Slide 13

swimlogisticsElectronics - Devices

Nov 26, 2013 (3 years and 6 months ago)

102 views

COSC 6114

Prof. Andy Mirzaian

References
:




[M. de Berge et al] chapter 14


Marshal Bern [1997],
“Triangulations,”

in

Handbook of Discrete and Computational Geometry,
J.E. Goodman & J. O’Rourke (editors), chapter 22,

413
-
428.

Applications
:



VLSI circuits in electrical devices



heat emission


a design issue



experiment with prototype



automated simulation by finite
-
element methods



MESH (uniform / non
-
uniform)

Applications & Methodology
:



VLSI circuits in electrical devices



heat emission


a design issue



experiment with prototype;


reject faulty design


(expensive, time consuming)



automated simulation by


finite
-
element methods



subdivide regions into sufficiently fine


triangles or quadrangles:


MESH (uniform/non
-
uniform)



what is the influence of one mesh


element on neighboring elements?



mesh should be more refined near the


“objects” and gracefully get coarser further away



avoid mesh elements that have extreme


aspect ratios (bounded angles)



mesh elements should respect object borders.

QUAD TREES

Non
-
Uniform Mesh Generation

Uniform vs Non
-
Uniform Mesh

Uniform Mesh



Sufficiently refined.



Too many mesh elements (time & space)

U

U

U = 2
j


integer

grid

Uniform vs Non
-
Uniform Mesh

Non
-
Uniform Mesh



Conforming



Respect Input



Well Shaped: bounded aspect ratio (e.g., angels


[45


: 90

])



Steiner Points

Uniform vs Non
-
Uniform Mesh

Unifrom Mesh



too many triangles

Uniform vs Non
-
Uniform Mesh

Non
-
Uniform mesh


based on:

Constrained

Delaunay Triangulation



small angels

Uniform vs Non
-
Uniform Mesh

Non
-
Uniform Mesh


based on:

QUAD TREE

Quad Trees for point sets

NE

1

NW

2

SE

4

SW

3

1

NE

2

NW

4

SE

3

SW

Quad Trees for point sets

1

21

22

23

242

241

243

244

31

32

33

34

4

1

4

2

3

21

22

23

24

31

32

33

34

241

242

243

244

Quad Trees for point sets

corner



square/node:
s




Point set P


[x
s

: x’
s

]


[y
s

: y’
s

]




x
mid

= (x
s

+ x’
s

)/2 ,
y
mid

= (y
s

+ y’
s

)/2




P
NE

= { p

P ⁼† p
x

> x
mid
, p
y

> y
mid

}



P
NW

= { p

P ⁼†⁰
x



x
mid
, p
y

> y
mid

}



P
SW

= { p

P ⁼ †p
x



x
mid
, p
y



y
mid

}



P
SE

= { p

P†簠 ⁰
x

> x
mid
, p
y



y
mid

}




Keep
splitting

a square if it contains
more than one

data point.

LEMMA:

P = a set of points in the plane,



S = side length of root square for Quad Tree of P,



C = smallest distance between pair of points in P,



D = depth of Quad Tree.

Then, D


log S/C + 3/2 .

Proof:




Any internal node contains at least 2 points of P.



Node at depth i has side
-
length S/2
i

, diagonal length (S

2)
/2
i



C



So, i


log (S

2)
/C = log S/C + ½ .



Deepest leaf has depth one more.



Balanced Quad Trees

Scene gives

un
-
balanced Quad Tree

neighbor squares with

un
-
balanced sides

resulting mesh has

tiny angles

To achieve conformity and satisfy angle
-
bounds (aspect ratio),

We need to
balance

the QuadTree.


Balanced QuadTree:



side
-
length ratio of any 2 neighboring squares = O(1).

Balanced Quad Trees



Two squares are neighbors if they have overlapping sides but disjoint interiors.



Quad Tree is
balanced

if any two
leaf

neighboring squares


differ in side
-
length by at most 2, i.e, have
depth difference


1
.

Balanced Quad Trees



Two squares are neighbors if they have overlapping sides but disjoint interiors.



Quad Tree is
balanced

if any two
leaf

neighboring squares


differ in side
-
length by at most 2, i.e, have
depth difference


1
.

BALANCING

Balanced Quad Trees



s

= a
leaf
of Quad Tree
T



NN(
s
⤠‽

North Neighbor of
s,
is a node

s
’ in
T

s.t.


