THE SQUARE THE TRIANGLE TRIANGULATING …loffl001/publications/slides/... · THE TRIANGLE...
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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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
3-2
Let P be a set of npoints in the plane.
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
3-3
Let P be a set of npoints in the plane.
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
3-4
Let P be a set of npoints in the plane.
A proximity structure
on P is ‘a structure
that stores some
kind of useful local
information’.
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
3-5
Let P be a set of npoints in the plane.
A proximity structure
on P is ‘a structure
that stores some
kind of useful local
information’.
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
3-6
Let P be a set of npoints in the plane.
A proximity structure
on P is ‘a structure
that stores some
kind of useful local
information’.
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
3-7
Let P be a set of npoints in the plane.
A proximity structure
on P is ‘a structure
that stores some
kind of useful local
information’.
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
4-1
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
4-2
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
All these structures
take Ω(n log n) time to
build.
4-3
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-4
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-5
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-6
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-7
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-8
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-9
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-10
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-11
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-12
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-13
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-14
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-15
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-16
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-17
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-18
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-19
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-20
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
4-21
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Intuitively, though,
once you built one,
you should be able to
derive the others.
All these structures
take Ω(n log n) time to
build.
Since information
is local, such a
conversion should
take O(n) time.
5-1
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
5-2
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Unfortunately, not
all point sets behave
themselves.
5-3
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Unfortunately, not
all point sets behave
themselves.
A single point may
have more than Ω(1)neighbours.
5-4
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Unfortunately, not
all point sets behave
themselves.
A single point may
have more than Ω(1)neighbours.
5-5
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Unfortunately, not
all point sets behave
themselves.
A single point may
have more than Ω(1)neighbours.
5-6
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Unfortunately, not
all point sets behave
themselves.
A single point may
have more than Ω(1)neighbours.
The scale of a point
set need not be
uniform.
5-7
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Unfortunately, not
all point sets behave
themselves.
A single point may
have more than Ω(1)neighbours.
The scale of a point
set need not be
uniform.
5-8
PROXIMITY STRUCTURES
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Unfortunately, not
all point sets behave
themselves.
A single point may
have more than Ω(1)neighbours.
The scale of a point
set need not be
uniform.
What can we do?
6-1
RESEARCHER’S DILEMMA
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
6-2
RESEARCHER’S DILEMMA
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Do we...
6-3
RESEARCHER’S DILEMMA
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
RESEARCHER’S DILEMMA
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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?
-OR-
6-5
RESEARCHER’S DILEMMA
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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?
-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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY
7-2
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
7-3
Delaunay
Triangulation
[Delaunay, 1934]
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Deterministic linear time
7-4
Delaunay
Triangulation
Gabriel
Graph
[Delaunay, 1934][Gabriel, 1969]
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Deterministic linear time
7-5
Delaunay
Triangulation
Gabriel
Graph
Minimum
Spanning Tree
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Deterministic linear time
7-6
Delaunay
Triangulation
Gabriel
Graph
Minimum
Spanning Tree
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour Graph
Deterministic linear time
7-7
Delaunay
Triangulation
Gabriel
Graph
Minimum
Spanning Tree
Compressed
Quadtree
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour Graph
Deterministic linear time
[Clarkson, 1983]
7-8
Delaunay
Triangulation
Gabriel
Graph
Minimum
Spanning Tree
Compressed
Quadtree
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
Deterministic linear time
[Clarkson, 1983]
7-9
Delaunay
Triangulation
Gabriel
Graph
Minimum
Spanning Tree
Compressed
Quadtree
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
Deterministic linear time
[Clarkson, 1983]
7-10
Delaunay
Triangulation
Gabriel
Graph
Minimum
Spanning Tree
Compressed
Quadtree
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
