THE SQUARE THE TRIANGLE TRIANGULATING …loffl001/publications/slides/... · THE TRIANGLE TRIANGULATING THE SQUARE & 1-2 University of California, Irvine ... Triangulation Gabriel

1-1

University of California,

Irvine

Freie Universitat Berlin

Maarten Loffler

Wolfgang Mulzer

SQUARING

THE TRIANGLE

TRIANGULATING

THE SQUARE&&

1-2

University of California,

Irvine

Freie Universitat Berlin

Maarten Loffler

Wolfgang Mulzer

QUADTREES AND DELAUNAYTRIANGULATIONS

EQUIVALENT

OR,

ARE

2-1

MAARTEN LOFFLER & WOLFGANG MULZER

: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT

INTRODUCTIONPART I

3-1

PROXIMITY STRUCTURES



3-2

Let P be a set of npoints in the plane.




3-3





3-4


A proximity structure

on P is ‘a structure

that stores some

kind of useful local

information’.




3-5




that stores some


information’.




3-6




that stores some


information’.




3-7




that stores some


information’.




4-1




4-2




All these structures

take Ω(n log n) time to

build.

4-3




Intuitively, though,

once you built one,

you should be able to

derive the others.



build.

4-4





once you built one,


derive the others.



build.

4-5





once you built one,


derive the others.



build.

4-6





once you built one,


derive the others.



build.

4-7





once you built one,


derive the others.



build.

4-8





once you built one,


derive the others.



build.

4-9





once you built one,


derive the others.



build.

4-10





once you built one,


derive the others.



build.

4-11





once you built one,


derive the others.



build.

4-12





once you built one,


derive the others.



build.

4-13





once you built one,


derive the others.



build.

4-14





once you built one,


derive the others.



build.

4-15





once you built one,


derive the others.



build.

4-16





once you built one,


derive the others.



build.

4-17





once you built one,


derive the others.



build.

4-18





once you built one,


derive the others.



build.

4-19





once you built one,


derive the others.



build.

4-20





once you built one,


derive the others.



build.

4-21





once you built one,


derive the others.



build.

Since information

is local, such a

conversion should

take O(n) time.

5-1




5-2




Unfortunately, not

all point sets behave

themselves.

5-3




Unfortunately, not


themselves.

A single point may

have more than Ω(1)neighbours.

5-4




Unfortunately, not


themselves.

A single point may


5-5




Unfortunately, not


themselves.

A single point may


5-6




Unfortunately, not


themselves.

A single point may


The scale of a point

set need not be

uniform.

5-7




Unfortunately, not


themselves.

A single point may



set need not be

uniform.

5-8




Unfortunately, not


themselves.

A single point may



set need not be

uniform.

What can we do?

6-1

RESEARCHER’S DILEMMA



6-2




Do we...

6-3




Do we...

... assume that in

practice, point sets

are well-behaved,

and design useable

algorithms that only

work (or are only

efficient) on certain

classes of point

sets?

6-4




Do we...

... assume that in


are well-behaved,

and design useable


work (or are only


classes of point

sets?

-OR-

6-5




Do we...

... assume that in


are well-behaved,

and design useable


work (or are only


classes of point

sets?

-OR-

... insist on being

general, and design

devilishly

complicated

algorithms that

nobody will ever use,

but that are able to

handle arbitrary

point sets?

7-1



A BRIEF HISTORY

7-2



A BRIEF HISTORY[Dirichlet, 1850]

Voronoi

Diagram

7-3

Delaunay

Triangulation

[Delaunay, 1934]




Voronoi

Diagram

Deterministic linear time

7-4

Delaunay

Triangulation

Gabriel

Graph

[Delaunay, 1934][Gabriel, 1969]




Voronoi

Diagram


7-5

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree


[Matsui, 1995]




Voronoi

Diagram


7-6

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree


[Matsui, 1995]

[Preparata & Shamos, 1985]




Voronoi

Diagram

Nearest

Neighbour Graph


7-7

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree


[Matsui, 1995]





Voronoi

Diagram

Nearest

Neighbour Graph


[Clarkson, 1983]