(i) north edge of
s

is shared by
s



(ii) depth(
s
’)


depth(
s
)


(i.e., square
s
’ not smaller than
s
)


(iii)
s
’ is the smallest such square in
T.



SN(
s
⤬⁅)(
s
⤬⁗丨
s
)

defined similarly.

EN
(s)

WN
(s)

SN
(s)

NN
(s)

EN
(s)

SN
(s)

WN
(s)

NN
(s)

s

s

Balanced Quad Trees

LEMMA:

Given a leaf
s

in
T

,

NN(
s
), NN(
s
), NN(
s
), NN(
s
) can be obtained in O( depth(
s
)).

Proof:
Let
s
’ = NN(
s
). From

s
, go up in
T

to lowest common ancestor


of
s

&
s
’, then come down to
s
’.

LEMMA:

Leaf
s

should split




leaf
s

in
T

, s.t.



s

is one of NN(
s

)
, NN(
s
’), NN(
s
’), or NN(
s
’), &




depth(
s
’)


2 +
depth(
s
).

These take O(D) time, D = height(
T

).

Balancing a Quad Tree

ALGORITHM
BalancedQuadTree(
T

)

Input
: Quad Tree
T

Output
: a balanced version of
T


1.

L


list of all leaves of
T

2.

while
L





do

3.



remove a leaf
s

from
L

4.



if

s

has to split
then do

5.



add 4 children
s
NE

,
s
NW

,
s
SE

,
s
SW

to
s

in
T
&



update their object contents

6.



insert
s
NE

,
s
NW

,
s
SE

,
s
SW

into
L


7.



check if
s
NE

,
s
NW

,
s
SE

,
s
SW

have neighbors that




should split & add them to
L

8.

end
-
if

9.

end
-
while

10.

return

T

end

THEOREM:

Let
T

be a quad
-
tree with m nodes and height D.


Then, the above algorithm constructs a balanced version


of
T

that has O(m) nodes in O(Dm) time.

THEOREM:

Let
T

be a quad
-
tree with m nodes and height D.


Then, the above algorithm constructs a balanced version


of
T

that has O(m) nodes in O(Dm) time.

Proof:
Step 1: takes O(m) time.


Steps 2
-
9: each iteration adds 4
-
1 = 3 nodes to
T
& takes O(D) time



to check O(1) neighboring nodes in
L

.

Therefore, total time for the while
-
loop is O(D


# new nodes added).

What is the number of added new nodes?

THEOREM:

Let
T

be a quad
-
tree with m nodes and height D.


Then, the above algorithm constructs a balanced version


of
T

that has O(m) nodes in O(Dm) time.

Proof:
Step 1: takes O(m) time.


Steps 2
-
9: each iteration adds 4
-
1 = 3 nodes to
T
& takes O(D) time



to check O(1) neighboring nodes in
L

.

Therefore, total time for the while
-
loop is O(D


# new nodes added).

What is the number of added new nodes?

NE

NW

SE

SW

s
2

s
3

Even though some leaf in
s
1

causes leaf
s
2


to split, that splitting cannot propagate to the

neighbor
s
3

of the same size.

[Proof by induction on node depth difference.]

s
1

THEOREM:

Let
T

be a quad
-
tree with m nodes and height D.


Then, the above algorithm constructs a balanced version


of
T

that has O(m) nodes in O(Dm) time.

Proof:
Step 1: takes O(m) time.


Steps 2
-
9: each iteration adds 4
-
1 = 3 nodes to
T
& takes O(D) time



to check O(1) neighboring nodes in
L

.

Therefore, total time for the while
-
loop is O(D


# new nodes added).

What is the number of added new nodes?

NE

NW

SE

SW

s
2

s
3

Even though some leaf in
s
1

causes leaf
s
2


to split, that splitting cannot propagate to the

neighbor
s
3

of the same size.

[Proof by induction on node depth difference.]

s
1

range of influence:

charge 1 to

neighbor of

equal size

that causes

splitting

each square is

charged


㠠瑩浥献

From Quad Trees to Meshes



Root square contains polygonal objects, all vertices at integer grid points (
U=2
j
),


all edges are at angels 0

, 45

, 90

, 135

.



Stop splitting when the square no longer intersects with any object edge, or when


it has unit size.



