maintenance of materialized

8/11/2019 Maintenance of Materialized

1/30

US

20070192283A1

( 1 9 ) United S t a t e s

( 1 2 ) Patent

Application

Publication ( 1 0 ) P u b .

N o . :

US 2 0 0 7 / 0 1 9 2 2 8 3 A 1

Larson e t a l .

( 4 3 ) P u b .

D a t e :

Aug.

1 6 ,

2007

( 5 4 )

MAINTENANCE

OF MATERIALIZED ( 5 2 ) US. l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707/2

OUTER-JOIN VIEWS

( 7 5 ) I n v e n t o r s :

P a r - A k e

L a r s o n ,

R e d m o n d ,

WA

U S ) ;

( 5 7 )

ABSTRACT

J i n g r e n

Z h o u , R e d m o n d , WA U S )

M a i n t e n a n c e

o f

m a t e r i a l i z e d

vieW f o r

q u e r y o p t i m i z a t i o n

i n

Correspondence Address: - - -

a d a t a b a s e .

The

a r c h i t e c t u r e p r o v i d e s t h e

? r s t p r a c t i c a l

AMIN' TUROCY & ALVIN L LP a l g o r i t h m s f o r i n c r e m en t a l ly m a i n t a i n i n g v i e W s W i t h

m u l

CENTER t i p l e o u t e r j o i n s a n d a g g r e g a t e d o u t e r - j o i n v i e W s . M a t e ri a l

CLEVELAND OH 4114

U

i z e d

v i e W s

W i t h a n y c o m b i n a t i o n

o f

i n n e r j o i n s ,

o n e - s i d e d

( ) o u t e r j o i n s a n d

f u l l o u t e r

o i n s , p l u s

a n

o p t i o n a l a g g r e g a t i o n

( 7 3 )

A s s i g n e e ' M i c r o s o f t

C o r p o r a t i o n Redmond WA o n

t o p c a n

b e p r o c e s s e d b y a l g o m h m ( s )

t h a t

C o n s t r u c t

incremental

maintenance expressions based on

converting

( 2 1 ) AP 1 N _ 1 1 / 3 5 4 375

t h e

vieW

d e ? n i t i o n a

j o i n - d i s j u n c t i v e

normal

form and

exploiting

database

constraints to reduce

maintenance

o v e r

- . ea . s stem com

rises a v1eW creation

com onent or

(22)

F i l e d . Feb.

1 5 ,

2 0 06 h d A Y P P f

c r e a t i n g

a m a t e r i a l i z e d vieW

d e ? n i t i o n

t h a t

i n c l u d e s o u t e r

P u b l i c a t i o n Cl a s s i ? c a t i o n

j o i n s a

c o n v e r s i o n

component

f o r c o n v e r t i n g

a n o u t e r - j o i n

vieW

e x p r e s s i o n

i n t o

a

normal

f o r m ,

and

a

maintenance

( 5 1 )

I n t , C l , c o m p o n e n t

f o r u p d a t i n g

t h e s t o r e d


v i e W

r e s u l t

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15/30

US

007/0192283 A 1

MAINTENANCE

OF

MATERIALIZED

OUTER-JOIN VIEWS

BACKGROUND

[ 0 0 0 1 ]

With

h e

a d v e n t

o f

t h e

I n t e r n e t ,

m a s s i v e

a m o u n t s

o f

d a t a a r e

becoming

more a c c e s s i b l e t o l a r g e

number

of s e r s ,

W h e t h e r i n i n t e r n a l c o r p o r a t e

e n v i r o n m e n t s

o r i n t h e p u b l i c

arena such a s Web-based d a t a b a s e s . Accordingly, more f f i

c i e n t a n d p e r f o r m a n c e - d r i v e n r e q u i r e m e n t s a r e b e i n g p l a c e d

on u c h s y s t e m s

a s

t h e number of s e r , and o n s e q u e n t l y , t h e

number

a n d c o m p l e x i t y o f

q u e r i e s

c o n t i n u e s t o i n c r e a s e .

[ 0 0 0 2 ] I n

a r e l a t i o n a l d a t a b a s e management

s y s t e m ,

a s an

e x a m p l e , r e l a t i o n s h i p s b e t W e e n da t a

( o r

? l e s ) a r e cre a t e d

b y

comparing d a t a ,

such

a s

a c c o u n t

numbers

and

names.

The

d a t a b e i n g compared a n d t h e

r e s u l t s

c a n b e s t r u c t u r e d i n t h e

f o r m

o f

a b l e s . A o i n p r o c e s s

i n c l u d e s

m a t c h i n g

r e c o r d s

i n

one t a b l e ( o r ? l e ) W i t h

r e c o r d s

i n a n o t h e r ,

b a s e d

on some

c o n d i t i o n ,

and

c r e a t i n g a

t h i r d t a b l e t h a t i n c l u d e s

d a t a

from

t h e

t a b l e s b e i n g

j o i n e d . I n one

e x a m p l e ,

a n Order

t a b l e

can

be

o i n e d

With

a

Customer

a b l e

t o

c r e a t e

a

t h i r d t a b l e f o r

a l l

c u s t o m e r s

Who p u r c h a s e d

a

p a r t i c u l a r p r o d u c t .

[ 0 0 0 3 ]

T h e d e f a u l t

t y p e

o f

j o i n i s t h e i n n e r

j o i n ,

W h i c h

p r o d u c e s a

r e s u l t i n g

r e c o r d Whenever tWo

r e c o r d s ,

one

from

e a c h i n p u t t a b l e ,

s a t i s f y

t h e

m a t c h i n g

c o n d i t i o n . C o n s i d e r

a n o t h e r

e x a m p l e t h a t m a t c h e s s h i p m e n t s W i t h

r e c e i p t s . T h e

j o i n Would

p r o d u c e

a t a b l e

o f

s h i p m e n t s

t h a t

h a v e b e e n

r e c e i v e d b u t o m i t s h i p m e n t s

t h a t

h a v e n o t been

r e c e i v e d .

C o n t r a r i W i s e ,

an

o u t e r j o i n Would c r e a t e a t a b l e o f e v e r y

s h i p m e n t , Whether o r n o t i t Was r e c e i v e d . Data f o r r e c e i v e d

i t e m s

i s

a t t a c h e d t o t h e s h i p m e n t s .

Empty

( o r n u l l ) ? e l d s

Would b e a t t a c h e d t o s h i p m e n t s

W i t h o u t

r e c e i p t s .

[ 0 0 0 4 ] M a t e ri a l i Z e d v i e W s ,

a l s o

c a l l e d i n d e x e d v i e W s , a re

a

mechanism f o r s p e e d i n g

up

p r o c e s s i n g

o f

d a t a b a s e q u e

r i e s .

A

a t e r i a l i z e d

vieW

s

a

precomputed

e s u l t t h a t

m a y

be

u s e d

to compute p a r t o r

a l l o f

q u e r y

i n s t e a d

o f computing

t

d i r e c t l y from b a s e t a b l e s .

M a t e r i a l i Z e d

v i e W s c a n p r o v i d e

d r a m a t i c i m p r o v e m e n t s i n q u e r y p r o c e s s i n g t i m e .

T o

b e

u s e f u l , a

m a t e r i a l i Z e d vieW

must be p d a t e d W hen any o f t s

u n d e r l y i n g b a s e

t a b l e s i s u p d a t e d .

T h i s

i s n o r m a l l y r efer r ed

t o a s vieW

m a i n t e n a n c e .

E ? i c i e n t , i n c r e m e n t a l vieW main

t e n a n c e

a l g o r i t h m s a r e c u r r e n t l y

knoWn o n l y

f o r

SPJG

v i e W s ,

t h a t i s , v i e W s c o m p o s e d o f s e l e c t , p r o j e c t a n d i n n e r

j o i n o p e r a t o r s , W i t h a n o p t i o n a l a g g r e g a t i o n o p e r a t o r o n o p ,

t h e a g g r e g a t i o n

o p e r a t o r

s u m m a r i z i n g t h e r e s u l t s o f t h e

q u e r y .

[ 0 0 0 5 ] P r i o r Work

on

i n c r e m e n t a l m a i n t e n a n c e o f m a t e r i

a l i Z e d vieWs h a s f o c u s e d p r i m a r i l y

on

SPJG vieWs b u t

one

c o n v e n t i o n a l

s y s t e m

p r o v i d e s a mechanism f o r

m a i n t a i n i n g

o u t e r - j o i n v i e W s .

H o W e v e r , i t

d o e s n o t

c o v e r

a g g r e g a t e d

v i e W s , a n d

t h e

mechanism

may

become g r o s s l y i n e f ? c i e n t

f o r vieWs W i t h

more

t h a n

a

s i n g l e o u t e r j o i n .

SUMMARY

[ 0 0 0 6 ] The f o l l o W i n g p r e s e n t s a

s i m p l i ? e d summary

i n

o r d e r t o p r o v i d e a b a s i c u n d e r s t a n d i n g o f

some

s p e c t s o f t h e

d i s c l o s e d i n n o v a t i o n . T h i s summary i s n o t an e x t e n s i v e

o v e r v i e W , a n d i t i s n o t i n t e n d e d t o i d e n t i f y k e y / c r i t i c a l

elements

o r t o

d e l i n e a t e t h e scope t h e r e o f . I t s s o l e purpose

i s t o

p r e s e n t

some c o n c e p t s i n a

simpli?ed

form a s a p r e l u d e

t o

t h e

more d e t a i l e d

d e s c r i p t i o n t h a t

i s p r e s e n t e d l a t e r .

