Spatial Data Structures

1

• Snehal Thakkar
• Spatial Data Structures
• Hanan Samet
• Computer Science Department
• University of Maryland

2
Spatial Data Structures

• Introduction
• Spatial Indexing
• Region Data
• Point Data
• Rectangle Data
• Line Data
• Conclusion

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Introduction

• Spatial Objects
• Points, Lines, Regions, Rectangles ..
• Spatial Indexing
• Unlike conventional data sort has to be
  on space occupied by data
• Hierarchical Data Structures
• Based on recursive decomposition, similar to
  divide and conquer method

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Spatial Indexing

• Mapping Spatial Data into Point
• - Same, Higher or Lower Dimension
• - Good storage purposes, queries like intersect
• - Problems with queries like nearest
• Bucketing Methods
• - Grid file, BANG file, LSD trees, Buddy
  trees.
• - Buckets based on not the representative
  point, but based on actual space.

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R-tree

• Based on Minimum Bounding Rectangle

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R-Trees (Continued)

• Organize spatial objects into d-dimensional
  rectangles.
• Each node in the tree corresponds to smallest
  d-dimensional rectangle that encloses child
  nodes.
• If an object is spatially contained in several
  nodes, it is only stored in one node.
• Tree parameters are adjusted so that small number
  of pages are visited during a spatial query
• All leaf nodes appear at same level
• Each leaf node is (R,O) where R is smallest
  rectangle containing O, e.g. R3,R4

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R-trees (Continued)

• Each non-leaf node is (R,P) where R is smallest
  rectangle containing all child rectangles, e.g.
  R1,R2
• R-tree of order (m,M) means that each node in
  the tree has between floor M/2 and M nodes, with
  exception of root node. Root node has two entries
  unless it is a leaf node.
• R-tree is not unique, rectangles depend on how
  objects are inserted and deleted from the tree.
• Problem is that to find some object you might
  have to go through several rectangles or whole
  database.

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R - Trees

• Decomposition of Space into Disjoint Cells

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R Trees (Continued)

• R-tree and Cell Trees used approach of
  discomposing space into cells
• R-trees deals with collection of objects bounded
  by rectangles
• Cell tree deals with collection of objects
  bounded by convex polyhedra
• R-trees is extension of k-d-B-tree.
• Try not to overlap the rectangles.
• If object is in multiple rectangles, it will
  appear multiple times.

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RTrees(Continued)

• Multiple paths to object from the root
• Height of the tree is increased
• Retrieval times are smaller
• When summing the objects, needs eliminate
  duplicates
• It is not possible to guarantee that all
  properties of B-trees is fulfilled without going
  through difficult insert and deletion routines.
• It is data-dependent, so depending on how you
  insert or delete records R-tree will be
  different.

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More Spatial Indexing

• Uniform Grid
• - Ideal for uniformly distributed data
• - More data-independence then R-trees
• - Space decomposed on blocks on uniform size
• - Higher overhead
• Quadtree
• - Space is decomposed based on data points
• - Sensitive to positioning of the object
• - Width of the blocks is restricted to power of
  two
• - Good for Set-theory type operations, like
  composition of data.

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Region Data

• Focus on Interior Representation
• Represented as Image array of pixels
• Runlength Code
• - Break array into 1m blocks, row
  representation
• Metal Axis Transformation (MAT)
• - Union of Maximal Square blocks
• - Blocks may overlap
• - Block are specified by center and radius

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More Region Data

• Region Quadtree
• - Is Metal Axis Transformation
• - Whose blocks are required to be disjoint
• - To have standard sizes(squares whose sides
  are power of two)
• - To be at standard locations
• - Based on successive subdivision of image
  array into four equal size quadrants.

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Region Quadtree
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Region Quadtree (Continued)

• Each leaf node is either Black or White
• All non-leaf nodes are Gray(Circle is previous
  example
• You can also use it for non-binary images
• Resolution of the decomposition may be governed
  by data or predetermined
• Can be used for several object representations.

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Variations of Quadtree

• Point Quadtree
• - Quadtree with rectangular quadrants
• - Adoption of Binary Search Tree to two
  dimensions or more
• - Useful for location based queries like where
  is nearest theatre from the location.
• - Descending the tree till you find the node
  for location based queries.
• - For nearest neighbor, search is continued in
  the neighborhood of the node containing
  object.
• - Feature based queries tough because index is
  based on spatial occupancy not on features.