7-8

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree


[Matsui, 1995]


[Callahan & Kosaraju, 1995]




Voronoi

Diagram

Nearest

Neighbour GraphWell-Separated

Pair

Decomposition


[Clarkson, 1983]

7-9

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree


[Matsui, 1995]






Voronoi

Diagram

Nearest


Pair

Decomposition


[Clarkson, 1983]

7-10

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree


[Matsui, 1995]



[Chazelle, 1991]




Voronoi

Diagram

Nearest


Pair

Decomposition


[Chin & Wang, 1998]

[Clarkson, 1983]

7-11

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree

DT

On Superset


[Matsui, 1995]



[Chazelle, 1991]

[Bern,




Voronoi

Diagram

Nearest


Pair

Decomposition

Eppstein & Gilbert, 1990]


[Chin & Wang, 1998]

[Clarkson, 1983]

7-12

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree

DT

On Superset


[Matsui, 1995]



[Chazelle, 1991]

[Bern,

[Chazelle, Devillers,




Voronoi

Diagram

Nearest


Pair

Decomposition


Hurtado, Mora,Sacristan & Teillaud, 2001]


Randomised linear time

[Chin & Wang, 1998]

[Clarkson, 1983]

7-13

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree

DT

On Superset

QT Sequence

(Skip Quadtree)


[Matsui, 1995]



[Chazelle, 1991]

[Bern,


[Eppstein,




Voronoi

Diagram

Nearest


Pair

Decomposition



Goodrich & Sun, 2005]Deterministic linear time


[Chin & Wang, 1998]

[Clarkson, 1983]

7-14

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree

DT

On Superset

QT Sequence

(Skip Quadtree)


[Matsui, 1995]



[Chazelle, 1991]

[Bern,


[Eppstein,

WSPD

Sequence

NNG

Sequence




Voronoi

Diagram

Nearest


Pair

Decomposition





[Chin & Wang, 1998]

[Clarkson, 1983]

7-15

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree

DT

On Superset

QT Sequence

(Skip Quadtree)


[Matsui, 1995]



[Chazelle, 1991]

[Bern,


[Eppstein,

WSPD

Sequence

NNG

Sequence

[Buchin & Mulzer, 2009]




Voronoi

Diagram

Nearest


Pair

Decomposition





[Chin & Wang, 1998]

[Clarkson, 1983]

7-16

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree

DT

On Superset

QT Sequence

(Skip Quadtree)

c-ClusterQuadtree


[Matsui, 1995]



[Chazelle, 1991]

[Bern,


[Eppstein,

WSPD

Sequence

NNG

Sequence


[Krznaric &




Voronoi

Diagram

Nearest


Pair

Decomposition



Goodrich & Sun, 2005]

Levcopoulos, 1998]



Linear time with floor operation

[Chin & Wang, 1998]

[Clarkson, 1983]

7-17

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree

DT

On Superset

QT Sequence

(Skip Quadtree)

c-ClusterQuadtree


[Matsui, 1995]



[Chazelle, 1991]

[Bern,


[Eppstein,

WSPD

Sequence

NNG

Sequence


[Krznaric &




Voronoi

Diagram

Nearest


Pair

Decomposition

[Us, 2011]




Levcopoulos, 1998]





[Chin & Wang, 1998]

[Clarkson, 1983]

7-18

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree

DT

On Superset

QT Sequence

(Skip Quadtree)

c-ClusterQuadtree


[Matsui, 1995]



[Chazelle, 1991]

[Bern,


[Eppstein,

WSPD

Sequence

NNG

Sequence


[Krznaric &




Voronoi

Diagram

Nearest


Pair

Decomposition

[Us, 2011]




Levcopoulos, 1998]





[Chin & Wang, 1998]

[Clarkson, 1983]

7-19

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree

DT

On Superset

QT Sequence

(Skip Quadtree)

c-ClusterQuadtree


[Matsui, 1995]



[Chazelle, 1991]

[Bern,


[Eppstein,

WSPD

Sequence

NNG

Sequence


[Krznaric &




Voronoi

Diagram

Nearest


Pair

Decomposition

[Us, 2011]




