest neighbors.



Divide and Conquer(Divide)

Input: A set of points (sites)

Output: A partitioning of the plane into regions of equal nearest neighbors.

Divide: Divide the point set into two halves



Divide and Conquer (Conquer)

Conquer: Recursively compute the Voronoi diagrams for the smaller point sets Abort condition: Voronoi diagram of a single point is the entire plane.



Divide and Conquer (Merge)

Merge the diagrams by a (monoton) sequence of edges)



The ResultThe finished Voronoi Diagram

Running time: With n given points is O(n log n)



Geometrical Divide and Conquer

Problem: Determine all intersecting pairs of segments

A

BC

DE

A

BC

DE

S1 S2

S

EC

D

A

B



DAC - Construction of the Voronoi diagramDivide:Divide P by a vertical dividing line T into 2 equal size subsets say P1 and P2. If |P| = 1 completed.Conquer:Compute VD(P1 ) and VD(P2 ) recursively.

Merge: Compute the edge sequence K separating P1 and P2 Cut VD(P1) and VD(P2 ) by means of K starting from VD(P1 ) and VD(P2 ) and K P1

P2

T

Theorem: If K can be computed in time O(n), then the running time of the D&C-algorithm is T(n) = O(n log n)

Proof : T(n) = 2 T(n/2) + O(n), T(1) = O(1)



Computation of K

P1 P2

First edge in K

Last edge in K

4 tangential points P1 P2 Observation: K is y - monotonous

Determines intersection s1 of m with Vor(p1) below l Determines intersection s2 of m with Vor(p2) below l

Incremental (sweep line) construction(p1 in P1 and p2 in P2 perpendicular with m, Sweep l)

Extend K by line segment l si

Set l = si

Compute new K defining pair p1, p2

Theorem: Running time O(n) Proof: Vor(pi) are convex, therefore each one‘s forward - edge are only visited once.



Example



Fortune´s Algorithm

Beach - line

Sweep - line

Observations: Intersection of the parabolas define edgesNew "telephones" ( ) define new parabolas Parabola intersection disappear, if C(P, q) has 3 points



Applications 1(Static Point Set)

Closest pair of points: Go through edge list for VD(P) and determine minimum

All next neighbors :Go through edge list for VD(P) for all points and get next neighbors in each case

Minimum Spanning tree (after Kruskal) 1. Each point p from P defines 1-element set of 2. More than a set of T exists 2.1) find p,p´ with p in T and p´ not in T with d(p, p´)

minimum. 2.2) connect T and p´ contained in T´ (union)

Theorem: The MST can be computed in time O(n log n)



Applications (dynamic object set)Search for next neighbor :Idea : Hierarchical subdivision of VD(P) Step 1 : Triangulation of final Voronoi regions Step 2 : Summary of triangles and structure of a search tree

Rule of Kirkpatrick : Remove in each case points with degree < 12,its neighbor is already far.

Theorem: Using the rule of Kirkpatrick a search tree of logarithmic depth develops.

A a

b c a b c

A

1 Lecture 8: Voronoi Diagram Computational Geometry Prof. Dr. Th. Ottmann Voronoi Diagrams...

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