Balance the resulting Quad Tree



for

each leaf square
s

do:


-

if there is a diagonal object edge, fine; otherwise, add one such diagonal.


-

to make
s

conform to a neighboring smaller leaf, add O(1) horizontal or


vertical or diagonal lines also.

all angels

45


or 90

.

THEOREM:

Let S = a set of disjoint polygonal components inside


the square [0 : U]

[0 : U] with properties stated earlier, &


p(S) = total perimeter length of components in S.

Then, there is a non
-
uniform triangular mesh for S that is:



conforming,



respects the input,



triangle angles are 45


or 90

,



# triangles = O(p(S) log U),



Construction time = O(p(S) log
2

U).

Proof:




# triangles incident to an input segment of length L is


2(L+2).



Thus, # cells in the quad
-
tree at the same depth is O(p(S)).



Depth of quad
-
tree is at most O(log U).



Thus, total # nodes in quad
-
tree is O(p(S) log U).



The rest follows from previous discussion:


m = O(p(S) log U)


D = O(log U)


O(Dm) = O(p(S) log
2

U).

Notes & Comments



[Bern 1997]: any polygonal domain with n vertices & no obtuse angles


has a mesh consisting of O(n) non
-
obtuse triangles (i.e., all angels


90

).




Other applications of quad
-
trees (oct
-
trees in

3
):



computer graphics



image analysis



range queries



hidden surface removal



ray tracing



medial axis transforms



overlay of raster maps



nearest neighbor query processing










Exercises

1.
Suppose a triangular mesh is needed inside a rectangle whose sides have length 1
and length k > 1. Steiner points may not be used on the sides, but they may be used
inside the rectangles. Also assume that all triangles msut have angles between 30


and
90

. Is it always possible to create a triangular mesh with these properties?
Suppose it is possible to create a mesh for a particular input, what is the minimum
number of Steiner points needed?


2.
The algorithm described produces non
-
obtuse triangulated mesh (provided all angles
in the input have no obtuse angles). Prove that if a triangulation of a set P of points in
the plane contains only non
-
obtuse triangles, then it must be the Delaunay
triangulation of P.


3.
Describe an algorithm to construct an oct
-
tree of a given set P of n points in 3D.


4.
It is possible to reduce the size of a quad
-
tree of height D for a set of points (with real

coordinates) inside a square from O((D+1)n) to D(n). The idea is to discard any node
v that has only one child under which points are stored. The node is discarded by
replacing the pointer from the parent of v to v with the pointer from parent to the only
interesting child of v. Prove that the resulting tree has linear size. Can you also
improve upon the O((D+1)n) construction time?


5.
We called a quad
-
tree balanced if every two adjacent squares of the quad
-
tree
subdivision differ by no more that a factor of two in size. To save a constant factor in
the number of extra nodes needed to balance a quad
-
tree, we could weaken the
balance condition by allowing adjacent squares to differ by a factor of four in size.
Can you still complete such a weakly balanced quad
-
tree subdivision to a mesh such
that all angles are between
45


and
90


by using only O(1) triangles per square?

6.
Suppose we make the balancing condition for quad
-
trees more severe: we no longer allow

adjacent squares to differ by a factor two in sieze, but we require them to have exactly the

same size. Is the number of nodes in the new balanced version still linear in the number of

nodes of the original quad
-
tree? If not, can you say anything about this number?


7.
A quad
-
tree can also be used to store a subdivision for efficient point location. The idea is to

keep splitting a bounding square of the subdivision until all leaf nodes correspond to squares

that contain at most one vertex and only edges incident to that vertex, or no vertex and at most

one edge.

(a) Since a vertex can be incident to many edges, we need an additional data structure at the


quad
-
tree leaves storing vertices. Which data structure would you use?

(b) Describe the algorithm for constructing the point location data structure in detail, and


analyze its running time.

(c) Describe the query algorithm in detail, and analyze its running time.


8.
Quad
-
tree can be used to perform range queries. Describe an algorithm for querying a quad
-
tree

on a set P of points with a query region R. Analyze the worst
-
case query time for the case

where R is a rectangle, and for the case where R is a half
-
plane bounded by a vertical line.


9.
Compare quad
-
tree, kD
-
trees and range trees (studies previously). Discuss the advantages

and disadvantages of each of these data structures.



END