Levcopoulos, 1998]





[Chin & Wang, 1998]

[Clarkson, 1983]

7-20

Delaunay

Triangulation

Gabriel

Graph

Minimum

Spanning Tree

Compressed

Quadtree

DT

On Superset

QT Sequence

(Skip Quadtree)

c-ClusterQuadtree


[Matsui, 1995]



[Chazelle, 1991]

[Bern,


[Eppstein,

WSPD

Sequence

NNG

Sequence


[Krznaric &




c-CQTOn Superset

Voronoi

Diagram

Nearest


Pair

Decomposition

[Us, 2011]




Levcopoulos, 1998]





[Chin & Wang, 1998]

[Clarkson, 1983]

8-1



PRELIMINARIESPART II

9-1

DELAUNAY TRIANGULATION



9-2




The Delaunay

triangulation (DT)

is a triangulation

D of a point set

P that has an edge

between two points

if there is an empty

circle through these

points.

9-3




The Delaunay

triangulation (DT)

is a triangulation

D of a point set

P that has an edge

between two points



points.

9-4




The Delaunay

triangulation (DT)

is a triangulation

D of a point set

P that has an edge

between two points



points.

9-5




The Delaunay

triangulation (DT)

is a triangulation

D of a point set

P that has an edge

between two points



points.

9-6




The Delaunay

triangulation (DT)

is a triangulation

D of a point set

P that has an edge

between two points



points.

9-7




The Delaunay

triangulation (DT)

is a triangulation

D of a point set

P that has an edge

between two points



points.

9-8




The Delaunay

triangulation (DT)

is a triangulation

D of a point set

P that has an edge

between two points



points.

9-9




The Delaunay

triangulation (DT)

is a triangulation

D of a point set

P that has an edge

between two points



points.

9-10




The Delaunay

triangulation (DT)

is a triangulation

D of a point set

P that has an edge

between two points



points.

10-1

MINIMUM SPANNING TREE



10-2




The Euclidean minimum

spanning tree (EMST)

is the minimum

spanning tree of

the complete graph on

a point set P.

10-3






is the minimum

spanning tree of


a point set P.

10-4






is the minimum

spanning tree of


a point set P.

10-5






is the minimum

spanning tree of


a point set P.

The EMST is a

subgraph of the DT.

10-6






is the minimum

spanning tree of


a point set P.

The EMST is a

subgraph of the DT.

11-1

COMPRESSED QUADTREE



11-2

COMPRESSED QUADTREE



A quadtree T for

a set of points Pis a hierarchical

subdivision of

a square into

smaller squares that

separates P.

11-3

COMPRESSED QUADTREE



A quadtree T for


subdivision of

a square into


separates P.

11-4

COMPRESSED QUADTREE



A quadtree T for


subdivision of

a square into


separates P.

11-5

COMPRESSED QUADTREE



A quadtree T for


subdivision of

a square into


separates P.

11-6

COMPRESSED QUADTREE



A quadtree T for


subdivision of

a square into


separates P.

11-7

COMPRESSED QUADTREE



A quadtree T for


subdivision of

a square into


separates P.

For point sets

with large spread,

a quadtree can be

compressed.

11-8

COMPRESSED QUADTREE



A quadtree T for


subdivision of

a square into


separates P.

For point sets

with large spread,

a quadtree can be

compressed.

11-9

COMPRESSED QUADTREE



A quadtree T for


subdivision of

a square into


separates P.

For point sets

with large spread,

a quadtree can be

compressed.

11-10

COMPRESSED QUADTREE



A quadtree T for


subdivision of

a square into


separates P.

For point sets

with large spread,

a quadtree can be

compressed.

11-11

COMPRESSED QUADTREE



A quadtree T for


subdivision of

a square into


separates P.

For point sets

with large spread,

a quadtree can be

compressed.

12-1

c-CLUSTER QUADTREE



12-2

c-CLUSTER QUADTREE



A c-cluster in a

point set P is a

subset C ⊂ P whose

distance to the rest

of P is at least cits diameter.

