Tree structures for Spatial Data

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Tree structures for Spatial Data
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Abstract

• The spatial data are

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Spatial trees Quadtree

• Structure should be based on the spatial keys.
  The data on the space does not have specific
  record structure and its difficult to
  predetermine. Its efficient to construct on the
  fly.
• For example, a 2D point can be represented as 4D
  spatial key. (X,Y) can be changed to (X1,Y1) and
  (X2,Y2) to define a region.

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Example of splitting Spatial data as Bounding
rectangles
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Spatial Occupancy

• The space can be viewed as grids/cells /buckets.
  This can be formed as bounding rectangles or
  proximities.
• Overlapping of points in space can be allowed to
  store objects in different places. This leads to
  multiple search paths to make sure that the
  target object is fetched properly.
• Non-overlapping nodes are difficult to perform,
  since they need to be verified for duplicates.

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Types of spatial occupancy

• Region data
• Line data
• Rectangle data
• Point data

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Region data

• Region data can be represented as boundary or by
  its interior value.
• Quad tree decomposes based on the distribution of
  the data (arbitrarily distributed), Uniform
  grid(uniformly distributed data). - Both are good
  for performing set operations on diff sets and
  operations.
• The bit values are used to represent the
  occupancy of the object in the space.

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Example
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Point data

• PR quadtree is used to represent points structure
  adapted from region quadtree.
• More precious the nearby points in the structure
  means more depth is the tree.
• The search for the target can be focussed easily
  and it doesnt have to traverse many paths.

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(No Transcript)
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Rectangle Data

• It represents the enclosing boundary of the
  space.
• The centroid of the rectangles that occupy the
  space is used to find the line segment that
  intersects with each other with the centroid
  point is used to form the tree.
• The splitting stops if the size of the rectangle
  is same as the splitting block.

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Example
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Line data

• There are two types of representing the line
  segment. Either by vertex or edge based.
• Vertex based saves space by splitting according
  to the occupancy of the line in the block.The
  vertex based makes sure that no more than one
  edge or vertex occupies the block unless all the
  edges meet the same vertex or in adjacent side.
• But in edge based variable number of line
  segments are allowed. Line segment intersects or
  occupies entirely the block

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Vertex based split
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Edge based split
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BSP tree

• Used by Doom, First Person Shooting games.
• The tree is constructed before the execution not
  on the fly.
• The polygons are used to split the space based on
  their positions.

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Example
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Example of Polygon
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Formation of tree
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R tree

• This is used for efficient spatial data handling
  operations such as deletion or insertion.
• Used for CAD or geo-data applications. Its index
  based searching for multi-dimensions

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Properties of R tree

• Every leaf node contains between m and M index
  records unless it is the root
• For each index record (I, tuple -identifier) in a
  leaf node, I is the smallest rectangle that
  spatially contains the n-dimensional data object
  represented by the indicated tuple
• Every non-leaf node has between m and M children
  unless it is the root

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..contd

• For each entry (I, child -pointer) a non-leaf
  node, I is the smallest rectangle that spatially
  contains the rectangles m the child node
• The root node has at least two children unless it
  is a leaf
• All leaves appear on the same level

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Example of R tree
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..contd
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Searching

• The record is searched from the root. The child
  area that encloses the range that matches the
  searching query is used for traversal down the
  tree.
• The leaf node that matches the query is the final
  record. More than one sub tree needs to be
  searched since there may be overlaps.

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Insertion

• The leaf is searched for the inserting record to
  accommodate new record. If this overflows the
  existing leaf node, the splitting is informed to
  the upper layer and the new root is selected to
  balance the change.
• If there is any change that can accommodate the
  new entry, it can be done before splitting.

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Splitting of the space

• Instead of analysing different combinations of
  the area occupied by the rectangles, two points
  are chosen such that the area will be wasted by
  enclosing them as a rectangle.
• Then each record is inserted by analysing the
  existing node and added to corresponding group.

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R tree

• This is an extension of R tree taking the concept
  basically but it is dynamic insertions and
  deletions with no need to change the global
  organisation.
• A single search path does not guarantee the
  fetching of all records that corresponds to the
  query.

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Properties

• The area enclosed by the directory rectangle
  should be minimum leaving the dead space as max
  as possible.
• The overlap directory between the rectangles
  should be minimum.
• The margin of the directory rectangle should be
  minimum.
• The storage utilisation should be minimum.

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R tree , Guttmann, Greene and Rtree comparisons
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K d tree

• This is used to store points of k-dimensional
  space.
• Each node in the tree stores a point whereas BSP,
  R, R stores records only in the leaf nodes.
• As one moves down the tree, the splitting axis
  goes around the cycle.

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..contd

• The point chosen in each level is the median of
  all points. This would make the tree structure
  more height balanced.
• But it is not required to calculate the exact
  medium, and it may not optimal for all
  applications. Some heuristic calculation will
  ease the splitting.

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Example of splitting
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• http//cis.poly.edu/hakcan01/projects/kdtree/kdTr
  ee.html
• http//donar.umiacs.umd.edu/quadtree/points/kdtree
  .html
• 2 nd one is better