[Chazelle, 1991]
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
Deterministic linear time
[Chin & Wang, 1998]
[Clarkson, 1983]
7-11
Delaunay
Triangulation
Gabriel
Graph
Minimum
Spanning Tree
Compressed
Quadtree
DT
On Superset
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
[Chazelle, 1991]
[Bern,
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
Eppstein & Gilbert, 1990]
Deterministic linear time
[Chin & Wang, 1998]
[Clarkson, 1983]
7-12
Delaunay
Triangulation
Gabriel
Graph
Minimum
Spanning Tree
Compressed
Quadtree
DT
On Superset
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
[Chazelle, 1991]
[Bern,
[Chazelle, Devillers,
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
Eppstein & Gilbert, 1990]
Hurtado, Mora,Sacristan & Teillaud, 2001]
Deterministic linear time
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)
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
[Chazelle, 1991]
[Bern,
[Chazelle, Devillers,
[Eppstein,
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
Eppstein & Gilbert, 1990]
Hurtado, Mora,Sacristan & Teillaud, 2001]
Goodrich & Sun, 2005]Deterministic linear time
Randomised 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)
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
[Chazelle, 1991]
[Bern,
[Chazelle, Devillers,
[Eppstein,
WSPD
Sequence
NNG
Sequence
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
Eppstein & Gilbert, 1990]
Hurtado, Mora,Sacristan & Teillaud, 2001]
Goodrich & Sun, 2005]Deterministic linear time
Randomised linear time
[Chin & Wang, 1998]
[Clarkson, 1983]
7-15
Delaunay
Triangulation
Gabriel
Graph
Minimum
Spanning Tree
Compressed
Quadtree
DT
On Superset
QT Sequence
(Skip Quadtree)
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
[Chazelle, 1991]
[Bern,
[Chazelle, Devillers,
[Eppstein,
WSPD
Sequence
NNG
Sequence
[Buchin & Mulzer, 2009]
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
Eppstein & Gilbert, 1990]
Hurtado, Mora,Sacristan & Teillaud, 2001]
Goodrich & Sun, 2005]Deterministic linear time
Randomised linear time
[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
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
[Chazelle, 1991]
[Bern,
[Chazelle, Devillers,
[Eppstein,
WSPD
Sequence
NNG
Sequence
[Buchin & Mulzer, 2009]
[Krznaric &
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
Eppstein & Gilbert, 1990]
Hurtado, Mora,Sacristan & Teillaud, 2001]
Goodrich & Sun, 2005]
Levcopoulos, 1998]
Deterministic linear time
Randomised linear time
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
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
[Chazelle, 1991]
[Bern,
[Chazelle, Devillers,
[Eppstein,
WSPD
Sequence
NNG
Sequence
[Buchin & Mulzer, 2009]
[Krznaric &
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
[Us, 2011]
Eppstein & Gilbert, 1990]
Hurtado, Mora,Sacristan & Teillaud, 2001]
Goodrich & Sun, 2005]
Levcopoulos, 1998]
Deterministic linear time
Randomised linear time
Linear time with floor operation
Deterministic linear time
[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
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
[Chazelle, 1991]
[Bern,
[Chazelle, Devillers,
[Eppstein,
WSPD
Sequence
NNG
Sequence
[Buchin & Mulzer, 2009]
[Krznaric &
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
[Us, 2011]
Eppstein & Gilbert, 1990]
Hurtado, Mora,Sacristan & Teillaud, 2001]
Goodrich & Sun, 2005]
Levcopoulos, 1998]
Deterministic linear time
Randomised linear time
Linear time with floor operation
Deterministic linear time
[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
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
[Chazelle, 1991]
[Bern,
[Chazelle, Devillers,
[Eppstein,
WSPD
Sequence
NNG
Sequence
[Buchin & Mulzer, 2009]
[Krznaric &
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
[Us, 2011]
Eppstein & Gilbert, 1990]
Hurtado, Mora,Sacristan & Teillaud, 2001]
Goodrich & Sun, 2005]
Levcopoulos, 1998]
Deterministic linear time
Randomised linear time
Linear time with floor operation
Deterministic linear time
[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
[Delaunay, 1934][Gabriel, 1969]
[Matsui, 1995]
[Preparata & Shamos, 1985]
[Callahan & Kosaraju, 1995]
[Chazelle, 1991]
[Bern,
[Chazelle, Devillers,
[Eppstein,
WSPD
Sequence
NNG
Sequence
[Buchin & Mulzer, 2009]
[Krznaric &
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A BRIEF HISTORY[Dirichlet, 1850]
c-CQTOn Superset
Voronoi
Diagram
Nearest
Neighbour GraphWell-Separated
Pair
Decomposition
[Us, 2011]
Eppstein & Gilbert, 1990]
Hurtado, Mora,Sacristan & Teillaud, 2001]
Goodrich & Sun, 2005]
Levcopoulos, 1998]
Deterministic linear time
Randomised linear time
Linear time with floor operation
Deterministic linear time
[Chin & Wang, 1998]
[Clarkson, 1983]
8-1
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
PRELIMINARIESPART II
9-1
DELAUNAY TRIANGULATION
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
9-2
DELAUNAY TRIANGULATION
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
DELAUNAY TRIANGULATION
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
9-4
DELAUNAY TRIANGULATION
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
9-5
DELAUNAY TRIANGULATION
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
9-6
DELAUNAY TRIANGULATION
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
9-7
DELAUNAY TRIANGULATION
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
9-8
DELAUNAY TRIANGULATION
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
9-9
DELAUNAY TRIANGULATION
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
9-10
DELAUNAY TRIANGULATION
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
10-1
MINIMUM SPANNING TREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
10-2
MINIMUM SPANNING TREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
The Euclidean minimum
spanning tree (EMST)
is the minimum
spanning tree of
the complete graph on
a point set P.