[ 0 0 0 7 ] The s u b j e c t i n n o v a t i o n

i s

r e l a t e d

t o

q u e r y p r o c e s s

i n g i n

d a t a b a s e

s y s t e m s , a n d

p r o v i d e s

t h e

? r s t g e n e r a l a n d

A u g . 1 6 , 2 0 07

p r a c t i c a l a l g o r i t h m s f o r m a i n t a i n i n g

o u t e r - j o i n

v i e W s ,

i n c l u d i n g

a g g r e g a t e d

o u t e r - j o i n v i e W s . M a t e ri a l i Z e d v i e W s

W i t h a n y c o m b i n a t i o n o f s e l e c t i o n , p r o j e c t i o n , i n n e r j o i n s ,

o n e - s i d e d o u t e r j o i n s a n d f u l l o u t e r j o i n s , p l u s a n o p t i o n a l

a g g r e g a t i o n

on

o p ,

c a l l e d

SPOJG

i e W s ,

c a n b e h a n d l e d

by

t h e

d i s c l o s e d

a l g o r i t h m ( s ) .

[ 0 0 0 8 ] D e s c r i b e d

h e r e i n i s

a n a r c h i t e c t u r e f o r e x t e n d i n g

i n c r e m e n t a l

vieW

maintenance

t o SPOJG v i e W s . E x t e n s i o n

i s by

c r e a t i n g

one o r more a l g o r i t h m s

t h a t

c o n s t r u c t

i n c r e

m e n t a l

m a i n t e n a n c e

e x p r e s s i o n s f o r SP0]G i e W s . Conven

t i o n a l l y ,

SPOJ e x p r e s s i o n s can be c o n v e r t e d

t o

a normal

f o r m , f o r e x a m p l e , a j o i n - d i s j u n c t i v e n o r m a l f o r m . The

d i s c l o s e d a l g o r i t h m ( s ) u t i l i Z e t h i s n o r m a l f o r m f o r o u t e r - j o i n

e x p r e s s i o n s , a n d

a l s o e x p l o i t

d a t a b a s e

c o n s t r a i n t s , f o r

e x a m p l e ,

f o r e i g n - k e y

c o n s t r a i n t s ,

t o

r e du c e m a i n t e n a n c e

o v e r h e a d . E x p e r i m e n t s shoW

h a t



vieW need

n o t

be

more

e x p e n s i v e

t h a n m a i n t a i n i n g

an

i n n e r - j o i n

v i e W ; and

f o r


v i e W s i t

c a n

e v e n

b e

c h e a p e r .

[ 0 0 0 9 ] A c c o r d i n g l y , t h e i n v e n t i o n

d i s c l o s e d

a n d

c l a i m e d

h e r e i n , i n o n e a s p e c t t h e r e o f , c o m p r i s e s a

c o m p u t e r - i m p l e

mented s y s t e m t h a t f a c i l i t a t e s m a i n t e n a n c e o f m a t e r i a l i Z e d

o u t e r - j o i n v i e W s . The s y s t e m c a n c o m p r i s e

a

vieW c r e a t i o n

component f o r c r e a t i n g a m a t e r i a l i Z e d

vieW d e ? n i t i o n

t h a t

i n c l u d e s

o u t e r - j o i n s ,

a

c o n v e r s i o n

c o m p o n e n t

f o r c o n v e r t i n g

vieW e x p r e s s i o n d e ? n i t i o n

i n t o

a normal f o r m , and a

main

t e n a n c e

component f o r

u p d a t i n g

t h e s t o r e d

m a t e r i a l i Z e d

vieW r e s u l t a s s o c i a t e d W i t h t h e m a t e r i a l i Z e d vieW d e ? n i t i o n

W hen an u n d e r l y i n g b a s e

t a b l e

i s modi?ed.

[ 0 0 1 0 ]

I n

a n o t h e r s p e c t ,

c o m p u t a t i o n s a n d

m e t h o d o l o g i e s

a re

d i s c l o s e d f o r

i d e n t i f y i n g

d i r e c t l y -

a n d

i n d i r e c t l y - a f f e c t e d

terms

of

an expression

i n

normal

form

a s a means f o r

r e d u c i n g t h e

o v e r h e a d o f v i e W m a i n t e n a n c e

t h e r e b y m a k i n g

e x p r e s s i o n s

more

e f ? c i e n t

t o

e x e c u t e .

[ 0 0 1 1 ] I n

y e t

a n o t h e r

i n n o v a t i v e a s p e c t ,


a r e

p r o v i d e d f o r m a i n t a i n i n g a g g r e g a t e d v i e W s t h a t

c o n t a i n

o u t e r j o i n s .

[ 0 0 1 2 ]

T o t h e

a c c o m p l i s h m e n t

o f t h e

f o r e g o i n g

a n d

e l a t e d

e n d s , c e r t a i n i l l u s t r a t i v e a s p e c t s o f t h e d i s c l o s e d i n n o v a t i o n

a r e d e s c r i b e d h e r e i n i n c o n n e c t i o n

W i t h

t h e f o l l o W i n g

d e s c r i p t i o n a n d t h e a n n e x e d d r a W i n g s . T h e s e

a s p e c t s

a r e

i n d i c a t i v e ,

h o W e v e r ,

o f

b u t a feW

o f

t h e v a r i o u s Ways

i n

Which t h e p r i n c i p l e s

d i s c l o s e d h e r e i n

c a n

b e

employed a n d

i s i n t e n d e d t o i n c l u d e a l l s u c h a s p e c t s

a n d

t h e i r e q u i v a l e n t s .

O t h e r

a d v a n t a g e s a n d n o v e l f e a t u r e s W i l l become a p p a r e n t

f r o m


d e t a i l e d

d e s c r i p t i o n When c o n s i d e r e d i n

c o n j u n c t i o n W i t h

t h e

d r a W i n g s .

BRIEF

DESCRIPTION OF

THE

DRAWINGS

[ 0 0 1 3 ]

F I G .

1 i l l u s t r a t e s a

c o m p u t e r - i m p l e m e n t e d

s y s t e m

t h a t f a c i l i t a t e s maintenance of a m a t e r i a l i Z e d v i e W .

[ 0 0 1 4 ] F I G . 2

i l l u s t r a t e s a

methodology

o f


a

m a t e r i a l i Z e d

vieW i n a c c o r d a n c e W i t h

a n o v e l a s p e c t .

[ 0 0 1 5 ] F I G . 3 i l l u s t r a t e s

s u b s u m p t i o n

g r a p h s

f o r

t h e

e x a m p l e SPOJ v i e W s V1

a n d

v 2 .

[ 0 0 1 6 ]

F I G . 4 5 i l l u s t r a t e s a methodology o f g e n e r a t i n g a

s u b s u m p t i o n g r a p h and n e t c o n t r i b u t i o n s t o t h e v i e W .

[ 0 0 1 7 ]

F I G . 5 6

i l l u s t r a t e s

s h o W s t h r e e


g r a p h s

f o r v i e W Vl h a t a r e d e r i v e d f r o m t h e


g r a p h f o r

V1 o f F I G . 3 .


16/30

US

007/0192283 A 1

[ 0 0 1 8 ] F I G .

6 i l l u s t r a t e s a

methodology

o f c o n s t r u c t i n g a

m a i n t e n a n c e g r a p h f o r i d e n t i f y i n g t e r m s a f f e c t e d

by

a n

u p d a t e , i n

a c c o r d a n c e W i t h

t h e d i s c l o s e d i n n o v a t i o n .

[ 0 0 1 9 ]

F I G . 7


a m e t h o d o l o g y

o f p e r f o r m i n g

vieW maintenance i n

a c c o r d a n c e W i t h an i n n o v a t i v e a s p e c t .

[ 0 0 2 0 ] F I G . 8 i l l u s t r a t e s V2


g r a p h s

f o r

u p d a t e s t o

t a b l e

C n d

f o r

u p d a t e s

t o

t a b l e O .

[ 0 0 2 1 ] F I G . 9

i l l u s t r a t e s t h e

r e d u c e d

V2


g r a p h s

o f

F I G .

8 b y e x p l o i t i n g t h e f o r e i g n k e y c o n s t r a i n t s

betWeen L and O , and betWeen

O

nd C .

[ 0 0 2 2 ] F I G . 1 0


How diagram

o f a

methodol

o g y o f

c r e a t i n g

a r e d u c e d m a i n t e n a n c e

g r a p h

b y

e l i m i n a t i n g

a d d i t i o n a l n o d e s , t h e r e b y

s i m p l i f y i n g

m a i n t e n a n c e a n d

i n c r e a s i n g p e r f o r m a n c e .

[ 0 0 2 3 ]

F I G .

1 1 i l l u s t r a t e s


g r a p h s

f o r

u p d a t e s

to t a b l e s T

and R .

[ 0 0 2 4 ] FIG. 1 2

i l l u s t r a t e s a system t h a t c o n s t r u c t s an

e x p r e s s i o n

f o r c o m p u t i n g

t h e

o u t e r u n i o n o f

c h a n g e s

t o

d i r e c t l y - a f f e c t e d

t e r m s .

[ 0 0 2 5 ]

F I G .

1 3

i l l u s t r a t e s a m e t h o d o l o g y

o f c o n s t r u c t i n g

e f ? c i e n t m a i n t e n a n c e


f o r


v i e W .

[ 0 0 2 6 ]

F I G . 1 4


How diagram

o f a

methodol

o g y o f m a i n t a i n i n g a n a g g r e g a t e d o u t e r - j o i n v i e W .

