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The B-Tree I ndex

Prof. Sin-Min Lee

Department of Computer Science

San Jose State University

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Outline

Introduction The B-tree shape

Dynamic changes in the B-tree

Properties of the B-tree Create index statement syntax

More about the B-tree

Summary

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Introduction

A B-tree is a keyed index structure, comparable to anumber of memory resident keyed lookup structures

such as balanced binary tree, the AVL tree, and the 2-3

tree.

The difference is that a B-tree is meant to reside on disk,being made partially memory-resident only when entries

int the structure are accessed.

The B-tree structure is the most common used index typein databases today.

It is provided byORACLE, DB2, and INGRES.

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The B-Tree Shape

A B-tree is built upside down with the root at the top andthe leaves at the bottom.

All nodes above the leaf level, including the root, are

called directory nodes or index nodes.

Directory nodesbelow the root are calledinternal

nodes.

The root node is known aslevel 1 of the B-tree and

successively lower levels are given successively largerlevel numbers with the leaf nodes at the lowest level.

The total number of levels is called thedepthof the B-

tree.

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Balanced and Unbalanced Trees

Trees can be balancedor unbalanced.

In a balanced tree, every path from the route to a leaf node is

the same length.

A tree that is balanced has at most logorder

nlevels. This is

desirable for an index.

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The Problem of Unbalanced

Trees a. A Troublesome Search Tree

b. A More Troublesome Search Tree

1

2

5

3

4

6

9

7

8

5

4

3

2

1

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Disadvantage of unbalanced tree

Searching an unbalanced tree may

require traversing an arbitrary and

unpredictable number of nodes and

pointers.

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Unbalanced Tree (cont.)

Problems:

1. The levels of the tree are onlysparsely filled

2. Resulting in long

3. Deep paths and defeating thepurpose of binary trees in the first

place.

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The general name for B-trees is multiwaytrees. However,

the best known of them have their very own names: 2-3

trees,B* trees, andB+ trees. Here we will look only at 2-3trees. They are the simplest. And or purposes of learning

the underlying principles this is good. Multiway trees can

get pretty complicated pretty fast. Keep in mind that 2-3

trees as we study them are rarely built. Their larger

cousins, B* and B+ trees are often used for medium and

large database applications. The general idea of a

multiway tree of order nis that each node can hold up to n

- 1key values and each node can have up to nchildern.

So, a 2-3 tree is actually a multiway tree of order 3. Thatmeans that a 2-3 tree has nodes that can hold 1 or 2 values

and that a node can have 0, 1, 2, or 3 children. Thus the

name 2-3 tree

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The B-Tree Shape (c.)

level1: root node, level2: directory nodes, level3: leaf nodes

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B-Trees

For a binary search tree, the search time is O(h), where h is

the height of the tree.

The height cannot be less than roughly log2

n for a tree

with n nodes.

If the tree is too large for internal memory, each access of a

node is an I/O.

For a tree with 10,000 nodes:

log210000 = 100 disk accesses

We need a structure that would require only 3 or 4 disk

accesses.

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B-Trees - Definition

A B-tree of order M is an M-ary tree

with the following properties:

(1) The data items are stored at leaves.

(2) The nonleaf nodes store up to M - 1

keys to guide the searching; key i represents

the smallest key in subtree i + 1.

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(3) The root is either a leaf or has between 2

and M children.

(4) All nonleaf nodes (except the root) havebetween ceiling(M/2) and M children.

(5) All leaves are at the same depth and

have between ceiling(L/2) and L data items,for some L.

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B-tree of order n

Every B-tree is of some "order n", meaning nodes

contain from n to 2n keys (so nodes are always at

least half full of keys), and n+1 to 2n+1 pointers,and n can be any number.

Keys are kept in sorted order within each node. A

corresponding list of pointers are effectively

interspersed between keys to indicate where tosearch for a key if it isn't in the current node.

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A B-tree of order n is a multi-way search

tree with two properties:

1.All leaves are at the same level

2.The number of keys in any node lies

between n and 2n, with the possible

exception of the root which may have fewerkeys.

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Other definition

A B-tree of order m is a m-way tree that satisfies the

following conditions.

Every node has < m children.

Every internal node (except the root) has 2 children.

