B+ Tree

Index tuning--

overview

B+-Tree Scalability

• Typical order: 100. Typical fill-factor: 67%.– average fanout = 133

• Typical capacities (root at Level 1, and has 133 entries):– Level 5: 1334 = 312,900,700 records– Level 4: 1333 = 2,352,637 records

• Can often hold top levels in buffer pool:– Level 1 = 1 page = 8 Kbytes– Level 2 = 133 pages = 1 Mbyte– Level 3 = 17,689 pages = 133 MBytes

A Note on `Order’

• Order (d) concept replaced by physical space criterion in practice (`at least half-full’).– Index pages can typically hold many more entries than

leaf pages.– Variable sized records and search keys mean different

nodes will contain different numbers of entries.– Even with fixed length fields, multiple records with the

same search key value (duplicates) can lead to variable-sized data entries

Inserting a Data Entry into a B+ Tree• 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, 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.

Inserting 7 & 8 into Example B+ Tree

Root

2 3 5 7 14 16 19 20 22 24 27 29 33 34 38 39

2 3 5 7 8

5

(Note that 5 is copied up andcontinues to appear in the leaf.)

17 24 3013

17 24 3013

• Observe how minimum occupancy is guaranteed in both leaf and index pg splits.

Insertion (Cont)

• Note difference between copy-up and push-up; be sure you understand the reasons for this.

appears once in the index. Contrast

5 24 30

17

13

(Note that 17 is pushed up and only

this with a leaf split.)

2 3 5 7 8

5 17 24 3013

Example B+ Tree After Inserting 8

Notice that root was split, leading to increase in height.

In this example, we can avoid splitting by re-distributing entries; however, this is usually not done in practice. Why?

checking whether redistribution is possible may increase I/Os

2 3

Root

17

24 30

14 16 19 20 22 24 27 29 33 34 38 39

135

75 8

Deleting a Data Entry from a B+ Tree

• 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.

Example Tree After (Inserting 8, Then) Deleting 19

• Deleting 19 is easy.

2 3

Root

17

24 30

14 16 20 22 24 27 29 33 34 38 39

135

75 8

Example Tree After Deleting 20 ...

• Deleting 20 is done with re-distribution. Notice how middle key is copied up.

2 3

Root

17

30

14 16 33 34 38 39

135

75 8 22 24

27

27 29

... And Then Deleting 24

• Must merge.• Observe `toss’ of

index entry (on right), and `pull down’ of index entry (below).

30

22 27 29 33 34 38 39

2 3 7 14 16 22 27 29 33 34 38 395 8

Root30135 17

Example of Non-leaf Re-distribution (Delete 24)

• In contrast to previous example, can re-distribute entry from left child of root to right child.

Root

135 17 20

22

30

14 16 17 18 20 33 34 38 3922 27 292175 832

Root

135 17 20

22

30

14 16 17 18 20

33 34 38 3927 29

2175 832

22 24

27

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.

14 16 33 34 38 3922 27 2917 18 20 2175 82 3

Root

135

17

3020 22

B+tree deletions in practice

– Often, mergeing/coalescing is not implemented– Too hard and not worth it!

Duplicates

• Using overflow page

• Several leaf pages may contain entries with the same key– To retrieve all data entries with a given key,

we search for the left-most data entry with the key, then follow the leaf sequence pointers to find other pages.

B+-Tree Performance

• Tree levels– Tree Fanout

• Size of key• Page utilization

• Tree maintenance– Online

• On inserts• On deletes

– Offline

• Tree locking• Tree root in main memory

B+-Tree Performance

• Key length influences fanout– Choose small key when creating an index– Key compression

• Prefix compression (Oracle 8, MySQL): only store that part of the key that is needed to distinguish it from its neighbors: Smi, Smo, Smy for Smith, Smoot, Smythe.

• Front compression (Oracle 5): adjacent keys have their front portion factored out: Smi, (2)o, (2)y. There are problems with this approach:

– Processor overhead for maintenance

– Locking Smoot requires locking Smith too.

Prefix Key Compression• Key values in index entries only `direct traffic’; can often compress

them.• E.g., If we have adjacent index entries with search key values

Dannon Yogurt, David Smith and Devarakonda Murthy, we can abbreviate David Smith to Dav. (The other keys can be compressed too ...)– Is this correct? Not quite! What if there is a data entry Davey Jones?

(Can only compress David Smith to Davi)– In general, while compressing, must leave each index entry greater than

every key value (in any subtree) to its left.• Insert/delete must be suitably modified.

summarize