Voronoi Diagrams Computational Geometry Lecture · The Post O ce Problem Vor(P) = Voronoi diagram...

Tamara Mchledidze · Darren Strash · Computational Geometry Voronoi-Diagrams1

Tamara Mchledidze · Darren Strash

Computational Geometry Lecture

INSTITUT FÜR THEORETISCHE INFORMATIK · FAKULTÄT FÜR INFORMATIK

Voronoi Diagrams

03.06.2014

Tamara Mchledidze · Darren Strash · Computational Geometry Voronoi-Diagrams

Tamara Mchledidze · Darren Strash · Computational Geometry Voronoi-Diagrams2-1

The Post Office Problem



p

q



p

q

b(p, q)



p

q

b(p, q) = {x ∈ R2 : |xp| = |xq|}



p

q

b(p, q)

h(p, q)

= {x ∈ R2 : |xp| = |xq|}



p

q

b(p, q)

h(p, q)

= {x ∈ R2 : |xp| = |xq|}

= {x : |xp| < |xq|}



p

q

b(p, q)

h(p, q) h(q, p)

= {x ∈ R2 : |xp| = |xq|}

= {x : |xp| < |xq|}



p

q

b(p, q)

h(p, q) h(q, p)

= {x ∈ R2 : |xp| = |xq|}

= {x : |xp| < |xq|} = {x : |xq| < |xp|}



p

q



P = {p1, p2, . . . , pn}



Vor(P ) = Voronoi diagram of P



1) Define Voronoi cells, edges, and vertices!



1) Define Voronoi cells, edges, and vertices!

2) Are Voronoi cells convex?


The Voronoi Diagram

Let P be a set of points in the plane and let p, p′, p′′ ∈ P .


The Voronoi Diagram


Vor(P )Voronoi Diagramp

p′

p′′


The Voronoi Diagram

V({p}) =


Vor(P )Voronoi Diagram

Voronoi cell

p

p′

p′′


The Voronoi Diagram

V({p}) =


Vor(P )Voronoi Diagram

Voronoi cell

p

p′

p′′

=V(p)


The Voronoi Diagram

V({p}) =


Vor(P )

{x ∈ R2 : |xp| < |xq| ∀q ∈ P \ {p}

}Voronoi Diagram

Voronoi cell

p

p′

p′′

=V(p)


The Voronoi Diagram

V({p}) ==⋂

q 6=p h(p, q)


Vor(P )

{x ∈ R2 : |xp| < |xq| ∀q ∈ P \ {p}

}Voronoi Diagram

Voronoi cell

p

p′

p′′

=V(p)


The Voronoi Diagram

V({p}) ==⋂

q 6=p h(p, q)


Vor(P )

V({p, p′})

{x ∈ R2 : |xp| < |xq| ∀q ∈ P \ {p}

}Voronoi edges

Voronoi Diagram

Voronoi cell

p

p′

p′′

=

=V(p)


The Voronoi Diagram

V({p}) ==⋂

q 6=p h(p, q)


Vor(P )

V({p, p′})

{x ∈ R2 : |xp| < |xq| ∀q ∈ P \ {p}

}

{x : |xp| = |xp′| and |xp| < |xq| ∀q 6= p, p′}Voronoi edges

Voronoi Diagram

Voronoi cell

p

p′

p′′

=

=V(p)


The Voronoi Diagram

V({p}) ==⋂

q 6=p h(p, q)


Vor(P )

V({p, p′})∂V(p) ∩ ∂V(p′)=

{x ∈ R2 : |xp| < |xq| ∀q ∈ P \ {p}

}


Voronoi Diagram

Voronoi cell

p

p′

p′′

=

=V(p)


The Voronoi Diagram

V({p}) ==⋂

q 6=p h(p, q)


Vor(P )

V({p, p′})∂V(p) ∩ ∂V(p′)= int( )

{x ∈ R2 : |xp| < |xq| ∀q ∈ P \ {p}

}


Voronoi Diagram

Voronoi cell

p

p′

p′′

=

=

d.h. without endpoints,

V(p)


The Voronoi Diagram

V({p}) ==⋂

q 6=p h(p, q)


Vor(P )

V({p, p′})∂V(p) ∩ ∂V(p′)= int( )

{x ∈ R2 : |xp| < |xq| ∀q ∈ P \ {p}

}

{x : |xp| = |xp′| and |xp| < |xq| ∀q 6= p, p′}

Voronoi verticesV({p, p′, p′′})