[ 0 0 2 7 ] F I G . 1 5 i l l u s t r a t e s

a b l o c k

diagram o f

a computer

o p e r a b l e

t o

e x e c u t e t h e d i s c l o s e d m a t e r i a l i z e d v i e W

main

tenance

a r c h i t e c t u r e .

[ 0 0 2 8 ]

F I G .

1 6 i l l u s t r a t e s a s c h e m a t i c

b l o c k

diagram o f an

e x e m p l a r y

c o m p u t i n g

e n v i r o n m e n t o p e r a b l e t o

m a i n t a i n


v i e W s

i n a c c o r d a n c e W i t h a n o t h e r a s p e c t .

DETAILEDDESCRIPTION

[ 0 0 2 9 ]

The

i n n o v a t i o n i s no W d e s c r i b e d W i t h r e f e r e n c e to

t h e d r a W i n g s , Wherein

l i k e

r e f e r e n c e

n u m e r a l s a r e used

to

r e f e r

t o

l i k e

e l e m e n t s t h r o u g h o u t .

I n t h e f o l l o W i n g d e s c r i p

t i o n ,

f o r p u r p o s e s

o f

e x p l a n a t i o n ,

n u m e r o u s

s p e c i ? c d e t a i l s

a r e

s e t f o r t h

i n o r d e r to

p r o v i d e

a t h o r o u g h

u n d e r s t a n d i n g

t h e r e o f . I t ma y be

e v i d e n t , h o W e v e r ,

t h a t

t h e

i n n o v a t i o n c a n

b e p r a c t i c e d W i t h o u t t h e s e

s p e c i ? c

d e t a i l s . I n o t h e r

i n s t a n c e s , Well-knoWn

s t r u c t u r e s

and d e v i c e s a r e

shoWn

i n

b l o c k d i a g r a m form i n

o r d e r

t o

f a c i l i t a t e

a d e s c r i p t i o n

t h e r e o f .

[ 0 0 3 0 ] As s e d

i n

t h i s a p p l i c a t i o n , t h e t e r m s

c o m p o n e n t

and

system a r e

i n t e n d e d to r e f e r t o a c o m p u t e r - r e l a t e d

e n t i t y , e i t h e r h a r d W a r e , a

c o m b i n a t i o n

o f h a r d W a r e and

s o f t W a r e , s o f t W a r e ,

o r s o f t W a r e

i n

e x e c u t i o n .

F o r e x a m p l e ,

a

component

can b e ,

b u t i s

n o t

l i m i t e d

t o

b e i n g , a p r o c e s s

running

on a p r o c e s s o r , a p r o c e s s o r , a

hard

d i s k d r i v e ,

m u l t i p l e

s t o r a g e d r i v e s ( o f o p t i c a l

a n d / o r m a g n e t i c

s t o r a g e

medium), an

o b j e c t ,

an e x e c u t ab l e , a t h r e a d of e x e c u t i o n , a

p r o g r a m ,

a n d / o r

a c o m p u t e r . By Way o f l l u s t r a t i o n , b o t h an

application running on a s e r v e r and t h e s e r v e r

can

be a

component.

One r mo re

components

can r e s i d e Within a

p r o c e s s

a n d / o r t h r e a d of x e c u t i o n , and

a component can be

l o c a l i z e d

on

one computer

and/

r d i s t r i b u t e d betWeen

tWo

or

more computers.

[ 0 0 3 1 ] R e f e r r i n g i n i t i a l l y t o t h e d r a W i n g s ,

F I G . 1 i l l u s

t r a t e s a computer-implemented s y s t e m 100 t h a t f a c i l i t a t e s

maintenance of

a m a t er i a l i z e d v i e W . T he

system

100 can

A u g . 1 6 , 2 0 07

c o m p r i s e a v i e W c r e a t i o n component

1 0 2 f o r

c r e a t i n g mate

r i a l i z e d

v i e W s ,

i n c l u d i n g

o u t e r - j o i n v i e W s , a

c o n v e r s i o n

component 1 0 4 f o r c o n v e r t i n g v i e W e x p r e s s i o n s i n t o a

n o r m a l

f o r m ,

and a

m a i n t e n a n c e component 1 0 6

f o r

u p d a t

i n g t h e m at e r i a l i z e d

v i e W s

c r e a t e d

b y

t h e

v i e W c o m p o n e n t

1 0 2 .

[ 0 0 3 2 ]

The

s y s t e m

f a c i l i t a t e s i n c r e m e n t a l

m a i n t e n a n c e o f


SPOJG ( o r s e l e c t , p r o j e c t i o n ,

o u t e r - j o i n a n d

o p t i o n a l g r o u p - b y ) v i e W s , t h a t i s , v i e W s Where

some

r l l o f

t h e j o i n s a r e

o u t e r j o i n s . T h i s i s a c c o m p l i s h e d b y n o v e l

a l g o r i t h m s

t h a t

c o n s t r u c t i n c r e m e n t a l m a i n t e n a n c e e x p r e s

s i o n s

f o r

SPOJG i e W s . C o n v e n t i o n a l l y , SPOJ e x p r e s s i o n s

can be c o n v e r t e d t o

a

normal

f o r m ,

f o r e x a m p l e ,

a

j o i n

d i s j u n c t i v e

n o r m a l

f o r m . The

c o n v e r s i o n

c o m p o n e n t

1 0 4

i n c l u d e s one o r more a l g o r i t h m ( s )

f o r

c o n v e r t i n g o u t e r - j o i n

e x p r e s s i o n s t o t h i s n o r m a l f o r m a n d e x p l o i t s c o n s t r a i n t s ,

f o r

e x a m p l e ,

f o r e i g n - k e y

c o n s t r a i n t s ,

t o

r e du c e m a i n t e n a n c e

o v e r h e a d .

[ 0 0 3 3 ] T h e

s y s t e m

1 0 0 p r o v i d e s t h e

? r s t

g e n e r a l

a n d

p r a c t i c a l

a l g o r i t h m s f o r h a n d l i n g v i e W s

W i t h

m u l t i p l e

o u t e r

j o i n s a n d

a l s o

a g g r e g a t e d o u t e r - j o i n

v i e W s . M a t e r i a l i z e d

v i e W s W i t h a n y

c o m b i n a t i o n o f

s e l e c t i o n s , i n n e r j o i n s ,

o n e - s i d e d

o u t e r j o i n s a n d f u l l o u t e r j o i n s , p l u s a n

o p t i o n a l


on

t o p , c a n

b e

h a n d l e d

by t h e

s y s t e m

1 0 0 .

[ 0 0 3 4 ]

F I G . 2


a methodology

o f


a


vieW

n

a c c o r d a n c e W i t h

a

n o v e l a s p e c t . W h i l e ,

f o r

p u r p o s e s o f

s i m p l i c i t y of e x p l a n a t i o n , t h e one

o r

more


shoWn

e r e i n , e . g . , i n

t h e f o r m o f a

How h a r t

o r

How

iagram, a r e s ho W n and described

a s a

s e r i e s of c t s ,

i t

i s to

b e

u n d e r s t o o d

a n d a p p r e c i a t e d

t h a t t h e s u b j e c t

i n n o v a t i o n i s n o t l i m i t e d by t h e o r d e r

of

a c t s ,

a s

some a c t s

may, i n

accordance t h e r e W i t h ,

occur

i n a

d i f f e r e n t

o r d e r

a n d / o r c o n c u r r e n t l y W i t h

o t h e r

acts from t h a t shoWn

and

d e s c r i b e d

h e r e i n .

F o r

e x a m p l e ,

t h o s e

s k i l l e d

i n

t h e

a r t

W i l l

u n d e r s t a n d

a n d

a p p r e c i a t e t h a t a m e t h o d o l o g y c o u l d a l t e r

n a t i v e l y be r e p r e s e n t e d a s a

s e r i e s

of i n t e r r e l a t e d states o r

e v e n t s , such a s i n a state

d i a g r a m .

Moreover,

n o t

a l l i l l u s

t ra t e d a c t s may e r e q u i r e d t o i m p l e m e n t a methodology i n

a c c o r d a n c e

W i t h t h e

i n n o v a t i o n .

[ 0 0 3 5 ]

At 2 0 0 ,

u p d a t e s o f

a t a b l e T a r e

r e c e i v e d

f o r

p r o c e s s i n g .

The u p d a t e s c o n s i s t o f

a s e t

o f neW t u p l e s

i n s e r t e d i n t o

t a b l e T

o r a set of u p l e s

d e l e t e d

from t a b l e T .

At 2 0 2 ,

t h e ? r s t vieW

d e ? n i t i o n V

s r e t r i e v e d t o be c o n s i d

e r e d

f o r m a i n t e n a n c e .

At 2 0 4 , a c h e c k

i s

performed

t o

d e t e r m i n e W h e t h e r a l l v i e W s h a v e b e e n p r o c e s s e d . I f t h a t i s

t h e

c a s e ,

p r o c e s s i n g t e r m i n a t e s . O t h e r W i s e , p r o c e s s i n g c o n

t i n u e s a t 2 0 6 , Where i t i s d e t e r m i n e d

W h e t h e r

t h e

vieW

r e f e r e n c e s t h e

u p d a t e d

t a b l e

T .

I f i t d o e s n o t , t h e

n e x t

vieW

i s

r e t r i e v e d ,

i n 2 0 8 ,

a n d

p r o c e s s i n g c o n t i n u e s

a t

2 0 6 .

