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Quadtrees and Octrees
Dr. Randy M. Kaplan
Introduction
Quadtrees
Hierarchical
Spacial
Based on -
Recursive decomposition of space
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Quadtree
Representation of 2-dimensional space
Space is decomposed using separators
Parallel to the coordinate axis
Split a region into four(4) regions
Southwest
Northwest
Southeast
Northeast
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Quadtree
A node corresponds to the region
The decomposition of the region into four(4)
can be resprensented as a subtree whose parent is the original region
one(1) child for each decomposed region
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Quadtree
Region of 2D
Space
Region of 2D
Space
Region of 2D
Space
Region of 2D
Space
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Region of 2D
Space
Octree
3D Analog to Quadtree
Region is split into eight(8) sub-regions
using planes parallel to the coordinate axis
A region node will have 8 children
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Octree
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Octree
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Octree
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More Than 3 Dimensions
In the case where more than 3 dimensions need
to be represented the term for the data structure
is
HYPEROCTREE
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Quadtree/Octree
Allow space to be represented at various levels of
granularity
Allows quick focusing on regions of interest
Example
Find all points in a dataset that lie within a
given distance from a query point
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Quadtree/Octree
Find all points in a dataset that lie within a given
distance from a query point
AKA Spherical Region Query
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Spherical Region Query
In the absence of any data structure
all points in the dataset will need to be
analyzed
If a quadtree is available, large regions of points
can be eliminated from the search
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Quadtree Construction
Start with a square, cubic, or hypercubic region
(depending on dimensionality of the dataset)
Decompose the initial region according to one of
two strategies
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Quadtree Construction
Guide decomposition by input data
Guide decomposition by principal of equal
subdivision
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Quadtree Construction
Guide decomposition by input data
Resulting tree size will be proportional to input
If all data is available a priori it is possible to
make the tree height balanced
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Quadtree Construction
Guide decomposition by input data
Although the resulting tree has desirable
properties, it also has some disadvantage, i.e.,
Difficult to make the data structure dynamic
Deletion of data is difficult
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Quadtree Construction
Decomposition based on equal subdivision
Distribution of spacial data determines the
characteristics of the resulting tree
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Quadtree Construction
Decomposition based on equal subdivision
Tree will be height balanced and size will be
linear when the spatial data is distributed
uniformly
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Quadtree Construction
Decomposition based on equal subdivision
Height and size properties deteriorate as the
distribution becomes non-uniform
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Termination Criteria
In the subdivision process, when is it time to stop
subdividing?
Two criteria can be used in an OR or AND
configuration
Resolution Requirement
Condition on Region Requirement
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Ubiquity
The quadtree is a ubiquitous data structure much
as the binary tree is also a ubiquitous data
structure
It can be found applied to many different areas of
spatial computing
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Ubiquity
Computational Geometry
Computer-aided design
Computer graphics
Databases
Geographic Information Systems
Image processing
Pattern recognition
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Quadtrees for Point Data
Set of n points in d dimensional space
Principle application
Organize multidimensional data
Facilitate queries requiring spatial information
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Quadtrees for Point Data
Kinds of queries
Range query
Spherical region query
All nearest neighbors query
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Range Query
Given a range of values for each dimension,
find all points that lie within the range
Equivalent to finding all points lying within a
hyper-rectangular region
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Spherical Region Query
Given, point p, radius r
Find all the points that lie within a distance r
from p
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Nearest Neighbor Query
Given n points,
find the nearest neighbor of each point within
the set
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Point Quadtrees
Point quadtree is a natural generalization of the
binary search tree
We will consider the 2-dimensional case
Begin with a square region containing all of the
input points
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Point Quadtrees
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Point Quadtrees
How to
Make a
Quadtree
Choose an
arbitrary
point
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Point Quadtrees
The point
becomes
the root of
the tree
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Point Quadtrees
Use lines that
are parallel to
the
coordinate
axis
Draw these lines to intersect
with the chosen point to divide
the region
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Point Quadtrees
Each of the
subregions are
recursively
divided in a
similar way
In this way the point quadtree is
produced
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Point Quadtrees
When a point is on a boundary it is necessary to
adopt some conventions to specify to which
region the point belongs
Points lying on the left and bottom edges of a
region are considered included in the region
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Point Quadtrees
Points lying on the left and bottom edges of a
region are considered included in the region