An internal node with k children contains (k-1) ordered

keys. The leftmost child contains keys less than or equal tothe first key in the node. The second child contains keys

greater than the first keys but less than or equal to the

second key, and so on.

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A B-tree of order 2

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Dynamic changes in the B-Tree

A B-tree is an efficient self-modifying structure whennew entries are inserted pointing to new rows inserted in

the indexed table.

The nodes at every level are generally assumed not to be

full.

Space is left so that inserts are often possible to a node at

any level without new disk space being required.

An insert of a new entry always occurs at the leaf level,but occasionally the leaf node is too full to simply accept

the new entry. In this case, for additional space the leaf

level node is split into two leaf pages.

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Properties of the B-Tree

Assumptions: Entry key values can have variable length because of

variable-length column values appearing in the index

key. When a node split occurs, equal lengths of entry

information are placed in the left and right split node.

Rebalancing actions in the B-tree occur when entries are

deleted.

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Properties of the B-Tree (c.)

Properties: Every node is disk-page sized and resides in a well-

defined location.

Nodes above the leaf level contain directory entries, with

n-1 separator keys and n disk pointers to lower-level B-

tree nodes.

Nodes at the leaf level contain entries with (keyval,

rowid) pairs pointing to individual rows indexed. All nodes below the root are at least half full with entry

information.

The root node contains at least two entries.

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Insertion in B-Tree

1. 2.

a, g, f,b: k:

a b f g

a b g k

f

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Insertion (cont.)

3. 4.

d, h, m: j:

5. 6.

e, s, i, r: x:

f

a b d g h k m

f j

a b d g h k m

f j

a b d e k m r sg h i

f j r

g h i s xk ma b d e

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Insertion (cont.)

7.

c, l, n, t, u:

8.p:

c f j r

s t u xk l m ng h ia b d e

j

a b d e k l n p

m rc f

g h i s t u x

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Inserting into a B-Tree

To insert key value x into a B-Tree:

Use the B-Tree search to determine on which node

to make the insertion. Insert x into appropriate position on that leaf

node.

If resulting number of keys on that node < L, then

simply output that node to disk and return.

Otherwise, split the node.

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Inserting into a B-Tree: Splitting a Node

Allocate a new leaf node. Put about half (i.e.,

about L/2) of the keys on the new node and

leave about half of the keys on the existing node. Make appropriate changes to keys and pointers in

the parent node.

If the parent node was already full, then split theparent node.

The splitting of parents may continue all the way

back up to the root node.

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Insert 19,12, 22,15.

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Insertion

Insert the keys in the folowing order into a B-tree of order 5.

A, G, F, B, K, D, H, M, J, E, S, I, R, X, C, L, N, T, U, P.

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Searching

Searching for an Item in a B-Tree:

1. Make a local variable, i, equal to the first index such

that data[i] >= target. If there is no such index, then set i

equal to data_count, indicating that none of the entries is

grater than or equal to the target.

2. if (we found the target at data[i])

return true;

else if (the root has no children)

return false;

else

return subset[i]->contains (target);

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Searching (cont.)

Example: target = 10

2 3

19 22

6 17

1610 18 20 25

12

5

4

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Deletion form a B-Tree

1. detete h, r :

s promote s

and

delete form leaf

j

c f

g i

d ea b k l n p

m r

g h i

t u x

s t u x

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Deletion (cont.)

2. delete p :

t

pull s down; pull t up

j

g i n pk ld ea b

m sc f

t u x

n s

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Deletion (cont.)

3. delete d:

Combine:

j

c f

g id ea b k l n su x

m t

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Deletion (cont.)

combine :

f

j

u xn sk lg i

g i k l n s u x

m t

a b c e

f j m t

a b c e

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Deleting from a B-Tree

To delete a key value x from a B-tree, first searchto determine the leaf node that contains x.

If removing x leaves that leaf node with fewerthan the minimum number of keys, try to adopt a

key from a neighboring node. If thats possible,

then youre finished.

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Deleting from a B-Tree

(continued) If the neighboring node is already at its minimum,

combine the leaf node with its neighboring node,

resulting in one full leaf node. This will require restructuring the parent node

since it has lost a child

If the parent now has fewer than the minimum

keys, adopt a key from one of its neighbors. Ifthats not possible, combine the parent with its

neighbor.

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Deleting from a B-Tree

(continued) This process may percolate all the way to

the root.

