B-T Inverters : Bayer ' ? McCreight , 1970 Boeing Research Labs . - - - PowerPoint PPT Presentation

B-T

Inverters : Bayer

? McCreight ,

1970

Boeing Research Labs.

why B-

• tree ?

Bayer , Boeing , Broad , Bushy ,

? ?

McCreight , 2014

"

the Bin B-trees

means , the better you understand

B- trees .

(The paper introducing them

was initially rejected)

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

A

⇐

727

262

779

, 62

179

L 97

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

Order m B-Tree ( As per textbook)

5-

*

tree ,

• data Items stored at

leaves , keys in other nudes 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 Itt" subtree

• all leaves

are at the

same depth

L keys ( for

some

(unless there

are fewer than THE

L

fixed L )

keys

in the

tree

* in this version of B-tree , not an

defined on

previous

slide .

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 56

because of D.)

Find 49:\

µ

⑤

③

"

3

sieetetteightofttperfecttrees-Aperfeetm-arytree-isanm.ae

ry 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 -20=1

• I. It . Choose

some d

an assume that

the number of

nodes at depth d

is 2d

Is . We must show that the number of

nodes

• at depth

dH

is 2d?

✓↳ since the tree

is

a perfect binary tree

&

• f height

> dtl , every node at

:

depth

d

has

2 children .

d-

• /\o

So

nodes at depth dtt

dtt

is g. 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

every

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

:

⇒

This

"

leaves

so the B-tree

does also

Since each leaf

has at least hz 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

¥

*

After inserting 57 :

go

.

Now , insert 5

[

This leaf will be "overfull"

=D we spH

÷:÷÷÷:¥i⇒

After insertion of 554 splitting of leaf :

splittinganInternalhlodet-ns.at

407

←

* splitting the leaf

g④F%¥¥

" ¥

¥

:

Afterspliking

splittingtheRoot.orerfullroyt.j.IT/.y?Newroot

11111×117 EE

THI THI

A

1 b. b. x.

.

.

I b. b. ¥

, I k

Tree just got taller

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 u with more than MK7 children)l

shift

a key and child from it to v.

} else{

merge

v with a sibling

• f r

}

u ← parent of the merged

node

}

if I v

is root and has only

• ne child , c )

delete vj

c is the

new root .

(corrected.)

Ifkwasthesmadestheafinl.ir

①

,

⇒

'

If node v has too few keys

and

is

not root . .

• sometimes

can metre

v with

an adjacent sibling

1-

i¥¥⇒

'¥R

• ADD A

n

A

A

A

D B

• sometimes not

1-

merging

r with a

H¥

✓ I

1

I \

sibling makes

5¥ D; f I

a node

that is Overfull

fEff⇒)

I'T

remove 79

=D

→

in :

¥

it.sn.

y

"extra " means/

④ shift

a key (76)

more than the

+

min

T

Et

t t

↳

( corrected)

tIingunderfuHnod

• shifting

is cheaper than

merging ,

and never leaves the

parent under -full

possible

not possible

( both adjacent siblings

have

min . # keys)

then fix by

merging

tixinfuging.GL?fym=5)

12614116¥

remove

52

←

←

to

→

I

TITHE

HH

am

I

p

b

t

b I

*

I too small

merge with sibling

¢

d-

*

→

m

%

shiftingataninternaluodlnas.IE

:b

"

to 1 ↳ ←

too few

A T¥

shift a key+ child.

→

¥71

µ. aissi rotation)

4 I

✓

↳

e

Hn

*

i.

*

1¥

tool

*

whenamergeshrinksaninterndnode-T.IE

← ↳ I

← just

got

too

small

ADD A

A

because of a

T1

The

T3

T4 T5

deletion

$

merge

Y

9-

I

/

I

1

I

l

A DAD D

TI

T2

T3 T4

T5

whentherootgetstoosmali.FI

← ↳

T

Fol ← just

got

too

small

/

1

IT

17PM

17 17

$

merge

IBE

Tree height

④so¥

is

reduced

/

I

1

'

'

by

1

LOmparisonof2logznandlogmk.lv

• rst case height of Avhtree

• We choose

m= 1024 , for a concrete

example

R

Rogen Logan

• 10

7

I

100

13 I

1000

20 I 104

27

2

106

40

2

108

53

3

10"

66

4

10 " 80 5

(logout

End

Motivation