I f

t h e

v i e W

r e f e r e n c e s

t a b l e T ,

t h e

SPOJ ar t o f

t h e

v i e W

d e ? n i t i o n ,

i . e . ,

t h e n o n - a g g r e g a t e d p a r t o f t h e v i e W d e ? n i t i o n , i s

c o n v e r t e d

i n t o

n o r m a l f r o m ,

a t

2 1 0 . At 2 1 2 ,

t h e

i n c r e m e n t a l

m a i n t e n a n c e e x p r e s s i o n f o r t h e

v i e W

i s g e n e r a t e d .

A t 2 1 4 ,

t h e v i e W m a i n t e n a n c e e x p r e s s i o n i s

e v a l u a t e d

and h e re s u l t

a p p l i e d t o t h e

v i e W , t h e r e b y b r i n g i n g

t h e v i e W u p t o

d a t e

W i t h

t h e b a s e t a b l e s . P r o c e s s i n g

t h e r e a f t e r

c o n t i n u e s

W i t h

t h e

n e x t

v i e W , a t s t e p 2 0 8 , w i t h How back

to

2 0 4 .

[ 0 0 3 6 ] I n o p e r a t i o n , c o n s i d e r t h e f o l l o W i n g e x a m p l e . A

v i e W , o j _ v i e W ,

p r e s e n t e d b e l o W ,

c o n s i s t s o f o u t e r o i n s o f t h e

t a b l e s P a r t ,

L i n e i t e m

and O r d e r s

from

a

TPC-H ( t r a n s a c t i o n

p r o c e s s i n g

p e r f o r m a n c e

c o u n c i l ,

t y p e

H b e n c h m a r k ) d a t a

b a s e .

(TPC i s

a n

o r g a n i z a t i o n

t h a t b e n c h m a r k s t r a n s a c t i o n


17/30

US

007/0192283 A 1

p r o c e s s i n g s y s t e m s . The TPC-H benchmark m e a s u r e s d e c i

s i o n

s u p p o r t

p e r f o r m a n c e

t h a t r e ? e c t s m u l t i p l e a s p e c t s

o f

s y s t e m c a p a b i l i t y

t o p r o c e s s q u e r i e s . )

I t

i s g i v e n t h a t

p _ p a r t

k e y

s t h e

p r i m a r y

k e y

o f a r t a n d o _ o r d e r k e y

i s

t h e p r i m a r y

k e y o f

O r d e r s ,

a n d

t h e r e

i s

a

f o r e i g n

k e y

c o n s t r a i n t

b e t W e e n

L i n e i t e m and

P a r t ,

and

a l s o

one betWeen L i n e i t e m

and

O r d e r s .

c r e a t e vieW OjiVl?W a s

s e l e c t p i p a r t k e y , p i n a l n e , p i r e t a i l p r i c e , o i o r d e r k e y , o i c u s t k e y ,

l i l i n e n u m b e r ,

l i q u a n t i t y ,

l i e x t e n d e d p r i c e

f r o m

p a r t

l e ? o u t e r j o i n

( o r d e r s

l e ? o u t e r j o i n l i n e i t e m on liorderkey =

oiorderkey) on pipartkey =

lipartkey

[ 0 0 3 7 ] F i r s t , a n a l y Z e W h a t

t y p e s

o f t u p l e s t h e v i e W may

c o n t a i n .

The o i n

b e t W e e n

O r d e r s

a n d L i n e i t e m

W i l l

r e t a i n

a l l L i n e i t e m

t u p l e s

b e c a u s e t h e

j o i n m a t c h e s

t h e

f o r e i g n k e y

c o n s t r a i n t

b e t W e e n

l _ o r d e r k e y

a n d

o _ o r d e r k e y .

I f

t h e

O r d e r s

t a b l e

c o n t a i n s some o r p h a n e d O r d e r s t u p l e s W i t h o u t match

i n g

L i n e i t e m

t u p l e s , t h e o r p h a n e d t u p l e s W o u l d

o c c u r

i n t h e

r e s u l t

n u l l - e x t e n d e d

on l l L i n e i t e m

c o l u m n s . The u t e r j o i n

W i t h

P a r t

W i l l n o t e l i m i n a t e a n y

r e a l { L i n e i t e m ,

O r d e r s }

t u p l e s

b e c a u s e t h i s j o i n

i s

a l s o

a f o r e i g n - k e y

j o i n b u t

i t

W i l l

e l i m i n a t e a l l t u p l e s t h a t a r e n u l l - e x t e n d e d on

Lineitem

c o l u m n s .

P a r t

t u p l e s t h a t d o n o t j o i n W i t h a n y t h i n g W i l l b e

r e t a i n e d i n t h e re s u l t

b e c a u s e

t h e j o i n i s a n o u t e r o i n . H e n c e ,

t h e ? n a l r e s u l t W i l l c o n t a i n

o n e c o m p l e t e { L i n e i t e m ,

O r d e r s ,

P a r t } t u p l e f o r e a c h L i n e i t e m t u p l e a n d s o m e o r p h a n e d P a r t

t u p l e s

n u l l - e x t e n d e d

on c o l u m n s from

i n e i t e m a n d

O r d e r s ,

b u t no { L i n e i t e m , O r d e r s }

and

no

O r d e r s

t u p l e s .

[ 0 0 3 8 ] N e x t , c o n s i d e r hoW t o m a i n t a i n t h e v i e W

i n

r e s p o n s e t o i n s e r t i o n s i n t o one

of

h e

base

t a b l e s . Suppose a

s e t

o f

neW

a r t t u p l e s

i s

i n s e r t e d i n t o t h e P a r t

t a b l e .

The

vieW

c a n

t h e n b e

b r o u g h t

u p - t o - d a t e b y

s i m p l y i n s e r t i n g t h e

neW

p a r t t u p l e s ,

a p p r o p r i a t e l y

e x t e n d e d W i t h n u l l s , i n t o

t h e

v i e W .

N o t h i n g m o r e

i s

r e q u i r e d b e c a u s e t h e

f o r e i g n

k e y c o n s t r a i n t

betWeen

L i n e i t e m

and a r t g u a r a n t e e s t h a t a neW p a r t t u p l e

d o e s n o t

j o i n

W i t h a n y L i n e i t e m t u p l e s .

I f

t h e neW a r t t u p l e

d i d , t h e n t h e

j o i n i n g

L i n e i t e m

t u p l e s W o u l d h a v e

v i o l a t e d t h e

f o r e i g n

k e y c o n s t r a i n t .

[ 0 0 3 9 ]

I n s e r t i o n s

i n t o t h e O r d e r s

t a b l e

W i l l have no

effect

on h e vieW s o n o t h i n g n e e d s t o be

d o n e .

AneW r d e r s t u p l e

c a n n o t j o i n W i t h a n y

e x i s t i n g

L i n e i t e m

t u p l e s

b e c a u s e o f

t h e

f o r e i g n

k e y

c o n s t r a i n t

b e t W e e n

L i n e i t e m

a n d

O r d e r s . The

neW r d e r s t u p l e , a p p r o p r i a t e l y

e x t e n d e d W i t h

u l l s , W i l l n o t

b e

c o n t a i n e d

i n t h e vieW e i t h e r b e c a u s e , a s d i s c u s s e d ab o v e ,

o r p h a n e d O r d e r s

t u p l e s a r e

e l i m i n a t e d

from t h e

v i e W .

[ 0 0 4 0 ] F i n a l l y , c o n s i d e r i n s e r t i o n s i n t o t h e L i n e i t e m

t a b l e .

Suppose t h e n e W l i n e i t e m s a r e c o n t a i n e d i n a t a b l e

neW_li

n e i t e m s . The v i e W c a n t h e n b e u p d a t e d

u s i n g t h e f o l l o W i n g

s e q u e n c e o f SQL

s t a t e m e n t s .

s e l e c t p i p a r t k e y , p i n a l n e , p i r e t a i l p r i c e , o i o r d e r k e y , o i c u s t k e y ,

l i l i n e n u m b e r ,

l i q u a n t i t y ,

l i e x t e n d e d p r i c e

i n t o d e l t a l

f r o m

n e w i l i n e i t e r n s , o r d e r s ,

p a r t

Where

liorderkey

= oiorderkey and

lipartkey

= pipartkey

i n s e r t i n t o O j i V l ? W

s e l e c t

from #deltal

A u g . 1 6 , 2 0 07

- c o n t i n u e d

d e l e t e

f r o m

O j i V l ? W

W h e r e o i o r d e r k e y i s

n u l l

a n d p i p a r t k e y

i n ( s e l e c t

p i p a r t k e y f r o m

# d e l t a l )

[ 0 0 4 1 ] The rs t s t a t e m e n t

c o m p u t e s t h e s e t

o f

t u p l e s t o b e

i n s e r t e d

i n t o

t h e

vieW

and s a v e s

them

i n a temporary t a b l e

d el tal

( t o

a v o i d c o m p u t i n g t h e s a m e e x p r e s s i o n W i c e ) . T h e

second

s t a t e m e n t adds

t h e neW u p l e s i n t o t h e

v i e W . The

neW

L i n e i t e m t u p l e s ma y c a u s e some orphaned P a r t t u p l e s t o be

e l i m i n a t e d from

t h e v i e W .

The

t h i r d

s t a t e m e n t

d e l e t e s

a l l

o r p h a n e d P a r t

t u p l e s ,

i f a n y , Whose p _ p a r t k e y

v a l u e o c c u r s

among t h e

n e W l y i n s e r t e d

t u p l e s .

I n

o t h e r W o r d s , i t

de le t e s

a l l o r p h a n s W h o c e a s e t o b e o r p h a n s b e c a u s e o f

t h e

i n s e r t .