Voronoi edges

Voronoi Diagram

Voronoi cell

p

p′

p′′

=

=


V(p)


The Voronoi Diagram

V({p}) ==⋂

q 6=p h(p, q)


Vor(P )

V({p, p′})∂V(p) ∩ ∂V(p′)= int( )

{x ∈ R2 : |xp| < |xq| ∀q ∈ P \ {p}

}


Voronoi vertices= ∂V(p) ∩ ∂V(p′) ∩ ∂V(p′′)V({p, p′, p′′})

Voronoi edges

Voronoi Diagram

Voronoi cell

p

p′

p′′

=

=


V(p)


The Voronoi Diagram

V({p}) ==⋂

q 6=p h(p, q)


Vor(P )

V({p, p′})∂V(p) ∩ ∂V(p′)= int( )

{x ∈ R2 : |xp| < |xq| ∀q ∈ P \ {p}

}



Voronoi edges

Voronoi Diagram

Voronoi cell

p

p′

p′′

=

=

subdivision


V(p)


The Voronoi Diagram

V({p}) ==⋂

q 6=p h(p, q)


Vor(P )

V({p, p′})∂V(p) ∩ ∂V(p′)= int( )

{x ∈ R2 : |xp| < |xq| ∀q ∈ P \ {p}

}



Voronoi edges

Voronoi Diagram

Voronoi cell

p

p′

p′′

=

=

subdivision

geometric graph


V(p)


Properties

Theorem 1: Let P ⊂ R2 be a set of n points. If all points arecollinear, then Vor(P ) consists of n− 1 parallellines. Otherwise Vor(P ) is connected and itsedges are either segments or half lines.


Properties


How many edges and vertices Vor(P ) has?


Properties


Find a set P so that a cell in Vor(P ) has linear complexity.



Properties



Can this happen with (almost) all cells?



Properties



Theorem 2: Let P ⊂ R2 be a set on n points. Vor(P ) has atmost 2n− 5 nodes and 3n− 6 edges.




Properties



Theorem 2: Let P ⊂ R2 be a set on n points. Vor(P ) has atmost 2n− 5 nodes and 3n− 6 edges.



Exercise!


Characterization

Definition: Let q be a point. Define CP (q) to be the points inP that lie on the empty circle with center q.

q


Characterization


q

CP (q)


Characterization


q

CP (q)

Theorem 3: A point q is a Voronoi vertex⇔ |CP (q) ∩ P | ≥ 3,the bisector b(pi, pj) defines a Voronoi edge⇔ ∃q ∈ b(pi, pj) with CP (q) ∩ P = {pi, pj}.


Computing Vor(P )

How can we calculate Vor(P ) withmethods we already know?


Computing Vor(P )

For each p ∈ P is V(p) =⋂

p′ 6=p h(p, p′) is the intersection of

n− 1 half planes.



Computing Vor(P )



n− 1 half planes.


foreach p ∈ P docompute V(p) =

⋂p′ 6=p h(p, p

′)


Computing Vor(P )



n− 1 half planes.



⋂p′ 6=p h(p, p

′) O(n log n) [Lecture 4]


Computing Vor(P )



n− 1 half planes.



⋂p′ 6=p h(p, p


O(n2 log n)


Computing Vor(P )



n− 1 half planes.



⋂p′ 6=p h(p, p


O(n2 log n)

Is O(n2 log n) running time for a linear-size object necessary?


Computing Vor(P )



n− 1 half planes.



⋂p′ 6=p h(p, p


O(n2 log n)


Idea 2: Sweep line

`


Computing Vor(P )



n− 1 half planes.



⋂p′ 6=p h(p, p


O(n2 log n)


Idea 2: Sweep line

Problem:Vor(P ) over ` depends onpoints under `!

`


In the Direction of the Sweep Line

Obviously the intersection of Vor(P ) and sweep line ` at thecurrent time point is not known yet.




Instead, we store the part above ` that is already fixed!





