Delaunay triangulations: properties, algorithms, and ... · 1-2 Delaunay triangulations:...

1 - 1

Delaunay triangulations: properties, algorithms, and complexity

Olivier Devillers

1 - 2

Delaunay triangulations: properties, algorithms, and complexity

Olivier Devillers

Extra question: what is the difference

between an algorithm and a program?

2

Algorithm

Recipe to go from the input to the output

Formalized description in some language

May use data structure

Proof of correctness

Complexity analysis

3 - 1

Delaunay Triangulation: definition, empty circle property

3 - 2


Point set

3 - 3


Point set

Query

3 - 4


Point set

Query

3 - 5


Point set

Query

3 - 6


Point set

Query

Nearest neighbor

3 - 7


Point set

Voronoi diagram

3 - 8


Point set

Voronoi diagram

3 - 9


Point set

Voronoi diagram

3 - 10


Point set

Voronoi diagram

3 - 11


Point set

Delaunay triangulation

Voronoi diagram

3 - 12


Point set

Empty circle property


3 - 13


Point set

Empty circle property


4 - 1

Delaunay Triangulation: size

4 - 2


Euler relation

Vertices

Edges

Faces

n e t+ 1

= 2� +

4 - 3


Euler relation

Vertices

Edges

Faces

n e t+ 1

= 2� +

triangular faces 3t+ k = 2e

4 - 4


t = 2n� k � 2 < 2n

e = 3n� k � 3 < 3n

Euler relation

Vertices

Edges

Faces

n e t+ 1

= 2� +


4 - 5


Euler relation

Vertices

Edges

Faces

n e t+ 1

= 2� +


X

p2S

d�(p) = 2e = 6n� 2k � 6

E(d�(p)) = 1n

X

p2S

d�(p) < 6

average on the choice of point p in set of points S

5 - 1

Delaunay Triangulation: max-min angle

5 - 2


Triangulation Delaunay

5 - 3



smallest angle

5 - 4



smallest angle

5 - 5



smallest angle

5 - 6



smallest angle

second smallest angle

6 - 1


Proof

6 - 2


Delaunay edgeDefinition

6 - 3


Delaunay edge

9 empty circle

Definition

6 - 4


Delaunay edgelocally w.r.t. a triangulationDefinition

6 - 5


Delaunay edgelocally w.r.t. a triangulation

9 circle

not enclosing the two neighbors

Definition

neighbor = visible from the edge

7 - 1


Lemma (8 edge: locally Delaunay) () Delaunay

7 - 2



Proof:

choose an edge

7 - 3



Proof:

choose an edge

edges of the quadrilateral

are locally Delaunay

7 - 4



Proof:

choose an edge



7 - 5



Proof:

choose an edge



Vertices visible through one edge are outside circle

7 - 6



Proof:

choose an edge



Vertices visible through one edge are outside circle

Induction ! all vertices outside circle

8 - 1


Lemma For four points in convex position

Delaunay () maximize the smallest angle

Two possible triangulation

8 - 2




Case 1: smallest angle in corner

�

8 - 3




Case 1: smallest angle in corner

9 a smaller angle 2 other triangulation

�< �

8 - 4




Case 2: smallest angle

�along diagonal

8 - 5





�along diagonal

�

8 - 6





9 a smaller angle 2 other triangulation

�

< �

along diagonal

�

9 - 1


Map: Triangulations �! R6n�3k�4 smallest angle ↵1

9 - 2



second smallest angle ↵2

9 - 3




third smallest angle ↵3

9 - 4




third smallest angle ↵3(↵1,↵2,↵3, . . . ,↵6n�3k�4)

9 - 5




third smallest angle ↵3(↵1,↵2,↵3, . . . ,↵6n�3k�4)

sort triangulations in lexicographic order

10 - 1


Theorem:

Delaunay maximizes minimum angles (in lexicographic order)

10 - 2


Theorem:


Proof:Let T be the triangulation maximizing angles

10 - 3


Theorem:



=) 8 convex quadrilateral (from 2 triangles 2 T )

the diagonal maximizes smallest angle (in quad)

10 - 4


Theorem:





=) 8 edge, it is locally Delaunay

10 - 5


Theorem:





=) 8 edge, it is locally Delaunay

=) T = Delaunay

11 - 1

Delaunay triangulation Lower bound

A stupid algorithm for sorting numbers

11 - 2



project on parabola

11 - 3



project on parabola

compute Delaunay triang.

11 - 4



project on parabola


find lowest point

11 - 5



project on parabola


find lowest point

enumerate x coordinatesin ccw CH order

11 - 6



project on parabola


find lowest point


O(n)

f(n)

O(n)

O(n)

Lower bound on sorting

=) f(n) +O(n) � ⌦(n log n)

11 - 7



project on parabola


find lowest point


O(n)

f(n)

O(n)

O(n)

Lower bound on sorting

=) f(n) +O(n) � ⌦(n log n)

12 - 1

Delaunay Triangulation: predicates

12 - 2


Orientation predicate

p

qr

pqr + ?

