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)
QUAD TREES
Non-Uniform Mesh Generation
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.
Uniform vs Non-Uniform Mesh
Uniform Mesh
• Sufficiently refined.
• Too many mesh elements (time & space)
U = 2j
U
integer
grid
U
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
NW
2
SW
3
NE
1
SE
4
1
NE
2
NW
3
SW
4
SE
Quad Trees for point sets
22
21
1
23
1
2
242 241
243 244
32
24
21
31
22
23
31
4
33
4
3
34
241 242
243
244
32
33
34
Quad Trees for point sets
• square/node: s
• Point set P  [xs : x’s ]  [ys : y’s ]
• xmid = (xs + x’s )/2 ,
• PNE =
ymid = (ys + y’s )/2
corner
{ p P | px > xmid , py > ymid }
• PNW = { p P | px  xmid , py > ymid }
• PSW = { p P | px  xmid , py  ymid }
• PSE
= { p P | px > xmid , py  ymid }
• 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/2i , diagonal length (S2)/2i  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), EN(s), WN(s) defined similarly.
EN(s)
s
SN(s)
SN(s)
EN(s)
s
NN(s)
WN(s)
NN(s)
WN(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.
3.
4.
5.
6.
7.
while L   do
remove a leaf s from L
if s has to split then do
add 4 children sNE , sNW , sSE , sSW to s in T &
update their object contents
insert sNE , sNW , sSE , sSW into L
check if sNE , sNW , sSE , sSW 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?
Even though some leaf in s1 causes leaf s2
to split, that splitting cannot propagate to the
neighbor s3 of the same size.
[Proof by induction on node depth difference.]
s1
NW
NE
SW
SE
s2
s3
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?
Even though some leaf in s1 causes leaf s2
to split, that splitting cannot propagate to the
neighbor s3 of the same size.
[Proof by induction on node depth difference.]
s1
range of influence:
charge 1 to
neighbor of
equal size
that causes
splitting
NW
NE
SW
SE
s2
s3
each square is
charged  8 times.
From Quad Trees to Meshes
 Root square contains polygonal objects, all vertices at integer grid points (U=2j),
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) log2 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) log2 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