1. Multilevel Indexing and B+ Trees

2. Indexed Sequential Files
Provide a choice between two alternative views of a file:
Indexed: the file can be seen as a set of records that is indexed by key; or
Sequential: the file can be accessed sequentially (physically contiguous records), returning records in order by key.

3. Example of applications
Student record system in a university:
Indexed view: access to individual records
Sequential view: batch processing when posting grades
Credit card system:
Indexed view: interactive check of accounts
Sequential view: batch processing of payments

4. The initial idea Maintain a sequence set:
Group the records into blocks in a sorted way.
Maintain the order in the blocks as records are added or deleted through splitting, concatenation, and redistribution.
Construct a simple, single level index for these blocks.
Choose to build an index that contain the key for the last record in each block.

5. Maintaining a Sequence Set
Sorting and re-organizing after insertions and deletions is out of question. We organize the sequence set in the following way:
Records are grouped in blocks.
Blocks should be at least half full.
Link fields are used to point to the preceding block and the following block (similar to doubly linked lists)
Changes (insertion/deletion) are localized into blocks by performing:
Block splitting when insertion causes overflow
Block merging or redistribution when deletion causes underflow.

6. Example: insertion Block size = 4 Key : Last name ADAMS … BIXBY …
CARSON …
COLE …
Block 1
Insert “BAIRD …”:
Block 1
ADAMS …
BAIRD …
BIXBY …
CARSON ..
COLE …
Block 2

7. Example: deletion ADAMS … BAIRD … BIXBY … BOONE … Block 1 Block 2
BYNUM…
CARSON ..
CARTER ..
DENVER…
ELLIS …
Block 3
Block 4
COLE…
DAVIS
Delete “DAVIS”, “BYNUM”, “CARTER”,

8. Add an Index set BERNE 1 CAGE 2 DUTTON 3 EVANS 4 FOLK 5 GADDIS 6
Key Block
BERNE 1
CAGE 2
DUTTON 3
EVANS 4
FOLK 5
GADDIS 6

9. Tree indexes This simple scheme is nice if the index fits in memory.
If index doesn’t fit in memory:
Divide the index structure into blocks,
Organize these blocks similarly building a tree structure.
Tree indexes:
B Trees
B+ Trees
Simple prefix B+ Trees …

10. Separators 1 ADAMS-BERNE Block Range of Keys Separator BOLEN
2 BOLEN-CAGE
CAMP
3 CAMP-DUTTON
EMBRY
4 EMBRY-EVANS
FABER
5 FABER-FOLK
FOLKS
6 FOLKS-GADDIS

11. root 1 3 4 6 2 5 Index set EMBRY BOLEN CAMP FABER FOLKS ADAMS-BERNE
CAMP-DUTTON
EMBRY-EVANS
FOLKS-GADDIS 1 3 4 6
BOLEN-CAGE
FABER-FOLK 2 5

12. B Trees
B-tree is one of the most important data structures in computer science.
What does B stand for? (Not binary!)
B-tree is a multiway search tree.
Several versions of B-trees have been proposed, but only B+ Trees has been used with large files.
A B+tree is a B-tree in which data records are in leaf nodes, and faster sequential access is possible.

13. Formal definition of B+ Tree Properties
Properties of a B+ Tree of order v :
All internal nodes (except root) has at least v keys and at most 2v keys .
The root has at least 2 children unless it’s a leaf..
All leaves are on the same level.
An internal node with k keys has k+1 children 17

14. B+ tree: Internal/root node structure
P0 K1 P1 K ……………… Pn-1 Kn Pn
Each Pi is a pointer to a child node; each Ki is a search key value
# of search key values = n, # of pointers = n+1
Requirements:
K1 < K2 < … < Kn
For any search key value K in the subtree pointed by Pi,
If Pi = P0, we require K < K1
If Pi = Pn, Kn  K
If Pi = P1, …, Pn-1, Ki < K  Ki+1

15. B+ tree: leaf node structure
L K1 r1 K ……………… Kn rn R
Pointer L points to the left neighbor; R points to the right neighbor
K1 < K2 < … < Kn
v  n  2v (v is the order of this B+ tree)
We will use Ki* for the pair <Ki, ri> and omit L and R for simplicity

16. Example: B+ tree with order of 1
Each node must hold at least 1 entry, and at most 2 entries
10*
15*
20*
27*
33*
37*
40*
46*
51*
55*
63*
97* 20 33 51 63 40
Root 6

17. Example: Search in a B+ tree order 2
Search: how to find the records with a given search key value?
Begin at root, and use key comparisons to go to leaf
Examples: search for 5*, 16*, all data entries >= 24* ...
The last one is a range search, we need to do the sequential scan, starting from the first leaf containing a value >= 24.
Root 17 24 30 2* 3* 5* 7*
14*
15*
19*
20*
22*
24*
27*
29*
33*
34*
38*
39* 13 10