12-3

c-CLUSTER QUADTREE



A c-cluster in a

point set P is a




12-4

c-CLUSTER QUADTREE



A c-cluster in a

point set P is a




12-5

c-CLUSTER QUADTREE



A c-cluster in a

point set P is a




12-6

c-CLUSTER QUADTREE



A c-cluster in a

point set P is a




The c-clusters on Pform a hierarchy.

12-7

c-CLUSTER QUADTREE



A c-cluster in a

point set P is a





12-8

c-CLUSTER QUADTREE



A c-cluster in a

point set P is a





12-9

c-CLUSTER QUADTREE



A c-cluster in a

point set P is a





12-10

c-CLUSTER QUADTREE



A c-cluster in a

point set P is a





12-11

c-CLUSTER QUADTREE



A c-cluster in a

point set P is a





12-12

c-CLUSTER QUADTREE



A c-cluster in a

point set P is a





12-13

c-CLUSTER QUADTREE



A c-cluster in a

point set P is a





This c-cluster tree

can have linear

degree.

12-14

c-CLUSTER QUADTREE



A c-cluster quadtree

is a c-cluster tree

augmented with a

quadtree on its

high-degree nodes.

12-15

c-CLUSTER QUADTREE




is a c-cluster tree

augmented with a

quadtree on its

high-degree nodes.

12-16

c-CLUSTER QUADTREE




is a c-cluster tree

augmented with a

quadtree on its

high-degree nodes.

12-17

c-CLUSTER QUADTREE




is a c-cluster tree

augmented with a

quadtree on its

high-degree nodes.

13-1

WELL-SEPARATED PAIRS



13-2




A c-well-separatedpair is a pair of

point sets U, Vwhose distance to

each other is at

least c times their

diameters.

13-3






each other is at

least c times their

diameters.

U

V

13-4






each other is at

least c times their

diameters.

U

V

13-5




U

V

A c-well-separatedpair decomposition

(WSPD) of a set of

points P is a set of

well-separated pairs

of subsets of P such

that every pair of

points p, q ∈ P appear

in exactly one pair

of subsets.

13-6





(WSPD) of a set of




that every pair of


in exactly one pair

of subsets.

13-7





(WSPD) of a set of




that every pair of


in exactly one pair

of subsets.

13-8





(WSPD) of a set of




that every pair of


in exactly one pair

of subsets.

13-9




A WSPD with O(|P |)pairs always exists.

13-10




A WSPD with O(|P |)pairs always exists.

One way to construct

one is from a

hierarchical

subdivision (such

as a c-clusterquadtree).

14-1



THE PART YOU’VE

BEEN WAITING FORPART III

15-1

FROM WSPD TO MST



15-2

FROM WSPD TO MST



Suppose we have a

WSPD P on P.

15-3

FROM WSPD TO MST



Suppose we have a

WSPD P on P.

15-4

FROM WSPD TO MST



Suppose we have a

WSPD P on P.

15-5

FROM WSPD TO MST



Suppose we have a

WSPD P on P.

Let G be the graph

that contains the

shortest edge between

two points in any

pair of P.

15-6

FROM WSPD TO MST



Suppose we have a

WSPD P on P.

Let G be the graph

that contains the


two points in any

pair of P.

15-7

FROM WSPD TO MST



Suppose we have a

WSPD P on P.

Let G be the graph

that contains the


two points in any

pair of P.

Claim: G has linear

size and contains the

MST of P.

16-1



REDUCING THE WEIGHT

16-2

Problem: P has

quadratic weight.



REDUCING THE WEIGHT

16-3

Problem: P has

quadratic weight.

Idea: most heavy

pairs are far away

and don’t contribute

to the MST anyway.



REDUCING THE WEIGHT

16-4

Problem: P has

quadratic weight.

Idea: most heavy

pairs are far away


to the MST anyway.

Let Pφ be the pairs

in P with general

direction φ.



REDUCING THE WEIGHT

16-5

Problem: P has

quadratic weight.

Idea: most heavy

pairs are far away


to the MST anyway.


in P with general

direction φ.φ



REDUCING THE WEIGHT

16-6

Problem: P has

quadratic weight.