��

xq � xp xr � xpyq � yp yr � yp

��=

��

1 1 1

xp xq xryp yq yr

��

> 0

pqr - ?��

1 1 1

xp xq xryp yq yr

��

< 0

pqr 0 ?��

1 1 1

xp xq xryp yq yr

��

= 0

p

q

r

p

q

r

12 - 3


pqr ccw triangle

pq

r

query s

s

inside circumcircle

Incircle predicate

12 - 4


pqr ccw triangle

pq

r

query s

s

cocircular

12 - 5


pqr ccw triangle

pq

r

query s

s

outside circumcircle

12 - 6


pqr ccw triangle

pq

r

query s

12 - 7


Space of circles

p = (x, y) p? = (x, y, x2+ y2)

12 - 8


Space of circles

circle through pqr plane through p?q?r?

s inside/outside of

above/below s?

incircle predicate

3D orientation predicate

12 - 9


Space of circles

circle through pqr plane through p?q?r?

s inside/outside of

above/below s?

incircle predicate

3D orientation predicatesign

��

1 1 1 1

xp xq xr xsyp yq yr ys

x2p + y2p x2q + y2q x2r + y2r x2s + y2s

��

13

Delaunay Triangulation: data structure

Data structure for (Delaunay) triangulation

Representing incidences

Representing hull boundary

Representing user’s data put c in trianglesolors

14 - 1

Delaunay Triangulation: very naive algorithm

14 - 2


For each triple of points (p, q, r)

For each point s

If pqr ccw

If s in circle pqr

Delaunay = true;

Delaunay = false;

Output pqr

14 - 3



For each point s

If pqr ccw

If s in circle pqr

Delaunay = true;

Delaunay = false;

Output pqr

Correctness: easy

14 - 4



For each point s

If pqr ccw

If s in circle pqr

Delaunay = true;

Delaunay = false;

Output pqr

Correctness: easy

Complexity: O(n4)

14 - 5



For each point s

If pqr ccw

If s in circle pqr

Delaunay = true;

Delaunay = false;

Output pqr

Correctness: easy

Complexity: O(n4)

does not compute incidences

15 - 1

Delaunay Triangulation: incremental algorithm

15 - 2


New point

15 - 3


New point

Locate

15 - 4


New point

Locate

e.g.: visibility walk

15 - 5


New point

Locate


15 - 6


New point

Locate


15 - 7


New point

Locate


15 - 8


New point

Locate


15 - 9


New point

Locate


15 - 10


New point

Locate


15 - 11


New point

Locate


15 - 12


New point

Locate


not unique

15 - 13


New point

Locate


not unique

15 - 14

Visibility walk terminates


15 - 15



15 - 16



May loop

15 - 17



May loop

Not Delaunay

15 - 18

Delaunay Triangulation: pencils of circles

Power of a point w.r.t a circle

x2+ y2 � 2ax� 2by + c

blue yields smaller power

black yields smaller power

equal power

(x2+ y2 � 2a0x� 2b0y + c0)�

+(1� �) (

)

= 0



15 - 19




15 - 20




Green power Red power<

15 - 21




Green power Red power<Power decreases

15 - 22




Green power Red power<Power decreasesVisibility walk terminates

16 - 1


New point

Locate

Search conflicts

16 - 2


New point

Locate

Search conflicts

16 - 3


New point

Locate

Search conflicts

16 - 4


New point

Locate

Search conflicts

16 - 5


New point

Locate

Search conflicts

16 - 6


New point

Locate

Search conflicts

16 - 7


New point

Locate

Search conflicts

16 - 8


New point

Locate

Search conflicts

16 - 9


New point

Locate

Search conflicts

16 - 10


New point

Locate

Search conflicts

16 - 11


New point

Locate

Search conflicts

16 - 12


New point

Locate

Search conflicts

16 - 13


New point

Locate

Search conflicts

16 - 14


New point

16 - 15


New point

17 - 1

circle of new triangle



17 - 2

circle of new triangle



⇢ [

18 - 1


Complexity

Locate

Search conflicts

18 - 2


Complexity

Locate

Search conflicts ] triangles in conflict

] triangles neighboring triangles in conflict

18 - 3


Complexity

Locate

Search conflicts ] triangles in conflict

] triangles neighboring triangles in conflict

degree of new point in new triangulation

< n

18 - 4


Complexity

Locate

Search conflicts

degree of new point in new triangulation

< n

Walk may visit all triangles< 2n

18 - 5


Complexity

Locate

Search conflicts

O(n) per insertion

18 - 6


Complexity

Locate

Search conflicts

O(n) per insertion

O(n2) for the whole construction

18 - 7


Complexity

Locate

Search conflicts

18 - 8


Complexity

Locate

Search conflicts

Insertion: ⌦(n)

Whole construction: ⌦(n2)

19 - 1

Delaunay Triangulation: sweep-line algorithm

Discover the points from left to right

19 - 2



19 - 3


Discover the points from left to rightCertified Delaunay triangles

19 - 4



19 - 5



19 - 6


Discover the points from left to rightCertified Delaunay trianglesCertified Delaunay edges

19 - 7



19 - 8


Discover the points from left to rightNew point

I

H

Locate vertically

I

H

19 - 9


Discover the points from left to rightNew point

I

H

Locate vertically

I

H

Create edge

19 - 10



Closing a triangle ?

