search for 8\h is 779 262 727 L 97 < 27 179 , 62 - Search is - - PowerPoint PPT Presentation

m-arysearchtree.sn

rooted trees where

• each node may have up to m children

• a node with

K children has k-1 keys

• search is

a natural generalization of BSTsearch :

search for8\h

is

⇐

727

262

779

, 62

179

L 97

• Search is a generalization of BST search
• Generalization of in - order traversal visits keys in order

Order m B-Tree ( As per textbook)

5.

m- ar

search tree

• data Items stored at

leaves , keys in other nodes guide search

• non- leaf nodes , other thanthe root have

between TM1N and

m children

• the root has between 2 and

M children , or

is a leaf

• in a

non - heat

node , the

ith key

is the smallest

key stored in the ittst subtree

are at the

same depth

L keys ( for

some

(unless there

are fewer than Hd

(

fixed L )

keys

in the

tree

B-Tree Search : Generalization of BST search .

• Example , in an order -5 B-tree

:(

This tree :

• must

have

• rder 75 because of

*

• must

not

be of order

> 6 because of D.)

Find 49:\

⑤ ¥

"

3

sieetetteight.fi/Perfect''Trees-Aperfeetm-arytree-

is

an

m - ary tree where :

• every node has Zero
• r

m children

• every

leaf has the same

depth

• A perfect m-ary tree of height h has the max # of

nodes for

a height

• h m-ary tree

Fact:

The number of nodes in a perfect

m - airy tree of

• height h

is :

mh

M - I

for : The number of nodes in

a perfect binary tree

• f height

h

is

2h11

The number of nodes at depth d. in a perfect

binary tree

• f height at

least d

is

2d

• Proof By induction
• n d

Basis : D= 0

2d=2° = I

• I. It . Choose

some d 70 an assume that

the number of

nodes at depth d

is 2d

Is . We must show that the number of

nodes

at depth

dtt

is 2d?

Since the tree

is

a perfect binary tree

• f height

> dtl , every node at

depth

d

has

2 children .

So

nodes at depth dH

is 2. 2d

The number of

nodes at depth d

in a perfect m - ary

tree of height at

least d

is md

PI By induction

• n depth d-

Basis:

D= 0

There

is just the root

III.

Choose

some d > 0

and

assume the

number of nodes at depth d

is

md

II.

depth dtl

is mdtt

Since the tree is

a perfect m - ary tree of

height

at

least dtt

node at

depth d has

m children .

So

nodes at depth dtl

is

M

• Md

Claim : Every

• rder
• m B-tree of height h

contains

at

least (E)

h

keys

P¥ Observe that every m- ary B-tree of height

h

has a perfect (f)

• cry tree as

a

sub

• graph

:

n

'

⇒

This subtree has (E)

"

leaves

so the B-tree

does also

Since each leaf

has at least

HE keys , the B- tree contains at least CH2Xmath key

1¥ : The height of

an order - m

B-tree with

n

keys

is

at most

log m=n

If :

Let T be an order m B-tree with

n nodes

• and height h .

We have : ¥)h

±

n

So

h

E

log; n =0(log

n)

⇐ Searching for

a key in a B-tree with

n keys

can be done in time 0( log

B-treeinserti.org/L=m-- 5)

① Insert 577

I

*

After inserting 57 :

go

.

Now , insert 5

[

This leaf will be "overfull"

=D we spH

÷

⇒

After insertion of 554 splitting of leaf :

splittinganInternalhlodet-ns.at

407

←

* splitting the leaf

g④F%¥¥

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2

3

4

5

6

1

2

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4

5

6

Tree just got taller

• Still , leaves all

have the

same (larger) depth

BTreeInsertij

Find the leaf where

new key k belongs , call it

v.

v

while(v

is

• ver- full and not the root

) {

split

v.

v ← parent G)

¥ r is the

root and

• ver - full

v

in two ,

a

new root with two children

• Claim : B - tree

insertion can be done

in

time

0 ( log n) , where

n is the number of

keys

in the tree .

B-treeRemov

L with the key K

to be removed

• delete K from L
• if

K was the smallest key in L , correct the key in an ancestor

• V ← L

• ( r

has too few keys

and

is not root) {

if ( v has an adjacent sibling with more than

FY27 children)l

shift

a key and child

• rom th
• or

I

} else{

merge

v with a sibling

• f r

g

v ← parent of the merged

node

}

if I v

is root and has only

• ne child , c )

delete vj

c is the

new root .

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