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R* Trees. Presenters:- Twisha Surender. R* Trees. Variant of R-tree Supports point and spatial data efficiently at the same time Implementation cost only slightly higher than that of other R-trees. Supports map-overlay operation – Spatial Join

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r trees

R* Trees

Presenters:-

Twisha

Surender

r trees1
R* Trees
  • Variant of R-tree
  • Supports point and spatial data efficiently at the same time
  • Implementation cost only slightly higher than that of other R-trees.
  • Supports map-overlay operation – Spatial Join
      • E.g. of Spatial Join queries: Two spatial relations S1 and S2, find all pairs: x in S1, y in S2 s.t. x rel y = true where rel = intersect, inside etc.
  • Completely Dynamic
r tree
R*-tree
  • Optimization Criteria:
    • (O1) Area covered by an index MBR
    • (O2) Overlap between index MBRs
    • (O3) Margin of an index rectangle
    • (O4) Storage utilization
  • Sometimes it is impossible to optimize all the above criteria at the same time!
difference between r trees and r trees
Difference between R* Trees and R Trees
  • Optimization in ChooseSubTree module for leaf nodes
  • Revised Node-Split Algorithm
  • Forced Reinsertion at Node Overflow
choosesubtree for insertion
ChooseSubTree for Insertion
  • ChooseSubtree:
    • If next node is a leaf node, choose the node using the following criteria:
      • Least overlap enlargement
      • Least area enlargement
      • Smaller area
    • Else
      • Least area enlargement
      • Smaller area
  • Perform better especially inQueries with small query rectangles on datafiles with non-uniformly distributed small rectangles or points
split node
Split Node
  • SplitNode
    • Choose the axis to split
    • Choose the two groups along the chosen axis
  • ChooseSplitAxis
    • Along each axis, sort rectangles and break them into two groups (M-2m+2 possible ways where one group contains at least m rectangles). Compute the sum S of all margin-values (perimeters) of each pair of groups. Choose the one that minimizes S
  • ChooseSplitIndex
    • Along the chosen axis, choose the grouping that gives the minimum overlap-value, then area-value
forced reinsert
Forced Reinsert
  • Forced Reinsert:
    • defer splits, by forced-reinsert, i.e.: instead of splitting, temporarily delete some entries, shrink overflowing MBR, and re-insert those entries
  • Which ones to re-insert?
  • How many? A: 30%
forced reinsert1
Forced Reinsert
  • OverflowTreatment( parameter level)
    • If level is not root level and this is first call to OverflowTreatment for this level
      • Then, invoke Reinsert
      • Else, invoke Split
  • Reinsert
    • Sort entries on their distance from center
    • Remove first p entries, adjust BR
    • Invoke insert to reinsert the entries
forced reinsert2
Forced Reinsert
  • Forced reinsert changes entries between neighboring nodes and decreases the overlap
  • Storage utilization is improved
  • Due to restructuring, less splits occur
  • Since outer rectangles are re-inserted, more quadratic directory rectangles.
r trees why robust
R* Trees – Why Robust?
  • For every query file and every data file less disk accessed are required than any other variants.
  • Highly robust against ugly data distributions
performance
Performance
  • Likely significant improvement over other R tree variants. In spite of forced reinsertion, average insertion cost is not increased but essentially decreased regarding the R-tree variants.
  • Efficiently supports point and spatial data at the same time
references
References:
  • Norbert Beckmann, Hans-Peter Kriegel, Ralf Schneider, Bernhard Seeger: The R*-Tree: An Efficient and Robust Access Method for Points and Rectangles
  • www.corelab.ntua.gr/courses/ds.grad/lect2NTUA07.ppt