1. R* Tree By Rohan Sadale Akshay Kulkarni

2.  Motivation  Optimization criteria for R* Tree  High level Algorithm  Example  Performance Agenda

3. Motivation  R Tree  Every parent completely covers its children  A child MBR may be covered by more than one parent. It is stored under only one of them.  Minimizes only area.  Multiple nodes should be visited because of overlap (MBRs).

4. Motivation  R+ Tree  Disjoint MBRs  Prevents overlapping by duplicating objects across leaf nodes

5. Motivation  R* Tree  Improved query performance over R-Tree  But, higher construction cost  Incorporates a combined optimization of: -area -margin -overlap

6.  Follows an engineering approach to find best possible combinations of the criteria mentioned below:  Minimization of area covered by each MBR  Minimization of overlap between MBRs  Minimization of MBR margins  Maximization of storage utilization But, can’t do all at once. Need a heuristic approach. Optimization criteria for R* Tree

7. High Level Algorithm  Same algorithm as the R-tree for query and delete operations.  When inserting, the R*-tree uses a combined strategy.  For inner nodes (non-leaf), enlargement and area are minimized.  For leaf nodes, overlap is minimized.  Deletion and reinsertion of entries allows them to "find" a place in the tree that may be more appropriate than their original location.  When a node overflows, a portion of its entries are removed from the node and reinserted into the tree.  This has the effect of producing more well-clustered groups of entries in nodes, reducing node coverage.

8. R*-Tree example a b c e d f g i h j N3 N4 N2 N1 N6 N5 N6 N1 N2 N3N4 j ihgfedcba

9. R*-Tree example a b c e d k f g i h j N3 N4 N2 N1 N6 N5 N6 N1 N2 N3N4 j ihgfedcba Insert k

10. R*-Tree example a b c e d k f g i h j N3 N4 N2 N1 N6 N5 N6 N1 N2 N3N4 j ihgfedcba Insert k Check for minimum area enlargement at root node. N5 or N6 ? - choose N5 k ?

11. R*-Tree example a b c e d k f g i h j N3 N4 N2 N1 N6 N5 N6 N1 N2 N3N4 j ihgfedcba Insert k Check for minimum area enlargement at next non leaf node. N1 or N2 ? - choose N1 k ?

12. R*-Tree example a b c e d k f g i h j N3 N4 N2 N1 N6 N5 N6 N1 N2 N3N4 j ihgfedcba Insert k If node capacity is 3., then where will ‘k’ go? Reinsertion of nodes from nearest MBRs (strategy - overlap minimization) k ?

13. R*-Tree example a b c e d k f g i h j N3 N4 N2 N1 N6 N5 N6 N1 N2 N3N4 j ihgfedcba Insert k If node capacity is 3., then where will ‘k’ go? k Reinsert nodes from N1 and N2

14. R*-Tree example a b c e d k f g i h j N3 N4 N2 N1 N6 N5 N6 N1 N2 N3N4 j ihgfedkba Insert k If node capacity is 3., then where will ‘k’ go? - Reinsertion !!! c

15. R-Tree vs R+ Tree vs R* Tree j a b c e fg h i 1 2 3 1 st level index leafs 123 a bc efg hi d d new data x new data: x Initial data: {a,b,…,h,i} Legend Initial R-tree

16. R-Tree j a b c e fg h i 1 2 3 1 st level index leafs d new data x new data: x Initial data: {a,b,…,h,i} Legend

17. R+ Tree j a b c e fg h i 1 2 3 1 st level index leafs d new data x new data: x Initial data: {a,b,…,h,i} Legend

18. R* Tree j a b c e fg h i 1 2 3 1 st level index leafs d new data x new data: x Initial data: {a,b,…,h,i} Legend

19. Performance  Improved split heuristic produces pages that are more rectangular and thus better for many applications.  Reinsertion method optimizes the existing tree, but increases complexity.  Efficiently supports point and spatial data at the same time.

20. References  Encyclopedia of GIS: R*-tree, H. Kriegel, P. Kunath, page 987- 992.  R-Trees: Theory and Applications, Yannis Manolopoulos, Alexandros Nanopoulos, Apostolos N. Papadopoulos and Yannis Theodoridis  Wikipedia - https://en.wikipedia.org/wiki/R*_treehttps://en.wikipedia.org/wiki/R*_tree

21. Thank you :)