Lossless Join

8/9/2019 Lossless Join

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Jan. 2012 Yangjun Chen 1

Lossless-join

Outline: Lossless-joinChapter 15 3r! e!. "Chap. 15 # th$ 5th e!.% Chap. 1&$ &th e!.'

()asi* !e+inition o+ Lossless-join

(, a ples

(/esting algorith


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Lossless-join

()asi* !e+inition o+ Lossless-join

A !e*o position 1$ 2$...$ 4 o+ has the lossless join propert 6ith respe*t to the set o+ !epen!en*ies 7 on i+$+or e8er relation r o+ that satis+ies 7$ the +ollo6ing hol!s$

"1

"r'$ ...$ "r'' r$

6here is the natural join o+ all the relations in .

/he 6or! loss in lossless re+ers to loss of information $ not toloss o+ tuples.


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Lossless-join

(, a ple: !e*o posion-1

SS ;< hours , A


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Jan. 2012 Yangjun Chen #

Lossless-join

(, a ple: !e*o position-2

SS ;< hours , A


5/18


Lossless-join

(!e*o posion-1

A1SS A2, A21

>31

>12

>22

>32

>13

>23

>33

>1#

>2#

>3#

>15

>25

>35

>1&

>2&

>3&

1

2

3

a1

>21

a1

a2

>22

>32

>13

a3

a3

>1#

a#

>3#

>15

a5

>35

>1&

>2&

a&

1

2

3


6/18

Jan. 2012 Yangjun Chen &

Lossless-join

a1

>21

a1

a2

>22

a2

>13

a3

a3

>1#

a#

>3#

>15

a5

>35

>1&

>2&

a&

1

2

3

a1

>21

a1

a2

>22

a2

>13

a3

a3

>1#

a#

a#

>15

a5

a5

>1&

>2&

a&

1

2

3

SS , A


7/18Jan. 2012 Yangjun Chen ?

Lossless-join

(!e*o position-2

A1SS A2, A21

>12

>22

>13

>23

>1#

>2#

>15

>25

>1&

>2&

1

2

>11

a1

a2

>22

>13

a3

>1#

a#

a5

a5

>1&

a&

1

2

/he atri *an not >e *hange!@

SS , A


8/18Jan. 2012 Yangjun Chen

Lossless-join

a1 a2 >13 >1# >15 >1& >21 >22 a3 a# a5 a&

a1 a2 a3 a# a5 a&

,< B OJ

h D

a1 a2

>21 >22

1 >13 >1# >15a3 a# a5

a1 >13 >1& >21 a3 >2&a1 a3 a&

2

3

e*o position-1:


9/18Jan. 2012 Yangjun Chen 9

Lossless-join

1 3 13

h D

e*o position-1:

a1 a2 >13 >1&

a1 a2 a3 a&

>21 >22 a3 >2&

13 2

a1 a2 >13 >1# >15 >1&

>21 >22 a3 a# a5 a&

a1 a2 a3 a# a5 a&



Lossless-join

>11 a2 >13 >1# a5 >1&a1 >22 a3 a# a5 a&

,< B OJ

h D

a2 a5

>22 a5

1

>11 >13 >1# a5 >1&

a1 a3 a# a5 a&

2

e*o position-2:



Lossless-join

h D

e*o position-2:

1 2

>11 a2 >13 >1# a5 >1&a1 a2 a3 a# a5 a&

>11 >22 >13 >1# a5 >1&

a1 >22 a3 a# a5 a&Spurious tuples



Lossless-join

stu!entBno *ourseBno instrBno

Instructors teach onecourse only

Student takes a courseand has one instructor

stu!entBno$ *ourse4 instrBnoinstrBno *ourseBno

Stu!ent-*ourse-instru*tor:



Lossless-join


stu!entBno instrBno

CourseBno instrBno1

2

A1stu-no

A2*ourse-no

A3instr-no

>11

>21

>12

>22

>13

>23

1

2

A1stu-no

A2*ourse-no

A3instr-no

>11

a1

a2

>22

a3

a3

1

2

>11

a1

a2

a2

a3

a3

1

2



14/18Jan. 2012 Yangjun Chen 1#

Lossless-join


stu!entBno *ourseBno

CourseBno instrBno1

2

A1stu-no

A2*ourse-no

A3instr-no

>11

>21

>12

>22

>13

>23

1

2

A1stu-no

A2*ourse-no

A3instr-no

>11

a1

a2

a2

a3

>23

1

2

>11

a1

a2

a2

a3

>23

1

2

instrBno *ourseBno




Lossless-join


stu!entBno *ourseBno

stu!entBno instrBno1

2

A1stu-no

A2*ourse-no

A3instr-no

>11

>21

>12

>22

>13

>23

1

2

A1stu-no

A2*ourse-no

A3instr-no

a1

a1

>12

a2

a3

>23

1

2

a1

a1

>12

a2

a3

>23

1

2



16/18Jan. 2012 Yangjun Chen 1&

Lossless-join

Testing algorithminput: A relation $ a !e*o position 1$ 2$...$ 4 o+ $ an!

a set 7 o+ +un*tion !epen!en*ies.

1. Create an initial atri S 6ith one ro6 i +or ea*h relation i in$ an! one *olu n j +or ea*h attri>ute A j in .

2. Set S" i, j ' : > ij +or all atri entries.3. For ea*h ro6 i representing relation s*he a i Do

for ea*h *olu n j representing A j doif relation i in*lu!es attri>ute A j then

set S" i, j ' : a j%4

#. epeat the +ollo6ing loop until a *o plete loop e e*ution resultsin no *hanges to S.


17/18Jan. 2012 Yangjun Chen 1?

Lossless-join

#. epeat the +ollo6ing loop until a *o plete loop e e*ution resultsin no *hanges to S.

for ea*h +un*tion !epen!en* E Y in 7 dofor all ro6s in S 6hi*h ha8e the sa e s >ols in the

*olu ns *orrespon!ing to attri>utes in E doaFe the s >ols in ea*h *olu n that *orrespon! to

an attri>ute in Y >e the sa e in all these ro6s as +ollo6s:i+ an o+ the ro6s has an GaH s >ol +or the *olu n$set the other ro6s to the sa e GaH s >ol in the *olu n.=+ no GaH s >ol e ists +or the attri>ute in an o+ thero6s$ *hoose one o+ the G>H s >ols that appear in oneo+ the ro6s +or the attri>ute an! set the other ro6s tothat sa e G>H s >ol in the *olu n%44

5. =+ a ro6 is a!e up entirel o+ GaH s >ols$ then the !e*o po-sition has the lossless join propert % other6ise it !oes not.


18/18Jan 2012 Yangjun Chen 1

Lossless-join

a1

>21

a1

a2

>22

>32

>13

a3

a3

>1#

a#

>3#

>15

a5

>35

>1&

>2&

a&

a2 >21a1

>22 >32

1ISS $ , A15a3 a# a5a3 >3# >35

a1 >13 >1& >21 a3 >2&a1 a3 a&

3ISS $ ;3#$ >35$ a1$ a3$ a&

;< A

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Lossless Join