What does it consist of?






p(px, py)

`(`y)






p(px, py)

`(`y)

Points closer to p than ` are already processed.






p(px, py)

`(`y)

fixed

not fixed






p(px, py)

`(`y)

fixed

not fixed

q = (x, f(x))






p(px, py)

`(`y)

Enforcing the equality |pq| = |q`| gives

f(x) =1

2(py − `y)(x− px)2 +

py + `y2

fixed

not fixed

q = (x, f(x))






p(px, py)

`(`y)

Enforcing the equality |pq| = |q`| gives

f(x) =1

2(py − `y)(x− px)2 +

py + `y2

fixed

not fixed

q = (x, f(x))

f `p(x) =


The Beach Line

Definition: The beach line β` is the lower envelope ofparabolas f `p for the points already found.

The Beach Line


The Beach Line


β`

The Beach Line


The Beach Line


β`

What does it have to do with Vor(P )?

The Beach Line


The Beach Line


β`


Obs.: The beach line is x-monotoneIntersection points in the beach line lie on VoronoiedgesAs the sweep-line goes down, intersection pointsrun along Vor(P )

The Beach Line


The Beach Line


β`


Obs.: The beach line is x-monotoneIntersection points in the beach line lie on VoronoiedgesAs the sweep-line goes down, intersection pointsrun along Vor(P )

The Beach Line

Goal: Store (implicit) contour β` instead of Vor(P ) ∩ `


Before we proceed...

Demohttp://www.diku.dk/hjemmesider/studerende/duff/Fortune/


Point Events


Point Events

If ` meets a point, then a new parabola is added to β`


Point Events


The two intersection points generate a new part of an edge.


Point Events



Lemma 1: New arcs on β` only come from point events.


Point Events




Corollary: β` is at most 2n− 1 parabolic arcs


Point Events




Corollary: β` is at most 2n− 1 parabolic arcs

More about this in exercises...


Circle Events

pi

pj

pk


Circle Events

A parabolic arc disappears from f `pi , f`pj , f

`pk

at a commonpoint q

pi

pj

pk pi

pj

pk

q


Circle Events


`pk

at a commonpoint q

pi

pj

pk pi

pj

pk

q

The circle CP (q) that touches pi, pj , pk and `⇒ q is a Voronoi vertex


Circle Events


`pk

at a commonpoint q

pi

pj

pk pi

pj

pk pi

pj

pk

q



Circle Events


`pk

at a commonpoint q

pi

pj

pk pi

pj

pk pi

pj

pk

q


Def.: The lowest point of the circle defined by three pointsof consecutive arcs in β` defines a circle event.


Circle Events


`pk

at a commonpoint q

pi

pj

pk pi

pj

pk pi

pj

pk

q


Lemma 2: Arcs of β` only disappear through circle events.



Circle Events


`pk

at a commonpoint q

pi

pj

pk pi

pj

pk pi

pj

pk

q


Lemma 2: Arcs of β` only disappear through circle events.


Lemma 3: For each Voronoi vertex there is a circle event.


Data Structures

Double-connected edge list (DCEL) D for Vor(P )Warning: Include a bounding box to avoid half-lines


Data Structures


Balanced binary search tree T for implicit beach line– Leaves represent parabolic arcs from left to right– Interior nodes 〈pi, pj〉 represent intersection points of fpi and fpj– Pointers from interior nodes to the corresponding edges in D

p1

p2

p3

p4

p5

〈p4, p5〉

〈p2, p3〉

〈p1, p2〉〈p3, p4〉

〈p5, p4〉

p1 p2 p3 p4

p5 p4


Data Structures


Balanced binary search tree T for implicit beach line– Leaves represent parabolic arcs from left to right– Interior nodes 〈pi, pj〉 represent intersection points of fpi and fpj– Pointers from interior nodes to the corresponding edges in D

p1

p2

p3

p4

p5

〈p4, p5〉

〈p2, p3〉

〈p1, p2〉〈p3, p4〉

〈p5, p4〉

p1 p2 p3 p4

p5 p4

Priority queue Q for the point and circle events– Pointer from circle event to corresponding leaf in T and vice versa


Fortune’s Sweep Algorithm

VoronoiDiagram(P ⊂ R2)Q ← new PriorityQueue(P ) // Point events sorted by yT ← new BalancedBinarySearchTree() // sweep status (β)D ← new DCEL() // DS for Vor(P )while not Q.empty() do

p← Q.ExtractMax()if p point event then

HandlePointEvent(p)else

α← arcs of β to be removedHandleCircleEvent(α)

Handle interior remaining nodes of T (half-lines of Vor(P ))return D


Handing Point Events

HandlePointEvent(point p)

p




Search in T for the arc α above p.If α has a pointer to a circle event in Q, remove it from Q.

p

α




Search in T for the arc α above p.If α has a pointer to a circle event in Q, remove it from Q.

p

α

How?