[ 0 0 4 2 ] T h e f o l l o W i n g d e s c r i p t i o n c o n t a i n s p r e l i m i n a r y

m a t e r i a l

a n d i n t r o d u c e s c o n c e p t s

u s e d

l a t e r .

[ 0 0 4 3 ]

A

e l e c t i o n o p e r a t o r W i l l

b e d e n o t e d

i n

t h e

n o r m a l

W a y

a s o p , Where p i s a

p r e d i c a t e . A

r e d i c a t e p

r e f e r e n c i n g

some set S of

columns

s s a i d

t o

be s t r o n g o r n u l l - r e j e c t i n g

i f i t

evaluates

t o f a l s e

or unknoWn

n

a tuple

a s

soon

a s

one

o f t h e c o l u m n s i n S

i s

n u l l . P r o j e c t i o n

( W i t h o u t

d u p l i c a t e

e l i m i n a t i o n ) i s

d e n o t e d

by

o ,

Where c i s a l i s t o f c o l u m n s .

B o r r o w i n g

f r o m

SQL s t r u c t u r e d

q u e r y

l a n g u a g e ) ,

t h e s h o r t

hand . i s used o d e n o t e a l l columns o f t a b l e T . An p e r a t o r

i s

a l s o n e e d e d t h a t r e m o v e s d u p l i c a t e s

( s i m i l a r

to

SQLs

s e l e c t d i s t i n c t ) ,

W h i c h

i s d e n o t e d b y 6 .

[ 0 0 4 4 ] A

p e c i a l p r e d i c a t e

N u l l ( T )

a n d i t s

n e g a t i o n

~ N u l l ( T ) W i l l b e

u s e d .

N u l l ( T ) e v a l u a t e s t o t r u e

i f a t u p l e

i s

n u l l - e x t e n d e d on

t a b l e

T .

N u l l ( T )

c a n

be implemented

i n

SQL s T . C s n u l l , Where C s any

column

o f T t h a t

c a n n o t c o n t a i n

a n u l l

v a l u e .

W h e n

p p l y i n g

t h e s e p r e d i c a t e s

to

a

s e t

o f

t a b l e s

T ,

t h e p r e d i c a t e s

NN(T)

and

N(T)

c a n

be

used Which

a r e

de?ned

a s

N N ( T ) = / \ t T ~ N u l l ( t ) and

N(T)=

A K T N u l l ( t ) .

[ 0 0 4 5 ] A chema S i s a s e t o f

a t t r i b u t e s

( c o l u m n n a m e s ) .

L e t T

and

2 b e

t a b l e s W i t h schemas

S

and 2, r e s p e c t i v e l y .

T h e o u t e r

u n i o n ,

d e n o t e d

b y

T

U T 2 ,

? r s t

n u l l - e x t e n d s ( p a d s

W i t h u l l s ) t h e

t u p l e s

o f e a c h o p e r a n d

t o

s c h e m a S l ( E S 2 a n d

t h e n

t a k e s t h e

u n i o n

o f t h e r e s u l t s ( W i t h o u t d u p l i c a t e e l i m i

n a t i o n ) . O u t e r

( s e t )

d i ? f e r e n c e ,

d e n o t e d b y 1 : 1 , i s d e ? n e d

i n

a

similar manner.

[ 0 0 4 6 ] A u p l e t l i s s a i d t o subsume a t u p l e t 2 i f t h e y a r e

de?ned on h e same schema, l a g r e e s With

2

on l l columns

Where t h e y

b o t h a r e

n o n - n u l l ,

a n d

t 1 c o n t a i n s f e W e r n u l l

v a l u e s

t h a n t 2 .

The o p e r a t o r

r e m o v a l o f

subsumed t u p l e s

o f

T ,

d e n o t e d

by

T j ,

r e t u r n s t h e t u p l e s

o f

T

t h a t

a r e

n o t

s u b s u m e d b y a n y o t h e r

t u p l e

i n T .

[ 0 0 4 7 ] The minimum n i o n o f a b l e s T 1

a n d T 2

i s d e ? n e d

a s

T l o T 2 = ( T l U T l ) j .

Minimum

n i o n

h a s l o W e r p r e c e d e n c e

t h a n j o i n .

I t

c a n b e shoWn t h a t minimum u n i o n i s b o t h

commutative and a s s o c i a t i v e .

[ 0 0 4 8 ] L e t

T 1

a n d

T 2 b e

t a b l e s

W i t h d i s j o i n t s c h e m a s

S 1

a n d

S 2 ,

r e s p e c t i v e l y , a n d

p a p r e d i c a t e r e f e r e n c i n g some

s u b s e t o f t h e c o l u m n s i n ( S 1 6 E 8 2 ) .

The ( i n n e r )

j o i n

o f

t h e

t a b l e s i s

d e ? n e d a s

T 1 l > < p T 2 = { ( t l , t 2 ) ] t 1 e T l , t 2 e T 2 ,

p ( t l , t 2 ) } .

T h e l e f t s e m i j o i n i s d e ? n e d a s

T l > < p l s T 2 = { l l l e T l ,

( 3 t 2 e T 2 ] p ( t l , t 2 ) ) } ,

t h a t i s ,

a

t u p l e

i n T 1

q u a l i ? e s

i f

i t

j o i n s

W i t h some

t u p l e i n T 2 .

The e f t

a n t i ( s e m i ) j o i n

i s

de?ned a s


18/30

US

007/0192283 A 1

T l l > < l p l a T 2 = { t l l t l e T l ,

( $ t 2 e T 2 ] p ( t l , t 2 ) ) } , t h a t i s , a t u p l e

i n

T 1

q u a l i ? e s i f

i t

d o e s n o t j o i n

W i t h

a n y t u p l e i n T 2 . The l e f t

o u t e r j o i n c a n

b e

e x p r e s s e d a s

T l l > < p 1 T 2 = T l l > < p T 2 o T 1 .

The

r i g h t

o u t e r

o i n c a n

b e

e x p r e s s e d

s T 1 > < 1 r T 2 = T 2 l > < 1 p 1 T l .

T h e

f u l l

o u t e r j o i n

c a n b e e x p r e s s e d

a s T 1 l > < 1 p f T 2 = T l

> < p T 2 o T l o T 2 .

[ 0 0 4 9 ]

As d e s c r i b e d h e r e i n , i t

i s assumed h a t e v e r y

b a s e

t a b l e

h a s

a u n i q u e k e y , Which

s u s u a l l y

t h e c a s e i n p r a c t i c e .

A l l

m a t e r i a l i z e d vieWs a r e

assumed o

have

a

u n i q u e key and

p r e d i c a t e s a r e assumed

t o be

n u l l - r e j e c t i n g on

l l

columns

r e f e r e n c e d .

[ 0 0 5 0 ] VieW i n c r e m e n t a l m a i n t e n a n c e

e x p r e s s i o n s a r e

d e r i v e d b a s e d on h e j o i n - d i s j u n c t i v e normal form f o r SP0]

e x p r e s s i o n s . T h r o u g h o u t t h i s d e s c r i p t i o n , t h e f o l l o w i n g

example

d a t a b a s e c o n t a i n i n g

t h r e e

t a b l e s W i l l b e u s e d

as

modeled on t h e t a b l e s

C u s t o m e r , O r d e r s , L i n e i t e m o f

t h e

TPC-H

d a t a b a s e .

C ( c k , c n , c n k ) , O ( o k , o c k , o d , o t p ) , L

l o k , l n , l p k , l q , l p )

[ 0 0 5 1 ] N u l l s

a r e

n o t

a l l o W e d i n

any

of

t h e

c o l u m n s .

U n d e r l i n e d c o l u m n s f o r m t h e k e y

o f e a c h t a b l e .

TWo

o r e i g n

key

c o n s t r a i n t s a r e

d e ? n e d :

O.ock e f e r e n c e s

C . c k and . l o k

r e f e r e n c e s

O . o k .

[ 0 0 5 2 ] C o n s i d e r a n e x a m p l e v i e W h a v i n g


d e ? n i t i o n ,

Q = C > < c k : o c k l o ( O > < o k : l o k l o ( 0 l p > 5 O O L ) )

[ 0 0 5 3 ]

The i e W c o n t a i n s a l l

c u s t o m e r s t o g e t h e r

W i t h

t h e i r

o r d e r s ,

i f a n y , and

l i n e

i t e m s W i t h a p r i c e

o v e r

5 0 0 , i f a n y .

The e s u l t

W i l l

c o n t a i n t u p l e s o f t h r e e d i f f e r e n t

t y p e s .

[ 0 0 5 4 ]

l . C O L - t u p l e s , t h a t i s ,

t u p l e s

f o r m e d b y c o n c a t

e n a t i n g a t u p l e

f r o m

C , a t u p l e

from

O

n d

a t u p l e

f r o m

L .

T h e r e

W i l l b e o n e COL-tuple f o r e v e r y L

t u p l e t h a t

s a t i s ? e s

t h e p r e d i c a t e

l p > 5 0 0 .

[ 0 0 5 5 ]

2 .

C O - t u p l e s ,

t h a t i s ,

t u p l e s

c o m p o s e d b y

c o n

c a t e n a t i o n o f

a

t u p l e from C ,

a

t u p l e from O nd u l l s

f o r a l l columns o f L . T h e r e W i l l be one s u c h

t u p l e

f o r

e v e r y O - t u p l e t h a t d o e s n o t j o i n

W i t h

a n y L - t u p l e

s a t i s f y i n g l p > 5 0 0 .