Idea: most heavy

pairs are far away


to the MST anyway.


in P with general

direction φ.φ

We construct a Gφ

separately from Pφ.



REDUCING THE WEIGHT

16-7

For each p ∈ P, find

the k closest pairs

in Pφ, and remove pfrom all other pairs.

φ



REDUCING THE WEIGHT

16-8


the k closest pairs


p

φ



REDUCING THE WEIGHT

16-9


the k closest pairs


p

φ



REDUCING THE WEIGHT

16-10


the k closest pairs


p

φ



REDUCING THE WEIGHT

16-11


the k closest pairs


p

φ



REDUCING THE WEIGHT

16-12


the k closest pairs


p

φ

Let P ′φ be the

resulting pair set.



REDUCING THE WEIGHT

16-13


the k closest pairs


p

P ′, the union of

P ′φ over all φ, is

no longer a pair

decomposition of P.But...

φ

Let P ′φ be the

resulting pair set.



REDUCING THE WEIGHT

16-14

p

Claim: P ′ has linear

weight.

φ



REDUCING THE WEIGHT

16-15

p


weight.

Claim: P ′ still

contains all edges of

the MST of P.

φ



REDUCING THE WEIGHT

16-16

p


weight.

Claim: P ′ still

contains all edges of

the MST of P.

Claim: and as if

that wasn’t cool

enough, P ′ can even

be computed in linear

time. φ



REDUCING THE WEIGHT

17-1

COMPUTING CLOSEST PAIRS



φ

17-2




Now, for each pair

(U, V ) ∈ P ′φ, we need

to find the closest

pair of points.

φ

17-3




Now, for each pair


to find the closest

pair of points.

φ

17-4




Now, for each pair


to find the closest

pair of points.

U

Vφ

17-5




Now, for each pair


to find the closest

pair of points.

Claim: if the points

would be sorted on

x-coordinate, we

could find it in

linear time.

U

Vφ

17-6




Now, for each pair


to find the closest

pair of points.

Claim: if the points

would be sorted on

x-coordinate, we

could find it in

linear time.

Problem: sorting

takes Θ(n log n) time.

U

Vφ

17-7




Claim: we only need

to sort the points

whose ‘upward cone’

is empty.

U

Vφ

17-8




Claim: we only need

to sort the points


is empty.

U

Vφ

17-9




Claim: we only need

to sort the points


is empty.

U

V ′ φ

17-10




Claim: we only need

to sort the points


is empty.

U ′

V ′ φ

17-11




Claim: we only need

to sort the points


is empty.

So, we only locally

need the correct

x-order.

U ′

V ′ φ

17-12




Claim: we only need

to sort the points


is empty.

So, we only locally

need the correct

x-order.

But we have that

information: we

started with a WSPD

of P!

U ′

V ′ φ

18-1

TOPOLOGICAL SORT



φ

18-2

TOPOLOGICAL SORT



Consider Pφ+ 12π, the

pairs perpendicular

to those in Pφ.

φ

18-3

TOPOLOGICAL SORT




pairs perpendicular

to those in Pφ.

φ+ 12π

φ

18-4

TOPOLOGICAL SORT




pairs perpendicular

to those in Pφ.

Let V be a set with

two copies of each

set in Pφ+ 12π.

φ+ 12π

φ

18-5

TOPOLOGICAL SORT




pairs perpendicular

to those in Pφ.

Let V be a set with

two copies of each

set in Pφ+ 12π.

φ+ 12π

φ

18-6

TOPOLOGICAL SORT




pairs perpendicular

to those in Pφ.

Let V be a set with

two copies of each

set in Pφ+ 12π.

φ+ 12π

φ

18-7

TOPOLOGICAL SORT




pairs perpendicular

to those in Pφ.

Let Γ be a graph

on V that follows

the hierarchical

structure and WSPD

pairs.

Let V be a set with

two copies of each

set in Pφ+ 12π.

φ+ 12π

φ

18-8

TOPOLOGICAL SORT




pairs perpendicular

to those in Pφ.

Let Γ be a graph

on V that follows

the hierarchical

structure and WSPD

pairs.