19 - 11



Next circle event

Close triangle

20 - 1


Complexity

Triangulation

List of events (x sorted)

List of boundary edges(ccw sorted)

Circle events Point events

Number

processed

20 - 2


Complexity

Triangulation




Number 2n ncreate2 triangles

createone edge

3 deletions 2 insertionsper event


replace2 edges by 1per event per event

locate, theninsert 2 edges

per event per event

processed

20 - 3


Complexity

Triangulation





createone edge





per event per eventO(

1

)

per operation

O(

l

o

g

n) per operation

O(

l

o

g

n) per operation

processed

20 - 4


Complexity

Triangulation





createone edge





per event per eventO(

1

)

per operation

O(

l

o

g

n) per operation

O(

l

o

g

n) per operation

O(n log n)

processed

21 - 1


21 - 2


x comparisons

21 - 3


x comparisons y comparisons

21 - 4


x comparisons y comparisons

more intricate than

orientation and incircle

22 - 1

Delaunay Triangulation: divide & conquer (sketch)

22 - 2


Divide

22 - 3


Recurse

Divide

22 - 4

Conquer


Recurse

Divide

22 - 5

Conquer


Recurse

O(n log n)

Divide

22 - 6

Conquer


Recurse

O(n log n)

Divide

balanced

linear time

easier conquer

23 - 1

Jump and walk Use randomness hypotheses

23 - 2


23 - 3


23 - 4

Jump and walk

Hopefully shorter walk

Designed for random points

O(

3pn) expected location time

Use randomness hypotheses

24 - 1

Jump and walk (no distribution hypothesis) Randomized

24 - 2

Jump and walk (no distribution hypothesis)E [] of in ] =

nk

Randomized

24 - 3

Jump and walk (no distribution hypothesis)

choose k =

2pn

Walk length = O�nk

�E [] of in ] =

nk

Randomized

24 - 4


choose k =

2pn

Delaunay hierarchy


�E [] of in ] =

nk

Randomized

24 - 5


choose k =

2pn

Delaunay hierarchynk1


�E [] of in ] =

nk

Randomized

24 - 6


choose k =

2pn


+

k1k2


�E [] of in ] =

nk

Randomized

24 - 7


choose k =

2pn


+

k1k2


�E [] of in ] =

nk

+

k2k3

+ . . .

Randomized

24 - 8


choose k =

2pn


+

k1k2


�E [] of in ] =

nk

+

k2k3

+ . . .

choose kiki+1

= ↵

Randomized

24 - 9


choose k =

2pn


+

k1k2


�E [] of in ] =

nk

+

k2k3

+ . . .

choose kiki+1

= ↵

point location in O (↵ log↵ n)

Randomized

24 - 10


choose k =

2pn


+

k1k2


�E [] of in ] =

nk

+

k2k3

+ . . .

choose kiki+1

= ↵

point location in O (↵ log↵ n)

point location in O (

p↵ log↵ n)

Randomized

25 - 1

Technical detail

Walk length = O ] of in( )= O�nk

�

25 - 2

Technical detail


�

random point

not a random point

25 - 3

Technical detail


�

random point

not a random point

E [d

�] =

1

n

X

q

d

�NN(q) =1

n

X

q

X

v=NN(q)

d

�v

=

1

n

X

v

X

q;v=NN(q)

d

�v 1

n

X

v

6d

�v 36

25 - 4

Technical detail


�

random point

not a random point

E [d

�] =

1

n

X

q

d

�NN(q) =1

n

X

q

X

v=NN(q)

d

�v

=

1

n

X

v

X

q;v=NN(q)

d

�v 1

n

X

v

6d

�v 36

Pd�

26

Drawbacks of random order

non locality of memory access

data structure for point location

Hilbert sort

Randomization

27 - 1

27 - 2

27 - 3

27 - 4

28

Drawbacks of random order

non locality of memory access

data structure for point location

Hilbert sort

Last point is not at all a random point

Walk should be fast

no control of degree of last point

29 - 1

Biased Random Insertion Order (BRIO)

29 - 2


29 - 3


29 - 4


29 - 5


29 - 6


30 - 1

Algorithm





Complexity analysis

30 - 2

Algorithm





Complexity analysis

Program

Implementation of an algorithm

Translation in a programming language (C++)

May use software library

Debugging

Running time

30 - 3

Algorithm





Complexity analysis

Program

Implementation of an algorithm

Translation in a programming language (C++)

May use software library

Debugging

Running time

may be difficult to transform into

too complicated

too complicated

inexistant

Computation model

big O notation

actualcomputer

(cache, predic

tion. .. )

31The end

Delaunay triangulations: properties, algorithms, and ... · 1-2 Delaunay triangulations:...

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