10-3
MINIMUM SPANNING TREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
The Euclidean minimum
spanning tree (EMST)
is the minimum
spanning tree of
the complete graph on
a point set P.
10-4
MINIMUM SPANNING TREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
The Euclidean minimum
spanning tree (EMST)
is the minimum
spanning tree of
the complete graph on
a point set P.
10-5
MINIMUM SPANNING TREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
The Euclidean minimum
spanning tree (EMST)
is the minimum
spanning tree of
the complete graph on
a point set P.
The EMST is a
subgraph of the DT.
10-6
MINIMUM SPANNING TREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
The Euclidean minimum
spanning tree (EMST)
is the minimum
spanning tree of
the complete graph on
a point set P.
The EMST is a
subgraph of the DT.
11-1
COMPRESSED QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
11-2
COMPRESSED QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A quadtree T for
a set of points Pis a hierarchical
subdivision of
a square into
smaller squares that
separates P.
11-4
COMPRESSED QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A quadtree T for
a set of points Pis a hierarchical
subdivision of
a square into
smaller squares that
separates P.
11-5
COMPRESSED QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A quadtree T for
a set of points Pis a hierarchical
subdivision of
a square into
smaller squares that
separates P.
11-6
COMPRESSED QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A quadtree T for
a set of points Pis a hierarchical
subdivision of
a square into
smaller squares that
separates P.
11-7
COMPRESSED QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A quadtree T for
a set of points Pis a hierarchical
subdivision of
a square into
smaller squares that
separates P.
For point sets
with large spread,
a quadtree can be
compressed.
11-8
COMPRESSED QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A quadtree T for
a set of points Pis a hierarchical
subdivision of
a square into
smaller squares that
separates P.
For point sets
with large spread,
a quadtree can be
compressed.
11-9
COMPRESSED QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A quadtree T for
a set of points Pis a hierarchical
subdivision of
a square into
smaller squares that
separates P.
For point sets
with large spread,
a quadtree can be
compressed.
11-10
COMPRESSED QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A quadtree T for
a set of points Pis a hierarchical
subdivision of
a square into
smaller squares that
separates P.
For point sets
with large spread,
a quadtree can be
compressed.
11-11
COMPRESSED QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A quadtree T for
a set of points Pis a hierarchical
subdivision of
a square into
smaller squares that
separates P.
For point sets
with large spread,
a quadtree can be
compressed.
12-1
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
12-2
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
12-4
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
12-5
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
12-6
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
The c-clusters on Pform a hierarchy.
12-7
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
The c-clusters on Pform a hierarchy.
12-8
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
The c-clusters on Pform a hierarchy.
12-9
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
The c-clusters on Pform a hierarchy.
12-10
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
The c-clusters on Pform a hierarchy.
12-11
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
The c-clusters on Pform a hierarchy.
12-12
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
The c-clusters on Pform a hierarchy.
12-13
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
The c-clusters on Pform a hierarchy.
