26cs157f_ha do - multivalued dependency

8/10/2019 26CS157F_Ha Do - Multivalued Dependency

1/14


2/14

Functional Dependency

b

c

4

Q

$

Domain (X) Range (Y)

a


3/14

Functional Dependency

Recall that if X uniquely determines Y, then Y isfunctionally dependent on X.

You may recall math the terms Domain andRange. The domain is the set of all valuespossible of X and the range is the set of allpossible values of Y.

The relation is a function because each of theelements of X maps exactly to one element of Y.


4/14

Multivalued Dependency

Smith

Jones

Cooper

Anna

John

Lila

Elsa

ChrisEmployee (X)

Dependent (Y)


5/14

Definition of MVD A multivalued dependency is a full constraint between two sets of attributes in a relation.

In contrast to the functional independency , themultivalued dependency requires that certaintuples be present in a relation. Therefore, amultivalued dependency is also referred as a

tuple-generating dependency. The multivalueddependency also plays a role in 4NFnormalization.


6/14

full constraint A constraint which expresses somethingabout all attributes in a database. (In contraryto an embedded constraint .) That amultivalued dependency is a full constraint

follows from its definition, where it sayssomething about the attributes R .

tuple-generating dependency A dependency which explicitly requires certaintuples to be present in the relation.


7/14

A Formal Definition

Let R be a relation schema and let and . Themultivalued dependency ->> holds on R if,in any legal relation r (R ), for all pairs of tuples t 1

and t 2 in r such that t 1[] = t 2[], there existtuples t 3 and t 4 in r such thatt 1[] = t 2[] = t 3[] = t 4[] t 3[] = t 1[] t 3[R ] = t 2[R ] t 4[] = t 2[] t 4[R ] = t 1[R ]


8/14

Definition of MVD (cont.)

A multivalued dependency on R , X ->>Y , saysthat if two tuples of R agree on all the attributesof X , then their components in Y may be

swapped, and the result will be two tuples thatare also in the relation.

i.e., for each value of X , the values of Y areindependent of the values of R - X -Y .


9/14

Tuples Implied by name->->phones

If we have tuples:

name addr phones beersLikedsue a p1 b1sue a p2 b2sue a p2 b1sue a p1 b2

Then these tuples must also be in the relation .


10/14

Example

Here is possible data satisfying these MVDs:

name areaCode phone beersLiked manfSue 650 555-1111 Bud A.B.Sue 650 555-1111 WickedAle Petes Sue 415 555-9999 Bud A.B.Sue 415 555-9999 WickedAle Petes

But we cannot swap area codes or phones by themselves.That is, neither name->->areaCode nor name->->phoneholds for this relation.


11/14

Properties of MVD

f ->> , Then ->> R If ->> and , Then ->> If ->> and If ->> , then ->> -

The following also involve functional dependencies:If ->> , then ->> If -> and -> , then ->

A decomposition of R into (X, Y) and (X, R-Y) is alossless-join decomposition if and only if X ->> Yholds in R.
http://en.wikipedia.org/w/index.php?title=Lossless-join_decomposition&action=edithttp://en.wikipedia.org/w/index.php?title=Lossless-join_decomposition&action=edithttp://en.wikipedia.org/w/index.php?title=Lossless-join_decomposition&action=edithttp://en.wikipedia.org/w/index.php?title=Lossless-join_decomposition&action=edithttp://en.wikipedia.org/wiki/Image:Twoheadrightarrow.gif


12/14

Decomposition Theorem

The split of relations is guaranteed to belossless if the intersection of the attributesof the new tables is a key of at least one ofthem.

The join connects tuples depending on theattribute (values) in the intersection. Ifthese values uniquely identify tuples in theother relation we do not lose information.


13/14

Example of lossy decomposition

A B C1 1 11 1 21 2 1

A B C

1 1 11 1 21 2 1

1 2 2

A B

1 1

1 2

A C

1 1

1 2

Original table Decomposition

Reconstruction


14/14

References

Silberschatz, Korth, Sudarshan. DatabaseSystem Concepts , 5th Editionhttp://en.wikipedia.org/wiki/Multivalued_dependency http://www.stanford.edu/~ullman/dscb/pslides/mvds.ppthttp://www.cs.sjsu.edu/faculty/lee/cs157/cs157a.html
http://en.wikipedia.org/wiki/Multivalued_dependencyhttp://en.wikipedia.org/wiki/Multivalued_dependencyhttp://en.wikipedia.org/wiki/Multivalued_dependencyhttp://en.wikipedia.org/wiki/Multivalued_dependency

26cs157f_ha do - multivalued dependency

Documents