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B-Trees

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MotivationWh e n data i s too la rge to fit in th e main m e mo ry, the nthe numb er of di sk acc esses be com es impo r tant.

A di sk acc ess is unb e lie vabl y ex pe ns ive compa re d to a

typical comput er ins tr uction (m e chanical limitation s) .

On e disk acc ess is wo r th 200 ,000 comput er ins tr uction s .

The numb er of di sk acc esses will dominat e the r unnin g time .

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Motivation (contd. )Se conda ry me mo ry (disk) is divide d intoeq ual -s ize d bloc ks (typical s ize a re 512 , 2048 ,4096 , or 8192 b ytes) .

The ba s ic I/O op er ation t r an s f ers the cont e nts of on e disk bloc k to/f r om RAM.

Ou r goal i s to d e vise multi wa y se a r chtree that will minimiz e file acc ess ( b y ex plo r ing disk bloc k re ad ).

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Multi wa y se a r ch t rees (of o r der m)

A ge ner alization of Bina ry Se a r ch Trees .

Each nod e ha s at mo s t m child re n.If k m is the numb er of child re n, the n th e nod e

ha s ex actl y k-1 keys .

The tree is or dere d.

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B-Trees A B-tree of o r der m is m-wa y se a r ch t ree .B-Trees a re balanc e d se a r ch t rees des igne d to

work we ll on di re ct acc ess se conda ry s tor a ge de vices .

B-Trees a re s imila r to R e d-Blac k Trees, but a re

be tter at minimizin g disk I/O op er ation s . All le av es a re at th e s am e le ve l.

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M

QTX

RS

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H e ight h = 42-le av es at d e pth 22-le av es at d e pth 31-le af at d e pth 4

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H e ight h = 26-le av es at d e pth 2

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B-Tree P r op er tiesB-Tree is a r oot e d t ree with root[T] with th e followin g pr op er ties:1 - Ev ery nod e x ha s the followin g fie lds .

a -n [ x ], the numb er of keys cu rre ntly s tore d in x .

b -The n [ x ] keys, the mse lves s tore d in non d e cre a s ing (As ce ndin g/Inc re a s ing) or der .

key 1[ x ] key 2 [ x ] key n [ x ].

c -L eaf [ x ], a Bool e an valu e that i s TRUE if x is le af , and fal se if x is inter nal nod e .

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P r op er ties Contd2 - if x is an int er nal nod e, it al s o contain s n [ x ]+1 point ers

to it s child re n. L e af nod e contain s no child re n.

3 -The keys key i [ x ] se pa r at e the r an ge of keys s tore d ine ach s ub t ree : if k 1 is an y key s tore d in th e s ub t ree with r oot c 1[ x], the n:

k1 key 1[ x ] k 2 key 2 [ x ] key n [ x ] [ x ] k n [ x ]+1

4 -Each l e af ha s the s am e de pth , which i s the he ight of the tree h .

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P r op er ties Contd5 -There a re lower and upp er bound on th e numb er of

keys a nod e can contain.

These bound s can b e ex presse d in t er ms of a fi xe dinteger t 2 , call e d th e minimum d egree of B-Tree .

Wh y t cant b e 1?

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P r op er ties Contda -Every nod e oth er than th e r oot mu s t hav e at l e a s t t -1

keys, Every inter nal nod e oth er than r oot , thu s ha s at

le a s t t child re n. If th e tree is non e mpt y, the r oot mu s thav e at l e a s t on e key .

b -Ev ery nod e can contain at mo s t 2 t -1 keys . There fore, an int er nal nod e can hav e at mo s t 2 t child re n. W e s a y a nod e is full if it contain s ex actl y 2 t -1 keys .

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He ight of a B-TreeWhat i s the ma ximum h e ight of a B-Tree with N

e ntr ies ?

This ques tion i s impo r tant , be cau se the ma ximumhe ight of a B-Tree will give an upp er bound on th e numb er of di sk acc esses .

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H e ight of a B-Tree

If n 1 , than fo r an y n -key B-Tree T of h e ight h andminimum d egree t 2 ,

$

21

logn

ht

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1

root[T]

t-1 t-1

t-1 t-1 t-1 t-1

t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1

# of nodes

1

2

2t

2t 2

t t

tt t t

A B-Tree of height 3 containing minimum possible keys

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P r oof Numb er of nod es is minimiz e d, whe n r oot

contain s on e key and all oth er nod es contain t -1 keys .

2 nod es at d e pth 1 , 2 t nod es at d e pth 2 ,

2 t 2 nod es at d e pth 3 and s o on.

At d e pth h , there a re 2 t h -1 nod es .

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P r oof( Contd. )Thu s numb er of keys (n ) s ati s fies the ineq ualit y:

$

$

u

u

u !

21

log

21

12

11

)1(21

2)1(11

1

nh

t n

t n

t

t t n

t t n

t

h

h

h

ih

i

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Num er ical E xampl eFor N= 2 ,000 ,000 (2 Million ), and m=100 ,

the ma ximum h e ight of a t ree of o r der mwill be onl y 3, wh ere a s a bina ry tree would

be of h e ight la rger than 20.

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Re adin gChapt er 19 B Trees of boo k Int r oductionto Algor ithm s By Thoma s H . Co r me n e t al