If the root is left with only one child, thenremove the root node and make its child the

new root.

Both insertion and deletion are O(h), whereh is the height of the tree.

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Delete 18

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Delete 5

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Delete 19

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Delete 12

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Properties of B-Trees

(1) In a B-tree of order m with n keys the

number of nodes, p, satisfies

and on average p ~1.44 n/m.

(2) On average less than

nodes are split per insertion

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Advantages of B-tree

Searching a balanced tree means that all leaves are

at the same depth. There is no runaway pointer

overhead. Indeed, even very large B-trees can

guarantee only a small number of nodes must be

retrieved to find a given key. For example, a B-

tree of 10,000,000 keys with 50 keys per node

never needs to retrieve more than 4 nodes to findany key.

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ORACLE Create Index Statement

Syntax:create[unique]index indexname ontablename

(columnname [asc| desc] {, columnname [asc| desc]})

[tablespace tblspacename]

[storage( [initialn] [nextn] [minextentsn]

[maxextentsn] [pctincreasen] ) ]

[pctfreen]

[other disk storage and transaction clauses not covered

or deferred]

[nosort];

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ORACLE Create Index Statement (c.)

Explanation:

The value of n inpctfree can range from 0 to 99 and this

number determines the percentage of each B-tree node

page.

The list of columnnames in parentheses on the secondline specifies a concatenation of column values that make

up an index key on the table specified.

Thenosortindicates that the rows already lie on disk insorted order by the key values for this index.

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DB2 Create Index Statement

Syntax:create[unique] indexindexname ontablename

(columname [asc| desc] {, columnname [asc| desc]})

[using . . .][freepagen]

[pctfreen]

[additional clauses not covered or deferred];

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DB2 Create Index Statement (c.)

Explanation: The usingspecifies how the index is to be constructed

from disk files.

The integer n of the freepagespecifies how frequently an

empty free page should be left in the sequence of pages

assigned to the index when it is loaded with entries by a

DB2utility.

One free page is left for every n index pages where nvaries from 0 to 255. The defaut value for n is 0 meaning

that no free pages are left. A value of n=1 means that

alternate disk page are left empty.

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INGRES Create Index Statement

Syntax:create[unique] indexindexname ontable

(columname {, columnname})

[with /* comas separate clauses following */[location= . . . ]

[structure= btree|isam |hash |. . .]

[key= (columnname {, columnname})]

[fillfactor= n]

[nonleaffill= n]

[additional clauses not covered or deferred] ];

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INGRES Create Index Statement (c.)Explanation:

The withkeyword must be present if any of the later

coma-separated clauses appear.

Thelocation specifies how the index is to be constructed

from disk files. Thestructureis unique to INGRESand names the

access structure that the index will be assigned when it is

created.

Thekey indicates that the index key value will be

constructed from the columnnames listed.

Thefillfactor andnonleaffill gives the percentage of

node space that should be filled.

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Index Node Layout and Free Space

Below is the schematic layout of a normal leaf-level index

node with unique key value.

Header info Free space

keyval rid keyval rid

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More about the B-Tree

The purpose of the B-tree index is to minimize the

number of disk I/Os needed to locate a row wit a given

index key value.

The depth of the B-tree bears a close relationship to the

number of disk I/Os used to reach the leaf-level entrywhere the rowid is kept.

The nodes of the B-tree are loaded in a left-to-right

fashion so that successive inserts normally occur to the

same leaf node held consistently in memory buffer.

When the leaf node splits, the successive leaf node is

allocated from the next disk page of the allocated extent.

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More about the B-Tree (c.)

Node splits at every level occur in a controlled way andallow us to leave just the right amount of free space on

each page.

It is common to estimate the fanout at each level to have

a value of n where n is expected number of entries thatappear in each node. Assuming that there are n directory

entries at the root node and every node below that, the

number of entries at the second level is n^2, at third level

is n^3, and so on. For a tree of depth K, the number of

leaf-level entries is n^K just before a root split occurs in

the tree to make it a tree of depth K+1.

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Summary

The B-tree is a tree-like structure that helps us to

organize data in an efficient way.

The B-tree index is a technique used to minimize the disk

I/Os needed for the purpose of locating a row with a

given index key value. Because of its advantages, the B-tree and the B-tree index

structure are widely used in databases nowadays.