Points lying on the right and top edges of a region
are considered not to be part of the region
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Point Quadtrees
When all of the points are known in advance, it is
possible to construct a balanced tree
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Point Quadtrees
A simple way to accomplish this is to sort the
points with one of the coordinates as the primary
key
Call this primary key x
The other coordinate, y, is the secondary tree
This key is also used to sort the data
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Point Quadtrees
The first subdivision point is the median of the
sorted data
This insures that none of the children of the root
node receives more than half the points
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Point Quadtrees
O(n) time will be used to sort the data contained
in each of the four subregions
The total work at every level of the tree is
bounded by O(n)
At most there will be O(log n) levels in the tree
A height balanced quadtree can be built in
O(nlogn) time
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Point Quadtrees
Search
To search for a point, compare the point at the
root with the point to find
If the points are different, then the point to
search for is used to determine where to go
next
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Point Quadtrees
Search
The search continues by recursively calling the
search procedure with the subregion’s root as
the root of the new tree to search
The search stops if the point is found or if a
leaf node is reached
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Point Quadtrees
Search
The algorithm for search is O(h) where h is the
height of the tree
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Point Quadtrees
Insertion
Insertion is accomplished by search for the
point in the tree
The search will end at a leaf
The leaf node now is a region that contains two
points
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Point Quadtrees
Insertion
The leaf node now is a region that contains two
points
One point is chosen of the two to subdivide the
region
The point becomes the child of the node
representing the region where the point
belongs
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Point Quadtrees
The run time for insertion is bounded by O(h)
In d dimensions the run time is bounded by O(dh)
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Point Quadtrees
Deletion
The node is located by search
To delete the node it is necessary to locate
another point that will take its place
This is where the complexity of the algorithm
for deletion lies
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Region Quadtrees
A region quadtree for n points in d dimensions is
defined as follows:
Consider that we have a hypercube large enough
to enclose all of the points
The region is represented by the root of the ddimensional quadtree
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Region Quadtrees
The region is subdivided into 2d subregions of
equal size
A subregion is subdivided by bisecting along each
dimension
Each of these regions containing at least one
point is represented as a child of the root node
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Region Quadtrees
Each of the regions that contain at least one point
is represented as the child of the root node
The same procedure is recursively applied to
each child of the root node
The process is terminated when a region contains
only a single point
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Region Quadtrees
This data structure is known as the
point region quadtree or
PR-quadtree
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Region Quadtrees
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2-D Polygonal Objects
Suppose we restrict ourselves to 2-D polygonal
objects
If we recursively divide the tree nodes yields
minimum size nodes all along the edges of the
polygon
The resulting tree is large and the number of nondivisible cells is proportional to the polygon’s
perimeter
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Algorithm
The algorithm for building the tree from boundary
information is simple
Recognition of boundaries from the quadtree is
complex
The algorithm must infer straight edges from
staircaselike contours
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A Different Quadtree
In this quadtree, EDGE nodes are considered,
including WHITE nodes
BLACK nodes
GRAY nodes
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EDGE Nodes
Partly inside and partly outside the object
The piece of the object boundary that is contained
in the node is a simple straight segment
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Features of Quadtrees
(1) The resulting trees are more compact and
shorter than the usual quadtree
(2) The recomputation of the boundary model is
easy and exact
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Quadtree Representation
The quadtree representation has two data
structures
First - a tree structure in which data is stored
linearly by a depth method
The second is a sequential list storing the edges
that define the polygonal object boundary
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An Example
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Compact Representation
We will be using a compact representation for the
tree - a so-called list representation
The primitive elements of the list will be X, W, G,
B
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Compact Representation
W
WHITE is used if the quadrant is entirely
outside the object the node represents
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Compact Representation
B
BLACK if the quadrant is entirely inside the
object boundary
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Compact Representation
X
EDGE if and only if one boundary edge and
only one edge crosses the quadrant
EDGE nodes have a second field storing a
pointer to the edge crossing the quadrant
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Compact Representation
G
A node is GRAY when more than one boundary
edge crosses the quadrant
Represent either terminal or non-terminal
nodes
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Compact Representation
((X(X(WWGW)XX)((WWWG)XWX)B)XX
((B(XXBG)BX)W(XXX(GWWW))W))
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Compact Representation
The coding of
quadrants is
done in this
order.
((X(X(WWGW)XX)((WWWG)XWX)B)XX
((B(XXBG)BX)W(XXX(GWWW))W))
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