[ 0 0 5 6 ] 3 . C - t u p l e s , t h a t

i s ,

t u p l e s c o m p o s e d o f a t u p l e

from C

i t h n u l l s

f o r a l l columns

o f

O

nd

L . There

W i l l

be o n e

s u c h t u p l e f o r

e v e r y C

u p l e t h a t

d o e s

n o t

j o i n

W i t h

a n y

t u p l e

i n

O .

[ 0 0 5 7 ] T h e e s u l t c o n t a i n s a l l t u p l e s i n t h e o i n C l > < C k = O Ck O

l > q o k = l o k ( o l p > s 0 0 L ) , a l l

t u p l e s i n

t h e

j o i n

C l > < 1 C k = O C k O , a n d a l s o

A u g . 1 6 , 2 0 07

a l l t u p l e s

i n

C .

Each f t h e s e t h r e e

s u b - r e s u l t s i s r e p r e s e n t e d

i n t h e

r e s u l t

i n a minimal W a y . F o r e x a m p l e , i f a t u p l e

( c l , n u l l , n u l l ) a p p e a r s i n

t h e r e s u l t ,

t h e n

t h e r e

e x i s t s

a t u p l e

c l

i n

C , b u t t h e r e i s

no

t u p l e

01

i n

0

u c h t h a t ( c l , o l )

a p p e a r s

[ 0 0 5 8 ] The e x p r e s s i o n can be r e w r i t t e n a s t h e

minimum

u n i o n o f

t h r e e t e r m s c o m p r i s e d s o l e l y

o f

i n n e r o i n s , W h i c h

i s t h e j o i n - d i s j u n c t i v e f o r m o f t h e o r i g i n a l

SPOJ

e x p r e s s i o n .

[ 0 0 5 9 ]

As l l u s t r a t e d b y t h e

e x a m p l e ,

a n SPOJ e x p r e s s i o n

E ver a s e t of a b l e s

U

an be

converted

t o a normal form

c o n s i s t i n g o f t h e

minimum u n i o n

o f t e r m s

composed

f r o m

s e l e c t i o n s and i n n e r j o i n s ( b u t no

o u t e r j o i n s ) .

More f o r

m a l ly , t h e j o i n - d i s j u n c t i v e

n o r m a l

form

o f E c a n b e

r e p r e

s e n t e d a s

t h e f o l lo W i n g ,

Where

c c p r o j e c t s

t h e

r e s u l t o n t o

t h e d e s i r e d columns C nd

e a c h t e r m

E i

i s

o f

t h e

form

T i l , T i 2 . . .

T i m

i s

a

s u b s e t of

h e

t a b l e s i n U. P r e d i c a t e p i i s

t h e

c o n j u n c t i o n

o f

a s u b s e t

o f

t h e

s e l e c t i o n a n d j o i n

p r e d i c a t e s

f o u n d i n t h e o r i g i n a l

form

o f t h e

q u e r y .

[ 0 0 6 0 ] The f o l l o W i n g t r a n s f o r m a t i o n

r u l e s a r e

u s e d

f o r

c o n v e r t i n g SPOJ e x p r e s s i o n t o


f o r m .

i f p ( l ) r e f e r e n c e s

o n l y

T 1

[ 0 0 6 1 ] R u l e ( 4 ) f o l l o W s

f r o m t h e o b s e r v a t i o n

t h a t a l l

t u p l e s o r i g i n a t i n g f r o m t h e t e r m T 2 i n ( T l > < 1 p T 2 o T 2 ) W i l l b e

n u l l - e x t e n d e d on l l

columns

o f T 1 . A l l t h o s e

t u p l e s

W i l l b e

d i s c a r d e d i f p ( l ) i s s t r o n g o n

T 1 . R u l e

( 5 ) f o l l o W s f r o m t h e

r u l e

o p ( ' l ) ( T l > 4 p T 2 ) = ( O ' p ( l ) T l ) ) q p T 2

b y

e x p a n d i n g t h e

t W o

o u t e r j o i n s .

[ 0 0 6 2 ] The

o l l o W i n g

e x a m p l e i l l u s t r a t e s c o n v e r s i o n o f

a n

SPOJ

e x p r e s s i o n u s i n g t h e a b o v e r u l e s .

p ( i , j )

d e n o t e s a

p r e d i c a t e

t h a t r e f e r e n c e s c o l u m n s

i n

t a b l e s

T i

a n d T i .

b y R u l e ( 1 )

( ( U ' M U T I ) p ( l , 2 ) T 2 ) ) ' \ ] ; f : i 2 , 3 ) T 3

b y R u l e

( 4 )

( T 1 ) 5 p m

mom

I i i w n

b y i n c l u d i n g s e l e c t i o n

p r e d i c a t e

i n

j o i n

( T 1 l p u u p u m T z q p m l m ' b y R u l e

( 2 )

( T l / i i p ( 1 > r p ( 1 . 2 > T 2 ) 6 T 3


19/30

US

007/0192283 A 1

[ 0 0 6 3 ]

Each t e r m

i n

t h e

j o i n - d i s j u n c t i v e form o f

a

SPOJ

e x p r e s s i o n

p r o d u c e s t u p l e s W i t h

a

u n i q u e

n u l l - e x t e n s i o n

p a t t e r n . S u p p o s e

t h e c o m p l e t e

s e t

o f

o p e r a n d

t a b l e s f o r t h e

e x p r e s s i o n

i s

U . A e r m i n t h e j o i n - d i s j u n c t i v e f o r m i s

de?ned

v e r a s u b s e t

S

o f

U

nd

e n c e ,

p r o d u c e s

t u p l e s

t h a t

a r e

n u l l extended on U-S. T he t a b l e s i n s u b s e t S

a r e

c a l l e d

t h e

terms

source t a b l e s .

[ 0 0 6 4 ]

A

u p l e

p r o d u c e d by a t e r m W i t h

s o u r c e

t a b l e

s e t

S

c a n

o n l y

b e

s u b s u m e d

b y t u p l e s p r o d u c e d

b y t e r m s

W h o s e

s o u r c e

t a b l e

set

i s

a s u p e r s e t of S.

The

subsumption r e l a

t i o n s h i p s among

t e r m s c a n be

modeled by

a DAG

d i r e c t e d

a c y c l i c g r a p h ) , Which

s

r e f e r r e d

t o

a s a s u b s u m p t i o n g r a p h

o f

t h e

SP0]


[0065] D e ? n i t i o n . Let E = J 1 C (E16 . . . oEn)

be

t h e

j o i n

d i s j u n c t i v e f o r m

o f

a

SPOJ


The s u b s u m p t i o n

g r a p h o f E

c o n t a i n s a

node i

f o r

e a c h t e r m E i i n

t h e normal

f o r m ,

and

t h e node s

l a b e l e d

W i t h t h e s o u r c e

t a b l e

s e t S i o f

E.

T h e r e

i s

an d g e from

node

i to

node

j ,

i f S i

i s

a

minimal

s u p e r s e t o f

i . S i i s

a

m i n i m a l

s u p e r s e t o f J -

i f v t h e r e d o e s n o t

e x i s t a node

n k

i n t h e g r a p h s u c h t h a t

S J - R S k R S 1 .

[ 0 0 6 6 ] C o n s i d e r t h e f o l l o W i n g e x a m p l e t h a t

s h o W s t W o

SPOJ v i e W s , V1 a n d V 2 ,

t h e i r n o r m a l

f o r m s . F I G . 3

i l l u s

t r a t e s

s u b s u m p t i o n g r a p h s f o r

t h e

e x a m p l e

SPOJ

v i e W s

V 1

a n d

V 2 .

)

m)

[ 0 0 6 7 ] T h e m i n i m u m - u n i o n o p e r a t o r s i n t h e j o i n - d i s j u n c

t i v e normal form have tWo f u n c t i o n s : t o

e l i m i n a t e subsumed

t u p l e s a n d

t o

t a k e t h e u n i o n o f t h e

r e m a i n i n g

t u p l e s .

By r s t

e l i m i n a t i n g s u b s u m e d t u p l e s f r o m e v e r y t e r m ,

t h e minimum

u n i o n s c a n

b e

r e p l a c e d by o u t e r u n i o n s . The

r e s u l t i n g f o r m

i s

u s e f u l

b e c a u s e i t c l e a rl y shoWs

What

e r m s

a r e a f f e c t e d by

an u p d a t e and hoW.

[ 0 0 6 8 ] S u b s u m p t i o n

among

t e r m s

i s

n o t

a r b i t r a r y ;

When

c h e c k i n g

W h e t h e r a t u p l e o f a t e r m i s s u b s u m e d ,

i t

i s

s u ? i c i e n t t o

c h e c k

a g a i n s t t u p l e s i n p a r e n t t e r m s . The

f o l l o W i n g

lemma

s h o W s

hoW o

e l i m i n a t e subsumed

t u p l e s

from

a term.

[ 0 0 6 9 ]

Lemma. L e t E i b e a t e r m i n t h e j o i n - d i s j u n c ti v e

normal form E f a n SPOJ

x p r e s s i o n .

Then

h e

s e t o f t u p l e s

g e n e r a t e d b y E i t h a t a re n o t s u b s u m e d b y a n y o t h e r t u p l e s i n

E can be computed s ,

Where E l ,

E 2 ,

. . . E i m a r e t h e p a r e n t t e r m s of

i and

t h e

j o i n

p r e d i c a t e q

s

a n e q u i j

o i n o v e r

c o l u m n s f o r m i n g a k e y o f

E.

