1. Multidimensional Index
K-D-B Tree
Multidimensional Index

2. Characteristics Multi-way branch Height-balanced tree
Repeatedly divide area of the domain into disjoint sub-area
A node in a tree corresponds to a (set of consecutive) disk page(s)
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K-D-B Tree

3. Example of Data Records
Table (stdntID, courseID, grade, year, smstr)
Table (accID, branchID, saving, name, addr)
Table (custID, age, gender, occupation, salary, children, promotion, since)
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K-D-B Tree

4. Nodes = Pages Region pages Point pages
Contain a set of <region, ptr. to page>
Internal nodes
Point pages
Contain a set of <point, ptr. to data record>
Leaf nodes
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K-D-B Tree

5. Region Pages Region The branching factor is determined by the
Xmax
Xmin
Ymax
Ymin …
Xmax
Xmin
Ymax
Ymin
Region
PAGE
PAGE
The branching factor is determined by the
page size and the size of each entry.
Jaruloj Chongstitvatana
K-D-B Tree

6. Point Pages X Y X Y X Y The branching factor of a point page is… X Y X Y X Y
DATA RECORD
DATA RECORD
DATA RECORD
POINT
The branching factor of a point page is
usually larger than that of a region page.
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K-D-B Tree

7. Example Point page Point page Point page Point page
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K-D-B Tree

8. Search Point query Point page Point page Point page Point page
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K-D-B Tree

9. Insert Insert a point here and the point page overflows.
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K-D-B Tree

10. Split Split a region r with page id p along xi
If r is on the right/page of xi then put <r, p> in the right/left page.
Otherwise;
For each children pc of p , split pc along xi
Split r along xi into rleft and rright.
Create 2 new pages with page id pleft and pright.
Move children of p in the left region into pleft and children in the right region into pright.
Return <rleft, pleft> and <rright, pright> .
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K-D-B Tree

11. Split: Example The page overflows, and is splitted.
This region is also splitted.
This region is splitted.
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K-D-B Tree

12. Split: Example The region page is splitted.
The point page is also splitted.
Children pages are transferred.
Create a new region page.
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K-D-B Tree

13. How to find split axis Cyclic: x -> y -> x -> y -> …
Priority: x -> x -> y -> x -> x -> y -> …
Possible one
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K-D-B Tree

14. Insert
Insert a record with point a and location l in a tree with root r
If r is NIL, then create a point page p and insert the record with <a,l> in p and return p.
Otherwise;
Search for a in the tree with root r until a point page, say p, is reached.
Insert the record in the point page p.
Jaruloj Chongstitvatana
K-D-B Tree

15. Insert (cont’d)
Insert a record with point a and location l in a tree with root r
If the point page p is overflowed, then find an appropriate axis to split p into pleft and pright.
If p is not the root, then change to p and pleft, and insert pright into the parent of p.
If p is the root, then create a new root node with two children of pleft and pright.
Jaruloj Chongstitvatana
K-D-B Tree

16. Insert: Example Insert here and split point page if overflows.
Search for the given point
until the point page is found.
Insert here and split
point page if overflows.
Divide region.
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K-D-B Tree

17. Insert: Example Parent page overflows, then split the page.
This region is splitted.
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K-D-B Tree

18. Insert: Example The point page is splitted.
The region page is splitted.
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K-D-B Tree

19. Insert: Example The root node is overflowed, and then splitted.
Insert the new region page in its parent.
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K-D-B Tree

20. Insert: Example Create the new root node Jaruloj Chongstitvatana
K-D-B Tree

21. Delete Simple, if storage utilization is ignored.
Otherwise, an underfull page should be merged with another page.
When 2 pages are merged, the region of the new page must be a valid region.
A number of regions are joinable if their union is also a region.
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K-D-B Tree

22. Joinable Regions
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K-D-B Tree

23. Unjoinable Regions
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K-D-B Tree

24. Delete (cont’d)
If a page p is underfull, merge sibling pages of p whose regions are joinable.
If the newly-created page is overflowed, then split the page.
Jaruloj Chongstitvatana
K-D-B Tree

25. Further Discussion

26. Splitting Criteria Combine the two decisions ? Axis Value Cyclic
Priority
Shape ?
Value
Area
Number of data points
Ratio ?
Random ?
Combine the two decisions ?
Jaruloj Chongstitvatana
K-D-B Tree