This c-cluster tree
can have linear
degree.
12-14
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A c-cluster quadtree
is a c-cluster tree
augmented with a
quadtree on its
high-degree nodes.
12-15
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A c-cluster quadtree
is a c-cluster tree
augmented with a
quadtree on its
high-degree nodes.
12-16
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A c-cluster quadtree
is a c-cluster tree
augmented with a
quadtree on its
high-degree nodes.
12-17
c-CLUSTER QUADTREE
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A c-cluster quadtree
is a c-cluster tree
augmented with a
quadtree on its
high-degree nodes.
13-1
WELL-SEPARATED PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
13-2
WELL-SEPARATED PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
WELL-SEPARATED PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A c-well-separatedpair is a pair of
point sets U, Vwhose distance to
each other is at
least c times their
diameters.
U
V
13-4
WELL-SEPARATED PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A c-well-separatedpair is a pair of
point sets U, Vwhose distance to
each other is at
least c times their
diameters.
U
V
13-5
WELL-SEPARATED PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
WELL-SEPARATED PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
13-7
WELL-SEPARATED PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
13-8
WELL-SEPARATED PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
13-9
WELL-SEPARATED PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
A WSPD with O(|P |)pairs always exists.
13-10
WELL-SEPARATED PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
THE PART YOU’VE
BEEN WAITING FORPART III
15-1
FROM WSPD TO MST
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
15-2
FROM WSPD TO MST
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Suppose we have a
WSPD P on P.
15-3
FROM WSPD TO MST
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Suppose we have a
WSPD P on P.
15-4
FROM WSPD TO MST
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Suppose we have a
WSPD P on P.
15-5
FROM WSPD TO MST
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
15-7
FROM WSPD TO MST
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
Claim: G has linear
size and contains the
MST of P.
16-1
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-2
Problem: P has
quadratic weight.
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-4
Problem: P has
quadratic weight.
Idea: most heavy
pairs are far away
and don’t contribute
to the MST anyway.
Let Pφ be the pairs
in P with general
direction φ.
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-5
Problem: P has
quadratic weight.
Idea: most heavy
pairs are far away
and don’t contribute
to the MST anyway.
Let Pφ be the pairs
in P with general
direction φ.φ
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-6
Problem: P has
quadratic weight.
Idea: most heavy
pairs are far away
and don’t contribute
to the MST anyway.
Let Pφ be the pairs
in P with general
direction φ.φ
We construct a Gφ
separately from Pφ.
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-7
For each p ∈ P, find
the k closest pairs
in Pφ, and remove pfrom all other pairs.
φ
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-8
For each p ∈ P, find
the k closest pairs
in Pφ, and remove pfrom all other pairs.
p
φ
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-9
For each p ∈ P, find
the k closest pairs
in Pφ, and remove pfrom all other pairs.
p
φ
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-10
For each p ∈ P, find
the k closest pairs
in Pφ, and remove pfrom all other pairs.
p
φ
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-11
For each p ∈ P, find
the k closest pairs
in Pφ, and remove pfrom all other pairs.
p
φ
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-12
For each p ∈ P, find
the k closest pairs
in Pφ, and remove pfrom all other pairs.
p
φ
Let P ′φ be the
resulting pair set.
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-13
For each p ∈ P, find
the k closest pairs
in Pφ, and remove pfrom all other 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.
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-14
p
Claim: P ′ has linear
weight.
φ
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-15
p
Claim: P ′ has linear
weight.
Claim: P ′ still
contains all edges of
the MST of P.
φ
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
16-16
p
Claim: P ′ has linear
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
REDUCING THE WEIGHT
17-1
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
φ
17-2
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Now, for each pair
(U, V ) ∈ P ′φ, we need
to find the closest
pair of points.
φ
17-3
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Now, for each pair
(U, V ) ∈ P ′φ, we need
to find the closest
pair of points.
φ
17-4
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Now, for each pair
(U, V ) ∈ P ′φ, we need
to find the closest
pair of points.
U
Vφ
17-5
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Now, for each pair
(U, V ) ∈ P ′φ, we need
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
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Now, for each pair
(U, V ) ∈ P ′φ, we need
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
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Claim: we only need
to sort the points
whose ‘upward cone’
is empty.
