1. CS522 Advanced database Systems
7/2/2018
CS522 Advanced database Systems
6. B+ tree
Huiping Guo
Department of Computer Science
California State University, Los Angeles

2. Outline Introduction to B+ tree indexing B+ tree inserts
7/2/2018
Outline
Introduction to B+ tree indexing
B+ tree inserts
B+ tree deletes
6. B+ Tree CS522_S16

3. B+ Tree: Most Widely Used Index
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B+ Tree: Most Widely Used Index
Adjust gracefully to inserts and deletes
Tree structure grows and shrinks dynamically
The leaf pages are linked. Why?
Index Entries
Data Entries
("Sequence set")
(Direct search)
6. B+ Tree CS522_S16 9

4. Characteristics of a B+ tree
Operations (insert, delete) on the tree keep it balanced
A minimum occupancy of 50% is guaranteed for each node except the root
Searching for a record requires just a traversal from the root to the appropriate leaf—the height of the tree
6. B+ Tree CS522_S16

5. B+ Tree parameters Order d Occupancy – at least half full Balance
A parameter of a tree
A measure of the capacity of a tree
Occupancy – at least half full
Each node contain m entries
d<=m<=2d
Exception: root node 1<=m<=2d
Balance
Keep the height of a tree from growing fast
Fan-out
The average number of children a node has
6. B+ Tree CS522_S16

6. B+ Trees in Practice Typical order: 100 Average fan-out = 133
7/2/2018
B+ Trees in Practice
Typical order: 100
Average fan-out = 133
Typical capacities:
Height 4: 1334 = 312,900,700 pages
Height 3: 1333 = 2,352,637 pages
Can often hold top levels in buffer pool:
Level 1 = page = Kbytes
Level 2 = pages = Mbyte
Level 3 = 17,689 pages = 133 MBytes
6. B+ Tree CS522_S16

7. Example B+ Tree (no duplicates)
7/2/2018
Example B+ Tree (no duplicates)
Search begins at root, and key comparisons direct it to a leaf (as in ISAM).
Search for 5*, 15*, all data entries >= 24* ...
Root
d=2
2<=m<=4 13 17 24 30 2* 3* 5* 7*
14*
16*
19*
20*
22*
24*
27*
29*
33*
34*
38*
39*
6. B+ Tree CS522_S16 10

8. B+ Tree Insert (no duplicates)
Find the appropriate leaf
Insert into the leaf
there’s room  we’re done
no room
split leaf node into two
Redistribute data entries evenly, copy up the middle key
Insert an index entry to its parent node
Recursively apply previous step if necessary
To split index node, redistribute entries evenly, but push up the middle key
6. B+ Tree CS522_S16

9. B+ Tree Insert Examples
(a) no split
space available in leaf
(b) leaf page split
(c) non-leaf page split
(c) new root
6. B+ Tree CS522_S16

10. (a) Insert key = 32 (no split)
1<=m<=3
100 30 4 5 11 30 31 32
6. B+ Tree CS522_S16

11. (b) Insert key = 7 (leaf page split)
100 30 7 3 5 11 30 31 3 5 7
6. B+ Tree CS522_S16

12. (c) Insert key = 160 (non leaf page split)
100
160
120
150
180
180
150
156
179
180
200
160
179
6. B+ Tree CS522_S16

13. (d) New root, insert 45 30 new root 10 20 30 40 1 2 3 10 12 20 25 3032 40 40 45
6. B+ Tree CS522_S16

14. Keys to B+ tree insertion
Observe how minimum occupancy is guaranteed in both leaf and index page splits.
Note difference between copy-up and push-up
Leaf page split
Non-leaf page split
Always copy(push) up the first key on the right affected page when there is a split.
6. B+ Tree CS522_S16

15. Exercise d=2 Insert a data entry with key 3. 6. B+ Tree CS522_S16
8 10
18 27
32 39
41 45
52 58
73 80
91 99
73 85 50
Insert a data entry with key 3.
6. B+ Tree CS522_S16

16. B+ Tree Delete Find the appropriate leaf Delete from the leaf
still at least half full  we’re done
below half full
borrow a <key,pointer> from one sibling node or
merge with a sibling node, and delete from a parent node
Recursively apply previous step if necessary
When do we need a new ROOT (or decrease the height of the tree)??
6. B+ Tree CS522_S16

17. B+ Tree Delete examples
a. The leaf page is still half full
b. The leaf page is below half full, borrow an entry from a sibling node – leaf page redistribution
c. The leaf page is below half full, merge with a sibling node
d. Non-leaf page redistribution
6. B+ Tree CS522_S16

18. B+ Tree Delete Example d=2
Root 17 5 13 24 30 2* 3* 5* 7* 8*
14*
16*
19*
20*
22*
24*
27*
29*
33*
34*
38*
39*
Delete 19*
Delete 20*
6. B+ Tree CS522_S16

19. Delete 20*: Re-distribution (leaf)
Root 17 5 13 24 27 30 2* 3* 5* 7* 8*
14*
16*
20*
22*
24*
27*
29*
33*
34*
38*
39*
6. B+ Tree CS522_S16

20. Then delete 24*, Merge (leaf)
Root 17 5 13 27 30 2* 3* 5* 7* 8*
14*
16*
22*
24*
27*
29*
33*
34*
38*
39*
6. B+ Tree CS522_S16

21. Delete 24* Pull down root 17 Root 6. B+ Tree CS522_S16 2* 3* 7* 14*
16*
22*
27*
29*
33*
34*
38*
39* 5* 8*
Root 30 13 5 17
6. B+ Tree CS522_S16

22. Another example of deletion
7/2/2018
Another example of deletion
d=2
Delete 24*
Non leaf page re-distribution
Root 22 5 13 17 20 25 30 3* 2* 5* 7* 8*
14*
16*
17*
18*
20*
21*
22*
24*
27*
29*
33*
34*
38*
39*
6. B+ Tree CS522_S16 17

23. After deleting 24* & leaf page merge
Root 13 5 17 20 22 30
14*
16*
17*
18*
20*
33*
34*
38*
39*
22*
27*
29*
21* 7* 5* 8* 3* 2*
6. B+ Tree CS522_S16

24. Non-leaf page redistribution
Alternative 1: move over 1 entry(20)
Root 20 5 13 17 22 30 2* 3* 5* 7* 8*
14*
16*
17*
18*
20*
21*
22*
27*
29*
33*
34*
38*
39*
6. B+ Tree CS522_S16

25. Non-leaf page redistribution
Alternative 2: move over 2 entry (17,20)
Root 17 5 13 20 22 30 2* 3* 5* 7* 8*
14*
16*
17*
18*
20*
21*
22*
27*
29*
33*
34*
38*
39*
6. B+ Tree CS522_S16

26. Exercise d=2
8 10
18 27
32 39
41 45
52 58
73 80
91 99
73 85 50
Delete a data entry with key 8. Only left sibling is checked
Delete a data entry with key 8. Only right sibling is checked
Delete a data entry with key 91 in the original tree
Successively delete 32, 39, 41, 45, 73 from the original tree
6. B+ Tree CS522_S16