[ 0 0 7 0 ] D i i s c a l l ed t h e n e t c o n t r i b u t i o n o f t e r m E i b e c a u s e

t h e

t u p l e s

o f

D i a re

n o t subsumed by

a n y

o t h e r t u p l e s , a n d

t h u s , a p p e a r e x p l i c i t l y i n t h e v i e W r e s u l t . E x a m p l e s o f t h e n e t

c o n t r i b u t i o n form a r e

p r o v i d e d i n f r a .

A u g . 1 6 , 2 0 07

[ 0 0 7 1 ] Theorem. L e t E be

a n

SPOJ

e x p r e s s i o n

W i t h n o r

mal

form

c c ( E l o E 2 o

. . .

oEn).

Then

E16E26

. .

6En=D1UD2U

. .

UDn

Where

a c h

D i

i s

computed from

i a s

de?ned

n

t h i s

Lemma.

[0072] T h e

form D l

U . .

UDn

s c a l l e d t h e

n e t - c o n t r i

b u t i o n ( n o r m a l ) f o r m o f t h e e x p r e s s i o n . T h i s

f o r m

i s i m p o r

t a n t

because

t h e r e i s no

i n t e r a c t i o n among

e t c o n t r i b u t i o n s

from d i f f e r e n t t e r m s , W hich

means

t h a t

each term can be

m a i n t a i n e d

i n d e p e n d e n t l y from o t h e r t e r m s .

[ 0 0 7 3 ] R e f e r r i n g noW t o F I G . 4 , t h e r e i s i l l u s t r a t e d a

m e t h o d o l o g y o f g e n e r a t i n g a s u b s u m p t i o n g r a p h a n d

n e t

c o n t r i b u t i o n s o f t e r m s t o t h e v i e W .

At

4 0 0 , t h e vieW

d e ? n i

t i o n

i s r e t r i e v e d .

At 4 0 2 ,

t h e

SP0] p a r t o f

t h e vieW

e x p r e s

s i o n i s c o n v e r t e d

i n t o j o i n - d i s j u n c t i v e

n o r m a l

form c o n s i s t

i n g o f

t h e minimum

n i o n

o f t e r m s composed f i n n e r

o i n s .

At

4 0 4 ,

a s u b s u m p t i o n g r a p h i s c r e a t e d

f r o m

t h e

n o r m a l

f o r m , W h i c h e v e a l s t h e s u b s u m p t i o n ( p a r e n t - c h i l d )

r e l a t i o n

s h i p s among

t h e t e r m s . At 4 0 6 , t h e t e r m e x p r e s s i o n s i n t h e

normal

form and

t h e

s u b s u m p t i o n g r a p h

a r e

u s e d

to c r e a t e

e x p r e s s i o n s d e ? n i n g

t h e n e t

c o n t r i b u t i o n o f e a c h t e r m t o t h e

v i e W . The

n e t

c o n t r i b u t i o n e x p r e s s i o n f o r

a

term

i s

con

s t r u c t e d a c c o r d i n g to t h e L emma

a b o v e . At 4 0 8 ,

t h e

e x p r e s

s i o n

d e ? n i n g t h e t o t a l n e t c o n t r i b u t i o n i s c o n s t r u c t e d b y

c o n n e c t i n g

a l l

n e t - c o n t r i b u t i o n


b y o u t e r

u n i o n

o p e r a t o r s .

[ 0 0 7 4 ] F o l l o W i n g i s a d e s c r i p t i o n o f hoW

o

i n c r e m e n t a l l y

m a i n t a i n

an SPOJ vieW W hen s i n g l e

b a s e

t a b l e i s modi?ed.

Only

i n s e r t i o n s o r d e l e t i o n s a r e

a d d r e s s e s

h e r e ; an u p d a t e

can be t r e a t e d a s a d e l e t i o n folloWed by an i n s e r t i o n . T he

f o c u s

h e r e i s

on d e r i v i n g c o r r e c t m a i n t e n a n c e e x p r e s s i o n s

and

n o t

n e c e s s a r i l y t h e

most e f ? c i e n t

o n e s .

More e f ? c i e n t

e x p r e s s i o n s a r e d e r i v e d i n f r a .

[ 0 0 7 5 ]

A

r s t

t a s k

i s

t o i d e n t i f y

Which

t e r m s

may

b e

a f f e c t e d

by an

n s e r t i o n o r

a d e l e t i o n . Consider a vieW V nd

suppose one of i t s b a s e t a b l e s T i s modi?ed. T h i s m a y

change t h e n e t c o n t r i b u t i o n of term

u t

o n l y i f T

c c u r s

i n

t h e e x p r e s s i o n

d e ? n i n g D. By

n s p e c t i o n o f t h e f o r m o f t h e

e x p r e s s i o n f o r

D,

t i s a p p a r e n t

t h a t

t h e c h a n g e may f f e c t

t h e

r e s u l t

i n one

of

t h r e e

Ways:

[ 0 0 7 6 ]

i n

T i ;

1 . D i r e c t l y ,

Which

c c u r s i f

T s among

h e

t a b l e s

[ 0 0 7 7 ]

2 . I n d i r e c t l y , Which

c c u r s i f

T

s

n o t

among

h e

t a b l e s i n T i b u t i t i s among t h e s o u r c e t a b l e s of one o r

more o f i t s

p a r e n t

n o d e s

( t e r m s ) ; and

[ 0 0 7 8 ]

3 . No f f e c t , o t h e r W i s e .

[ 0 0 7 9 ]

Based on

t h i s

c l a s s i ? c a t i o n

of

h o W

t e r m s

a r e

a f f e c t e d ,

a

vieW maintenance graph can

be c r e a t e d , a s

f o l l o W s .

[ 0 0 8 0 ] 1 . E l i m i n a t e

f r o m

t h e s u b s u m p t i o n

g r a p h

a l l

n o d e s ( t e r m s )

t h a t

a re u n a f f e c t e d b y t h e u p d a t e o f T .

[ 0 0 8 1 ]

2 . Mark t h e

n o d e s

o f

t h e

r e d u c e d g r a p h

by

D

r

I d e p e n d i n g

on

W h e t h e r t h e

node

( t e r m ) i s

a f f e c t e d

d i r e c t l y

o r

i n d i r e c t l y .

[ 0 0 8 2 ] 3 .

D r o p a l l

o u t g o i n g e d g e s f r o m n o d e s

m a r k e d

W i t h 1 .

[ 0 0 8 3 ] F I G . 5 s h o W s t h r e e m a i n t e n a n c e g r a p h s f o r v i e W

V l t h a t

a r e

d e r i v e d

f r o m t h e


g r a p h f o r V I o f


20/30

US

007/0192283 A 1

FIG. 3. C s a

member

f h e source s e t of l l t h r e e

nodes

s o

t h e

m a i n t e n a n c e g r a p h

( a ) f o r

u p d a t e s

t o C e t a i n s a l l t h r e e

n o d e s

a n d t h e y

a r e marked

W i t h

s u p e r s c r i p t D .

The

main

t e n a n c e

g r a p h ( b )

f o r

u p d a t e s

t o 0 l s o

r e t a i n s

a l l

t h r e e

n o d e s .

0 s

a

member

f

t h e s o u r c e

s e t s

ofnodes

{ C ,

O ,

L}

a n d

{ C ,

0 } ;

t h e y

a r e a f f e c t e d d i r e c t l y a n d m a r k e d W i t h

s u p e r s c r i p t e d D. O s n o t a member of h e s o u r c e s e t of node

{C}

b u t

i t i s

a

member o f

t h e s o u r c e

s e t o f i t s p a r e n t ; t h i s

node

s

a f f e c t e d

i n d i r e c t l y

and

marked W i t h

1 .

The m a i n t e

n a n c e g r a p h ( c ) f o r

u p d a t e s t o

L e t a i n s

o n l y

n o d e s

{ C , O , L }

a n d

{ C , O } . Node

{ C }

i s

d r o p p e d

b e c a u s e n e i t h e r

i t

n o r i t s

p a r e n t r e f e r e n c e t a b l e

L .

[ 0 0 8 4 ] R e f e r r i n g noW t o F I G . 6 , t h e r e i s

i l l u s t r a t e d

a

m e t h o d o l o g y

o f c o n s t r u c t i n g

a m a i n t e n a n c e

g r a p h

i d e n t i f y

i n g t e r m s a f f e c t e d b y a n u p d a t e , i n a c c o r d a n c e W i t h t h e

d i s c l o s e d i n n o v a t i o n . A t 6 0 0 , t h e e x p r e s s i o n d e ? n i n g t h e

vieW i s r e t r i e v e d . At 6 0 2 , t h e SP0] p a r t o f t h e v i e W

e x p r e s s i o n i s c o n v e r t e d

i n t o


n o r m a l

f o r m .

At 6 0 4 , t h e s u b s u m p t i o n g r a p h i s

c o n s t r u c t e d

i n t h e manner

d e s c r i b e d

e a r l i e r .

At

6 0 6 ,

t e r m s

u n a f f e c t e d

by

t h e

u p d a t e

a r e

e l i m i n a t e d f r o m

t h e s u b s u m p t i o n g r a p h .

A t 6 0 8 ,

e a c h

r e m a i n i n g n o d e i s

c l a s s i ? e d

a n d

marked

a c c o r d i n g t o

Whether i t

r e p r e s e n t s a d i r e c t l y - a f f e c t e d term o r

an

i n d i

r e c t l y - a l f e c t e d t e r m . A t 6 1 0 , a l l

o u t g o i n g

e d g e s a r e d r o p p e d

from t e r m s marked i n d i r e c t l y

a f f e c t e d .