18. B+ Trees in Practice Typical order: 100. Typical fill-factor: 67%.
average fanout = 133 (i.e, # of pointers in internal node)
Can often hold top levels in buffer pool:
Level 1 = page = Kbytes
Level 2 = pages = Mbyte
Level 3 = 17,689 pages = 133 MBytes
Suppose there are 1,000,000,000 data entries.
H = log133( /132) < 4
The cost is 5 pages read

19. How to Insert a Data Entry into a B+ Tree?
Let’s look at several examples first. 6

20. Inserting 16*, 8* into Example B+ tree
Root 13 17 24 30 2* 3* 5* 7* 8*
14* 15*
16*
You overflow 2* 5* 7* 3* 17 24 30 13 8*
One new child (leaf node) generated; must add one more pointer to its parent, thus one more key value as well. 10

21. Inserting 8* (cont.) Copy up the middle value (leaf split) 13 17 24 30
Entry to be inserted in parent node. 5
(Note that 5 is
s copied up and
continues to appear in the leaf.) 2* 3* 5* 7* 8*
You overflow! 12

22. Insertion into B+ tree (cont.)
Understand difference between copy-up and push-up
Observe how minimum occupancy is guaranteed in both leaf and index pg splits.
We split this node, redistribute entries evenly, and push up middle key. 
appears once in the index. Contrast
Entry to be inserted in parent node.
this with a leaf split.) 5 24 30 17 13
(Note that 17 is pushed up and only 12

23. Example B+ Tree After Inserting 8*
Root 17 5 13 24 30 2* 3* 5* 7* 8*
14*
15*
19*
20*
22*
24*
27*
29*
33*
34*
38*
39*
Notice that root was split, leading to increase in height. 13

24. Inserting a Data Entry into a B+ Tree: Summary
Find correct leaf L.
Put data entry onto L.
If L has enough space, done!
Else, must split L (into L and a new node L2)
Redistribute entries evenly, put middle key in L2
copy up middle key.
Insert index entry pointing to L2 into parent of L.
This can happen recursively
To split index node, redistribute entries evenly, but push up middle key. (Contrast with leaf splits.)
Splits “grow” tree; root split increases height.
Tree growth: gets wider or one level taller at top. 6

25. Deleting a Data Entry from a B+ Tree
Examine examples first … 14

26. Delete 19* and 20* Have we still forgot something? Root You underflow17 5 13 24 30 2* 3* 5* 7* 8*
14*
16*
19*
20*
22*
24*
27*
29*
33*
34*
38*
39*
22*
27* 29*
You underflow
22* 24*
Have we still forgot something? 13

27. Deleting 19* and 20* (cont.) Notice how 27 is copied up.
Root 17 5 13 27 30 2* 3* 5* 7* 8*
14*
16*
22*
24*
27*
29*
33*
34*
38*
39*
Notice how 27 is copied up.
But can we move it up?
Now we want to delete 24
Underflow again! But can we redistribute this time? 15

28. Deleting 24*
Merge with sibling!
You underflow
Observe the two leaf nodes are merged, and 27 is discarded from their parent, but …
Observe `pull down’ of index entry (below). 30
22*
27*
29*
33*
34*
38*
39*
New root 5 13 17 30 2* 3* 5* 7* 8*
14*
16*
22*
27*
29*
33*
34*
38*
39* 16

29. Deleting a Data Entry from a B+ Tree: Summary
Start at root, find leaf L where entry belongs.
Remove the entry.
If L is at least half-full, done!
If L has only d-1 entries,
Try to re-distribute, borrowing from sibling (adjacent node with same parent as L).
If re-distribution fails, merge L and sibling.
If merge occurred, must delete entry (pointing to L or sibling) from parent of L.
Merge could propagate to root, decreasing height. 14

30. Example of Non-leaf Re-distribution
Tree is shown below during deletion of 24*. (What could be a possible initial tree?)
In contrast to previous example, can re-distribute entry from left child of root to right child.
Root 22 5 13 17 20 30
14*
16*
17*
18*
20*
33*
34*
38*
39*
22*
27*
29*
21* 7* 5* 8* 3* 2* 17

31. After Re-distribution
Intuitively, entries are re-distributed by `pushing through’ the splitting entry in the parent node.
It suffices to re-distribute index entry with key 20; we’ve re-distributed 17 as well for illustration.
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* 18

32. Terminology
Bucket Factor: the number of records which can fit in a leaf node.
Fan-out : the average number of children of an internal node.
A B+tree index can be used either as a primary index or a secondary index.
Primary index: determines the way the records are actually stored (also called a sparse index)
Secondary index: the records in the file are not grouped in buckets according to keys of secondary indexes (also called a dense index)

33. Summary
Tree-structured indexes are ideal for range-searches, also good for equality searches.
B+ tree is a dynamic structure.
Inserts/deletes leave tree height-balanced; High fanout (F) means depth rarely more than 3 or 4.
Almost always better than maintaining a sorted file.
Typically, 67% occupancy on average.
If data entries are data records, splits can change rids!
Most widely used index in database management systems because of its versatility. One of the most optimized components of a DBMS. 23