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Multilevel Indexing andB+ Trees

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Indexed Sequential Files

Provide a choice between two alternativeviews of a file:

1. Indexed : the file can be seen as a set of records that is indexed by key; or

2. Sequential: the file can be accessedsequentially (physically contiguousrecords), returning records in order bykey.

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E xample 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

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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.

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Maintaining a Sequence Set

Sorting and re-organizing after insertions anddeletions is out of question. We organize thesequence 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 overflowBlock merging or redistribution when deletion causesunderflow.

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E xample: insertionBlock size = 4Key : Last name

ADAMS BIXBY CARSON COLE Block 1

Insert BAIRD :

ADAMS BAIRD BIXBY

CARSON .. COLE

Block 1

Block 2

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E xample: deletionADAMS BAIRD BIXBY BOONE Block 1

BYNUM CARSON .. CARTER ..

DENVER ELLIS

Block 2

Block 3

Delete DAVIS, BYNUM, CART E R,

COLE DAVISBlock 4

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Add an Index set

Key Block

BERNE 1CAGE 2DUTTON 3EVANS 4F OLK 5GADDIS 6

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Tree indexes

This simple scheme is nice if the index fits inmemory.If index doesnt fit in memory: Divide the index structure into blocks,

Organize these blocks similarly building a treestructure.

Tree indexes: B Trees

B+ Trees Simple prefix B+ Trees

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Separators

Block Range of Keys Separator1 ADAMS-BERNE

BOL E N2 BOLEN-CAGE

CAMP3 CAMP-DUTTON

E MBRY4 EMBRY-EVANS

FAB E R 5 F ABER- F OLK

FOLKS6 F OLKS-GADDIS

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ADAMS-BERNE F OLKS-GADDIS

FABER- F OLK BOLEN-CAGE

CAMP-DUTTON EMBRY-EVANS

BOLEN CAMP

EMBRY

FABER F OLKS

1

2

3

5

4 6

Index set

root

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B Trees

B-tree is one of the most important data structuresin 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 inleaf nodes, and faster sequential access is possible.

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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

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B+ tree: Internal/root node

structureP 0 K1 P 1 K2 P n-1 Kn P n

Requirements:

K1 < K 2 < < K n

For any search key value K in the subtree pointed by Pi,If Pi = P0, we require K < K1If Pi = Pn, Kn e KIf Pi = P1, , Pn-1, Ki < K e Ki+1

Each P i is a pointer to a child node; each K i is a search key value# of search key values = n, # of pointers = n+1

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Pointer L points to the left neighbor; R points tothe right neighbor

K1 < K2 < < K nv e n e 2v (v is the order of this B+ tree)

W e will use K i* for the pair and omit L andR for simplicity

B+ tree: leaf node structure

L K1 r 1 K2 Kn r nR

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E xample: B+ tree with order of 1

Each node must hold at least 1 entry, and at most2 entries

10* 15* 20* 27* 33* 37* 40* 46* 51* 55* 63* 97*

20 33 51 63

40

Root

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E xample: Search in a B+ tree order 2Search: 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 >= 2 4* ... The last one is a range search, we need to do the sequential scan, starting from

the first leaf containing a value >= 2 4 .Root

17 24 30

2* 3* 5* 7* 14* 15* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*

13

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B+ Trees in Practice

Typical order: 100. Typical fill-factor: 67% . average fanout = 1 33 (i.e, # of pointers in internal

node)

Can often hold top levels in buffer pool: Level 1 = 1 page = 8 Kbytes Level 2 = 1 33 pages = 1 Mbyte Level 3 = 1 7 ,68 9 pages = 1 33 MBytes

Suppose there are 1,000,000,000 data entries. H = log 133 (1000000000/1 3 2) < 4

The cost is 5 pages read

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H ow to Insert a Data E ntry into a

B+ Tree?

Lets look at several examples first.

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Inserting 16*, 8* into E xample B+ tree

Root 17 24 3013

2* 3* 5* 7* 8*

2* 5* 7*3*

17 24 3013

8*

You overflow

O ne new child (leaf node)generated; must add one morepointer to its parent, thus one morekey value as well.

14* 15* 16*

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Inserting 8* (cont.)

Copy up themiddle value(leaf split)

2*3* 5*

7* 8*

5Entry to be inserted in parent node.(Note that 5 iscontinues to appear in the leaf.)

s copied up and

13 17 24 30

You overflow!5 13 17 24 30

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(Note that 17 is pushed up and only

appears once in the index. Contrast

Entry to be inserted in parent node.

this with a leaf split.)

5 24 30

17

13

Insertion into B+ tree (cont.)

5 13 17 24 30

Understanddifference

between copy-upand push-up

Observe howminimumoccupancy isguaranteed in

both leaf and

index pg splits.

We split this node, redistribute entries evenly,and push up middle key.

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E xample B+ Tree After Inserting 8*

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

2* 3*

Root

17

24 30

14* 15* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*

135

7*5* 8*

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Deleting a Data E ntry from a B+

TreeExamine examples first

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Delete 19* and 20*

2* 3*

Root

17

24 30

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

135

7*5* 8*

22*

27* 29*22* 24*

Have we still forgot something?

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Deleting 19* and 20* (cont.)

Notice how 2 7 is c opied up .But can we move it up?Now we want to delete 2 4

Underflow again! But can we redistribute this time?

2* 3*

Root

17

30

14* 16* 33* 34* 38* 39*

135

7*5* 8* 22* 24*

27

27* 29*

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Deleting 24*Observe the two leaf nodes are merged, and27 is discarded fromtheir parent, but

Observe 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* 39*5* 8*

30135 17N ew root

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Deleting a Data E ntry from a B+ Tree:

SummaryStart 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 (adja c ent nodewith 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.

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E xample of Non-leaf Re-distribution

Tree is shown below during deletion of 2 4* . (Whatcould be a possible initial tree?)In contrast to previous example, can re-distribute entryfrom left child of root to right child.

Root

135 17 20

22

30

14* 16* 17* 18* 20* 33* 34* 38* 39*22* 27* 29*21*7*5* 8*3*2*

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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;weve re-distributed 1 7 as well for illustration.

14* 16* 33* 34* 38* 39*22* 27* 29*17* 18* 20* 21*7*5* 8*2* 3*

Root

135

17

3020 22

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Terminology

Bucket Factor: the number of records which canfit in a leaf node.Fan-out : the average number of children of aninternal node.A B+tree index can be used either as a primaryindex 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 secondaryindexes (also called a dense index)

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SummaryTree-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 managementsystems because of its versatility. One of the mostoptimized components of a DBMS.