Let V be a set with

two copies of each

set in Pφ+ 12π.

φ+ 12π

φ

18-9

TOPOLOGICAL SORT




pairs perpendicular

to those in Pφ.

Let Γ be a graph

on V that follows

the hierarchical

structure and WSPD

pairs.

Let V be a set with

two copies of each

set in Pφ+ 12π.

φ+ 12π

φ

18-10

TOPOLOGICAL SORT



Claim: Γ is acyclic.

φ+ 12π

φ

18-11

TOPOLOGICAL SORT




Claim: If a point

p is roughly in

direction φ + 12π as

seen from a point q,then q comes before pin Γ.

φ+ 12π

φ

18-12

TOPOLOGICAL SORT




Claim: If a point

p is roughly in

direction φ + 12π as

seen from a point q,then q comes before pin Γ.

φ+ 12π

φ

Claim: We can sort

P conforming Γ in

linear time.

19-1




Now, we can find

the closest edge ebetween U ′ and V ′ in

O(|U |+ |V |) time.

U ′

V ′ φ

19-2




Now, we can find


O(|U |+ |V |) time.

U ′

V ′ φ

19-3




Now, we can find


O(|U |+ |V |) time.

So, we can find Gφ

in O(n) time.

U ′

V ′ φ

19-4




Now, we can find


O(|U |+ |V |) time.

Yay! We found

G, a linear size

supergraph of the

MST of P, in linear

time!

So, we can find Gφ

in O(n) time.

U ′

V ′ φ

19-5




Now, we can find


O(|U |+ |V |) time.

Yay! We found

G, a linear size

supergraph of the

MST of P, in linear

time!

So, we can find Gφ

in O(n) time.

20-1



EXTRACTING THE MST

20-2



EXTRACTING THE MST

Problem: computing

a MST may take up to

O(nα(n)) time.

20-3



EXTRACTING THE MST

Problem: computing


O(nα(n)) time.

For planar graphs,

this is only O(n).

20-4



EXTRACTING THE MST

Unfortunately, Gneed not be planar.

Problem: computing


O(nα(n)) time.

For planar graphs,

this is only O(n).

20-5



EXTRACTING THE MST


Problem: computing


O(nα(n)) time.

For planar graphs,

this is only O(n).

20-6



EXTRACTING THE MST


Problem: computing


O(nα(n)) time.

For planar graphs,

this is only O(n).

20-7



EXTRACTING THE MST


Problem: computing


O(nα(n)) time.

For planar graphs,

this is only O(n).

20-8



EXTRACTING THE MST


Claim: any edge eof G crosses at most

O(1) edges of length

Ω(|e|).

Problem: computing


O(nα(n)) time.

For planar graphs,

this is only O(n).

21-1

EXTRACTING THE MST



21-2

EXTRACTING THE MST



Start with P and G.

21-3

EXTRACTING THE MST



Start with P and G.

21-4

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

21-5

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

21-6

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

21-7

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

21-8

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

21-9

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

21-10

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

21-11

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

21-12

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

21-13

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

21-14

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-15

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-16

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-17

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-18

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-19

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-20

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-21

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-22

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-23

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-24

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-25

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-26

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-27

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-28

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-29

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-30

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-31

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

21-32

EXTRACTING THE MST



Start with P and G.

Remember we have a

quadtree T on P.

We can process

the edges of G by

increasing length,

Boruvka-style.

Ignore edges within

components.

Done!

22-1

CONCLUSION



22-2

CONCLUSION



Given a quadtree on a

set of points P, we

can compute the MST

of P in linear time.

22-3

CONCLUSION



Given a quadtree on a

set of points P, we

can compute the MST

of P in linear time.

Many major proximity

structures on planar

point sets can be

derived from each

other in linear

deterministic time.

23-1



ANY QUESTIONS?

THANK YOU!

SQUARES ARE TRIANGLES

THE SQUARE THE TRIANGLE TRIANGULATING …loffl001/publications/slides/... · THE TRIANGLE TRIANGULATING THE SQUARE & 1-2 University of California, Irvine ... Triangulation Gabriel

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