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Variations of Quadtree

• Pyramid
• - Exponentially tapering stack of arrays, each
  one quarter size of previous
• - Useful for feature based queries like where
  does wheat grow in California.
• - Nodes that are not at maximum level of
  resolution contain summary information
• Octree
• - Three dimensional analog of quadtree
• - Recursively subdivide into eight octants

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More Variations of Quadtree

• Locational Code Based Quadtree
• - Treats image as a collection of leaf nodes,
  each encoded by pair of numbers
• - First is base 4 number, sequence of
  directional codes that locates leaf from the
  root
• - Second depth at which node is found or size
• DF-expression
• - Represents the image in form of traversal of
  nodes of its quadtree
• - Very Compact storage, each node type can be
  encoded with two bits.
• - Not easy to use when random access to nodes
  is required.

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Searching with Quadtree

• Useful for performing set operations
• When performing intersection, it only returns
  black node when both quadtrees have black nodes.
• Operation is performed using three quadtrees.
• Worst case scenario is sum of nodes in two
  quadtrees

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Algorithms with Quadtree

• Most algorithms are preorder traversals
• Execution time is linear function of number of
  nodes
• Quadtree Complexity Theorem
• - Number of nodes in quadtree representation is
  O(pq) for 2q2q image with perimeter p measured
  in pixel width.
• - It also holds for more dimensions.

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Point Data

• PR Quadtree
• - Regular decomposition of space into quadrants
• - Organized same way as the region quadtree
• - Leaf nodes are either empty or contain data
  point and its co-ordinates
• - A quadrant contains at most one data point
• - Shape of the tree is independent of the order
  in which points are inserted
• - If points are close together then
  decomposition can be deep
• - Can use quadrants with capacity c
• - Good for search within specified distance of
  given record

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PR-tree (Continued)
(50,50)
(75,75)
(25,25)
(75,25)
(20,88)
(0,100)
(100,100)
(88,65)
(52,15)
(92,1)
(0,0)
(100,0)
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Rectangle Data

• Used to approximate other objects in the image
  and in VLSI design rule checking
• If environment is static, solution is based on
  use of plane sweep paradigm
• Any addition to database forces re-execution of
  algorithm on whole database

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Rectangle Data (Continued)

• Grid File Based Approach
• - Each rectangle reduced to a point in higher
  dimension
• - Made up of Cartesian product of two one
  dimensional intervals
• - Each interval is represented by center and
  extent
• - Set of intervals is represented by Grid File
• - Grid File uses two dimensional array of grid
  blocks called Grid Directory

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Rectangle Data (Continued)

• Grid File Based Approach (Continued)
• - Grid Directory has address of the bucket
• - Set of linear scales is kept in the core to
  access grid block in the grid directory
• - Guarantees access to record in two operations
• - First operation to access the grid block
• - Second operation to access the grid bucket

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Rectangle Data (Continued)

• MX-CIF Quadtree
• - Based on Quadtree
• - Decomposition of space into rectangles
• - Each rectangle is associated with a quadtree
  node corresponding to the smallest block which
  contains it in its entirety
• - Subdivision stops when nodes block contains
  no rectangles or at predetermined size
• - Rectangles can be associated with terminal
  and non-terminal nodes

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MX-CIF Quadtree
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Line Data

• PM1 quadtree
• - Based on regular decomposition of space
• - Partitioning occurs as long as a block
  contains more than one line segment unless
  the line segments are incident at a vertex in
  the block
• - Vertex-based implementation
• - Useful because space requirements for
  polyhedral objects are smaller then
  conventional octree

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PM1 Quadtree(Continued)
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Line Data (Continued)

• PMR Quadtree
• - Edge-based variant of PM quatree
• - Uses probabilistic splitting rule
• - Block contains variable number of line
  segments
• - Each line segment is inserted into all blocks
  that it intersects or occupies
• - If block has more line segments than
  permitted, it is divided into four blocks once
  and only once
• - During deletion line segment is removed from
  all blocks and blocks are checked for merging

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PMR Quadtree
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PMR Quadtree
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PMR Quadtree
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Conclusion

• Questions ?
• Comments ?
• Email me at snehalth_at_usc.edu