Quadtrees and Octrees

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Quadtrees and Octrees

• Dr. Randy M. Kaplan

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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
Region of 2D Space
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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 subregions 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
  d-dimensional 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
  non-divisible cells is proportional to the
  polygons 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(GW
WW))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(GW
WW))W))