U
Vφ
17-8
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Claim: we only need
to sort the points
whose ‘upward cone’
is empty.
U
Vφ
17-9
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Claim: we only need
to sort the points
whose ‘upward cone’
is empty.
U
V ′ φ
17-10
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Claim: we only need
to sort the points
whose ‘upward cone’
is empty.
U ′
V ′ φ
17-11
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Claim: we only need
to sort the points
whose ‘upward cone’
is empty.
So, we only locally
need the correct
x-order.
U ′
V ′ φ
17-12
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Claim: we only need
to sort the points
whose ‘upward cone’
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
φ
18-2
TOPOLOGICAL SORT
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Consider Pφ+ 12π, the
pairs perpendicular
to those in Pφ.
φ
18-3
TOPOLOGICAL SORT
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Consider Pφ+ 12π, the
pairs perpendicular
to those in Pφ.
φ+ 12π
φ
18-4
TOPOLOGICAL SORT
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Consider Pφ+ 12π, the
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Consider Pφ+ 12π, the
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Consider Pφ+ 12π, the
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Consider Pφ+ 12π, the
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Consider Pφ+ 12π, the
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Consider Pφ+ 12π, the
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Claim: Γ is acyclic.
φ+ 12π
φ
18-11
TOPOLOGICAL SORT
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Claim: Γ is acyclic.
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Claim: Γ is acyclic.
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
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Now, we can find
the closest edge ebetween U ′ and V ′ in
O(|U |+ |V |) time.
U ′
V ′ φ
19-2
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Now, we can find
the closest edge ebetween U ′ and V ′ in
O(|U |+ |V |) time.
U ′
V ′ φ
19-3
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Now, we can find
the closest edge ebetween U ′ and V ′ in
O(|U |+ |V |) time.
So, we can find Gφ
in O(n) time.
U ′
V ′ φ
19-4
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Now, we can find
the closest edge ebetween U ′ and V ′ in
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
COMPUTING CLOSEST PAIRS
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Now, we can find
the closest edge ebetween U ′ and V ′ in
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
EXTRACTING THE MST
20-2
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
EXTRACTING THE MST
Problem: computing
a MST may take up to
O(nα(n)) time.
20-3
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
EXTRACTING THE MST
Problem: computing
a MST may take up to
O(nα(n)) time.
For planar graphs,
this is only O(n).
20-4
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
EXTRACTING THE MST
Unfortunately, Gneed not be planar.
Problem: computing
a MST may take up to
O(nα(n)) time.
For planar graphs,
this is only O(n).
20-5
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
EXTRACTING THE MST
Unfortunately, Gneed not be planar.
Problem: computing
a MST may take up to
O(nα(n)) time.
For planar graphs,
this is only O(n).
20-6
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
EXTRACTING THE MST
Unfortunately, Gneed not be planar.
Problem: computing
a MST may take up to
O(nα(n)) time.
For planar graphs,
this is only O(n).
20-7
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
EXTRACTING THE MST
Unfortunately, Gneed not be planar.
Problem: computing
a MST may take up to
O(nα(n)) time.
For planar graphs,
this is only O(n).
20-8
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
EXTRACTING THE MST
Unfortunately, Gneed not be planar.
Claim: any edge eof G crosses at most
O(1) edges of length
Ω(|e|).
Problem: computing
a MST may take up to
O(nα(n)) time.
For planar graphs,
this is only O(n).
21-1
EXTRACTING THE MST
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
21-2
EXTRACTING THE MST
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Start with P and G.
21-3
EXTRACTING THE MST
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Start with P and G.
21-4
EXTRACTING THE MST
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Start with P and G.
Remember we have a
quadtree T on P.
21-5
EXTRACTING THE MST
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Start with P and G.
Remember we have a
quadtree T on P.
21-6
EXTRACTING THE MST
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
22-2
CONCLUSION
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
Given a quadtree on a
set of points P, we
can compute the MST
of P in linear time.
22-3
CONCLUSION
MAARTEN LOFFLER & WOLFGANG MULZER
: QUADTREES AND DELAUNAY TRIANGULATIONS ARE EQUIVALENT
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.