[ 0 0 8 5 ] F o l l o w i n g i s a d e s c r i p t i o n o f

v i e W

m a i n t e n a n c e .

AlV

s

u s e d to d e n o t e t h e ( o u t e r ) u n i o n o f a l l c h a n g e s to

d i r e c t l y

a f f e c t e d

t e r m s ,

and A 2 V to d e n o t e

t h e same f o r

i n d i r e c t l y a f f e c t e d

t e r m s .

S i m i l a r l y ,

A I D i

a n d

A 2 D i

d e n o t e

t h e

change

i n t h e n e t c o n t r i b u t i o n D i

of

a

d i r e c t l y

o r an

i n d i r e c t l y a f f e c t e d

t e r m ,

r e s p e c t i v e l y .

VieW a i n t e n a n c e

c a n

b e

p e r f o r m e d

i n t h e

f o l l o W i n g

t h r e e s t e p s .

[ 0 0 8 6 ]

l .

The

a f f e c t e d b a s e

t a b le

i s

u p d a t e d

? r s t .

[ 0 0 8 7 ]

2 .

I f

t h e r e a r e

no

d i r e c t l y a f f e c t e d

t e r m s ,

We

r e

d o n e ; o t h e r W i s e ,

c o m p u t e

a n d a p p l y A 1 t o

t h e

v i e W .

[ 0 0 8 8 ] 3 . I f t h e r e a r e no n d i r e c t l y a f f e c t e d

t e r m s ,

We r e

d o n e ; o t h e r W i s e ,

c o m p u t e

a n d a p p l y

A 2 t o t h e

v i e W .

[ 0 0 8 9 ]

A c c o r d i n g l y ,

F I G .

7

i l l u s t r a t e s a m e t h o d o l o g y

o f

p e r f o r m i n g vieW m a i n t e n a n c e i n a c c o r d a n c e

W i t h

an i n n o

v a t i v e

a s p e c t .

At 7 0 0 ,

v i e W m a i n t e n a n c e

i s

i n i t i a t e d .

At 7 0 2 ,

t h e

a f f e c t e d

b a s e d t a b l e i s u p d a t e d ? r s t . As a

s i d e

e f f e c t o f

t h i s

o p e r a t i o n ,

t h e s e t o f

i n s e r t e d t u p l e s o r

t h e

s e t

o f

d e l e t e d

t u p l e s

i s

o b t a i n e d .

A t 7 0 4 , t h e m a i n t e n a n c e g r a p h i s c h e c k e d

f o r d i r e c t l y - a f f e c t e d t e r m s . I f no

such

terms a r e f o u n d , a t

7 0 6 , t h e p r o c e s s t e r m i n a t e s .

I f s u c h

t e r m s a r e f o u n d , a t 7 0 6 ,

How s t o

7 0 8 , Where t h e

o u t e r

union o f

c h a n g e s

t o t h e n e t

c o n t r i b u t i o n s

o f

d i r e c t l y - a f f e c t e d t e r m s

i s c o m p u t e d ,

a n d

t h e s e a r e a p p l i e d

t o

t h e v i e W .

At

7 1 0 , t h e

m a i n t e n a n c e g r a p h

i s c h e c k e d

f o r

i n d i r e c t l y - a f f e c t e d

t e r m s . A g a i n , a t

7 1 2 , no

such t e r m s a r e f o u n d , t h e p r o c e s s t e r m i n a t e s .

I f

such t e r m s

a r e f o u n d , a t 7 1 2 , How

p r o g r e s s e s

t o 7 1 4 , Where t h e o u t e r

u n i o n

o f c h a n g e s

t o

t h e

n e t

c o n t r i b u t i o n s

o f i n d i r e c t l y

a l f e c t e d

t e r m s

i s c o m p u t e d , and t h e s e a r e a p p l i e d to t h e

view.

[ 0 0 9 0 ] F o l l o w i n g a r e d e r i v a t i o n s o f m a i n t e n a n c e e x p r e s

s i o n s

f o r i n s e r t i o n s

and

f o r d e l e t i o n s . F i r s t , c o n s t r u c t

vieW

maintenance e x p r e s s i o n s

f o r

a vieW V n t h e c a s e o f i n s e r

t i o n s .

The s e t

of

neW

oWs

i n s e r t e d

i n t o

a t a b l e

T s denoted

b y A T . I f

E i

( D i ) i s a n e x p r e ss i o n , t h e e x p r e s s i o n

o b t a i n e d

f r o m

E i ( D i )

b y r e p l a c i n g e v e r y o c c u r r e n c e o f

T

b y AT

s

d e n o t e d

b y E i A T ( D i A T ) .

A u g . 1 6 , 2 0 07

[ 0 0 9 1 ]

C o n s i d e r t h e

f o l l o W i n g e x a m p l e . A e t o f

u p l e s A O

h a s been i n s e r t e d i n t o t a b l e 0 nd t i s d e s i r e d to m a i n t a i n

v i e W V 1 .

F I G .

5 ( b ) s h o W s t h e m a i n t e n a n c e

g r a p h f o r

t h i s

c a s e .

T Wo n o d e s

a re d i r e c t l y a f f e c t e d ,

n a m e l y

{ C , O , L } a n d

{ C ,

O } .

The

neW

t u p l e s

t o

b e

i n s e r t e d

i n t o

t h e

v i e W

c a n b e

c o m p u t e d b y

s u b s t i t u t i n g O i t h

A 0 n t h e e x p r e s s i o n f o r

t h e n e t c o n t r i b u t i o n o f e a c h t e r m . T h i s g i v e s t h e

f o l l o W i n g

A l - e x p r e s s i o n s ,

[ 0 0 9 2 ] P r e d i c a t e

q

o f

t h e

a n t i - s e m i j o i n e q u a l s ( l . o k = r . o k )

W h e r e

l . o k

( r . o k )

d e s i g n a t e s

t h e

column

ok f r o m t h e

l e f t

( r i g h t ) o p e r a n d o f

t h e

j o i n .

[ 0 0 9 3 ] N e x t , c o n s i d e r t h e i n d i r e c t l y

a f f e ct e d

t e r m A

new

0

u p l e may

o i n

W i t h

some C

t u p l e t h a t p r e v i o u s

d i d

n o t

j o i n

W i t h

a n y t h i n g . The

e r m may

u r r e n t l y

c o n t a i n s u c h

o r p h a n e d C u p l e s ,

Which must

noW be d e l e t e d . The vieW

t u p l e s

to

be

d e l e t e d

c a n be computed s ,

=

( a

5 1 5 %

W h e r e t h e

j o i n

p r e d i c a t e q e q u a l s ( l . c k = r . c k ) .

[ 0 0 9 4 ] C o m b i n i n g t h e p i e c e s , t h e

f o l l o W i n g c o m p l e t e

m a i n t e n a n c e e x p r e s s i o n c a n

be

o b t a i n e d .

V 1 ( C , 0 + A 0 , L ) = V I ( C , O , L ) H - J A I D C O L L 1 - J A I D C O

U A Z D C

[ 0 0 9 5 ] Lemma. Consider a term

i n t h e normal

form of a

SPOJ i e W Whose n e t co n t r i b u t i o n

i s

d e ? n e d b y

D i = E i l > < 1 q 1 a

( E i l U . . . UEim)

Suppose t a b l e

T

h a s been

updated by

i n s e r t i n g AT u p l e s . I f t h e

t e r m

i s d i r e c t l y affec ted b y

t h e

i n s e r t i o n ,

t h e

t u p l e s added o

t h e

n e t co n t r i b u t i o n

o f

t h e t e r m

can

be computed

s ,

AIDFDQALEQATIXILIla

( E H A T L T J

. . . k - J E i m A T ) .

I f t h e

t e r m

i s i n d i r e c t l y a f f e c t e d , t h e t u p l e s d e l e t e d f r o m t h e

n e t c o n t r i b u t i o n of t h e term can be computed s ,

A 2 n i = n i l > < l q l s ( E i l A T h - J

. H E F J E i

A T )

Where q s

a n

e q u i j o i n p r e d i c a t e o v e r a u n i q u e k e y o f E.

[ 0 0 9 6 ]

T h e

p r e v i o u s e x a m p l e c o n s i d e r e d

v i e W V 1

a n d

i n s e r t i o n s

i n t o

0;

no W c o n s i d e r

d e l e t i o n s from 0 . A s

i n

t h e

p r e v i o u s c a s e , t W o t e r m s ar e d i r e c t l y

a f f e c t e d ,

n a m e l y { C , O ,

L } a n d

{ C , O } ,

s e e F I G . 5 ( b ) .

T h e

A l - e x p r e s s i o n s f o r

t h e

d i r e c t l y - a f f e c t e d

terms

a r e t h e

same

a s f o r i n s e r t i o n

but noW

t h e r e s u l t i n g t u p l e s a r e

d e l e t e d

from t h e v i e W . T h e r e i s

one

i n d i r e c t l y - a f f e c t e d t e r m A

e l e t e d

0 u p l e may e

t h e

l a s t

t u p l e t o j o i n

W i t h

a C

u p l e .

I f s o , i t

s h o u l d a p p e a r i n t h e

r e s u l t o f

t h e

{ C }

t e r m a f t e r t h e u p d a t e . The t u p l e s t o b e

i n s e r t e d can

be computed

s ,

W h e r e t h e j o i n p r e d i c a t e q e q u a l s ( l . c k = r . c k ) .


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