Trees for spatial indexing

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Trees for spatial indexing

• Part 2 SAMs

2
SAMs
R-Tree
R-Tree
X
TV
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Answering question

• The Kd-Trie, is similar to kd-tree. In the
  article it was used for kd-tree.
• The split-axis isnt in the middle, but is
  choosen is the median point.
• Because, we work with points, we have no problem
  is separating the elements.

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UB-Tree range queries

• Algorithm is
• Find all region who intersects q
• IF this region is a page, all objects that
  intersects q is in the answer.
• After that we search for the last subcube in this
  region and we search the brother, and if it
  intersects q we make the same loop on it.
• After that we look the father of B and search
  again.

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R-Tree

• Special B-Tree for spatial indexing.
• The performance of the R-Tree is decreasing with
  the dimensionality.
• R-tree access method is prohibitively slow for
  dimensions higher than 5.

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Problems of (R-Tree based) Index Structures

• Because it has been shown that with the
  increasing of the dimensionality we have also
  more overlap.
• Overlap is intuitively when for some point
  queries, we have multiple paths to search.

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Definition of overlap

• Intuitively, overlap is the pourcentage of the
  volume that is covered by more than one directory
  hyperrectangle.
• This intuitive definition of overlap is directly
  correlated to the query performance.
• Because it implies multiple paths.

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Definition of the overlap (2)

• Overlap ( Ui,j, i?j Ri n Rj ) / ( Ui Ri
  )
• We add all the intersection of the MBR in volume
  and we divide it by the union of all the MBR in
  volume.
• But overlap in highly populated areas is much
  more critical than overlap in low population.
• WeightedOverlap pp Ui,j,i?j Ri n Rj ) /
  (pp Ui Ri )

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1
1
Overlap (¼)/(2) 1/8 12,5
WeightedOverlap (2)/(6) 1/3 33
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Overlap / WeightedOverlap

• Depending the kind of data the the measurement
  can be different.
• If we have uniformed distributed data points, we
  can use the overlap measure
• In the case of real data, when can have
  clustering, so the weightedOverlap is more
  accurate.

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X-Tree

• Avoid overlap in the directory.
• X-Tree hybrid of a linear array-like and a
  hierarchical R-Tree-like directory.
• In low dimensions the most efficient organization
  of the directory is hierarchical organization.
• For high dimensionality a linear organization is
  more efficient.

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X-Tree

• In the X-Tree we have 3 types of nodes data
  nodes,normal directory, and supernodes.
• The supernodes avoid splits in directory, so its
  more faster to search.
• Not the same as R-Tree with larger blocks,
  because it creates larger blocks only if
  necessary.

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X-Tree
Supernode Normal directory Data nodes
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Creation of supernodes

• They are only created if there is no other
  possibility to avoid overlap during insertion.

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TV-Tree (Telescopic-Vector tree)

• The basis of the tv-tree is to use dynamically
  contracting and extending feature vectors. ( Like
  in classification )

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TV-Tree

• A m-contraction of x, is a sequence of
• Amx where Am is a contraction matrix.
• A natural Am is
• ( 1 0 0 )( 0 1 0 0 )( . )(
  0 . 0 1)

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Multiple shapes

• We can use for example a sphere, because its
  only a center and a radius r. Represents the set
  of points with euclidean distance r.
• the euclidean distance is a special case of the
  Lp metrics with p2.
• For L1 metric (manhattan distance) it defines a
  diamond shape.
• The TV-tree is working with any Lp-sphere.

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Tv-Tree principle

• So the TV treats the attributs asymmetrically
  favoring the first few features over the rest.
• TV-Tree can use any type of MBR (minimum bounding
  region), rectangle,cube,sphere etc.
• TV-Tree can use any Lp-Sphere

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TV-Tree node structure

• Each node is represents the MBR of all its
  descendents ( say an Lp-sphere ).
• Each region is represented by a center which is a
  telescopic-vector and a radius.
• So we talk about TMBR.

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TV-1-Tree example
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TV-2-Tree example
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TMBR
Act. Dim y
Act. Dim x,z
Act. Dim z
Act. Dim x,y
Act. Dim x
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What is the best number of active dimensions ?

• They find out that the best number of active
  dimensions was two

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TV-Tree conclusion

• We accept overlap, so also multiple path to
  search.
• Branch choosen for new point is done with the
  following criteria