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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




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

    • 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


    • 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


    • 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