Search in T for the arc α above p.If α has a pointer to a circle event in Q, remove it from Q.Split α into α0 and α2.Let α1 be a new arc for p.

p

α

q





q

α

p

α

q

In T :





q 〈q, p〉

q 〈p, q〉

p q

α

p

α

q

α0 α1 α0

α2α1

In T :

2





q 〈q, p〉

q 〈p, q〉

p q

α

Intersection points

p

α

q

α0 α1 α0

α2α1

In T :

2





q 〈q, p〉

q 〈p, q〉

p q

α

Add edges 〈q, p〉 and 〈p, q〉 to D.

Intersection points

p

α

q

α0 α1 α0

α2α1

In T :

2





q 〈q, p〉

q 〈p, q〉

p q

α

Add edges 〈q, p〉 and 〈p, q〉 to D.Check 〈pl, q, p〉 and 〈p, q, pr〉 for circle events.

Intersection points

p

α

q

α0 α1 α0

α2α1

In T :

2pl

Neighbors of β





q 〈q, p〉

q 〈p, q〉

p q

α


Intersection points

p

α

q

α0 α1 α0

α2α1

In T :

2

Running time?pl

Neighbors of β





q 〈q, p〉

q 〈p, q〉

p q

α


Intersection points

p

α

q

α0 α1 α0

α2α1

In T :

2

Running time?

O(log n)

pl

Neighbors of β


Handling Circle Events

HandleCircleEvent(arc α)

ααleft

αrightp

p′

p′′




T .delete(α); Update intersection points in T

ααleft

αrightp

p′

p′′




T .delete(α); Update intersection points in TRemove all circle events with middle arc α from Q.

ααleft

αrightp

p′

p′′




T .delete(α); Update intersection points in TRemove all circle events with middle arc α from Q.

ααleft

αrightp

p′

p′′

falsealarms




T .delete(α); Update intersection points in TRemove all circle events with middle arc α from Q.Add node V({p, p′, p′′}) and edges 〈p, p′′〉, 〈p′′, p〉 to D.

ααleft

αrightp

p′

p′′

falsealarms




T .delete(α); Update intersection points in TRemove all circle events with middle arc α from Q.Add node V({p, p′, p′′}) and edges 〈p, p′′〉, 〈p′′, p〉 to D.Add potential circle events 〈pl, p, p′′〉 and 〈p, p′′, pr〉 to Q.

ααleft

αrightp

p′

p′′

Neighbors of β

falsealarms





ααleft

αrightp

p′

p′′Running time?

Neighbors of β

falsealarms





ααleft

αrightp

p′

p′′Running time?

O(log n)

Neighbors of β

falsealarms



Theorem 4: For a set P of n points, Fortune’s sweepalgorithm computes the Voronoi DiagramVor(P ) in O(n log n) time and O(n) space.



Theorem 4: For a set P of n points, Fortune’s sweepalgorithm computes the Voronoi DiagramVor(P ) in O(n log n) time and O(n) space.

Proof sketch:Each event requires O(log n) timen point events≤ 2n− 5 circle events (= #nodes of Vor(P ))False alarms are deleted from Q before they are processed.


Discussion

Are there other variants of Voronoi diagrams?


Discussion


Yes! For example, we can design an algorithm to compute the Voronoidiagram for line segments with the same running time and space.

Other metrics like Lp or additive/multiplicativeweighted Voronoi diagrams are possible.


Discussion




Voronoi diagrams for polygons define theso-called medial axis, (important in imageprocessing).

Also farthest-point Voronoi diagrams arepossible.


Discussion



What happens in higher dimensions?





Discussion



What happens in higher dimensions?

The complexity of Vor(P ) increases to Θ(ndd/2e) and the running timeto O(n log n+ ndd/2e).





From Voronoi to Art...

Geogebra

The Post Office ProblemThe Voronoi DiagramPropertiesCharacterizationComputing $\vd(P)$In the Direction of the Sweep LineThe Beach LinePoint EventsCircle EventsData StructuresFortune's Sweep AlgorithmHanding Point EventsHandling Circle EventsFortune's Sweep AlgorithmDiscussion

Voronoi Diagrams Computational Geometry Lecture · The Post O ce Problem Vor(P) = Voronoi diagram...

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