oct 28, 2003murali mani relational algebra b term 2004: lecture 10, 11

Oct 28, 2003 Murali Mani

Relational Algebra

B term 2004: lecture 10, 11


Basics Relational Algebra is defined on bags, rather

than relations. Bag or multiset allows duplicate values; but

order is not significant. We can write an expression using relational

algebra operators with parentheses: we need closure – an operator on bag returns a bag.

Relational algebra includes set operators, and other operators specific to relational model.


Set Operators Union, Intersection, Difference, cross product Union, Intersection and Difference are

defined only for union compatible relations. Two relations are union compatible if they

have the same set of attributes and the types (domains) of the attributes are the same.

Eg of two relations that are not union compatible: Student (sNumber, sName) Course (cNumber, cName)


Union: ∪ Consider two bags R1 and R2 that are union-

compatible. Suppose a tuple t appears in R1 m times, and in R2 n times. Then in the union, t appears m + n times.

A B

1 2

3 4

1 2

R1

A B

1 2

3 4

5 6

R2 A B

1 2

1 2

1 2

3 4

3 4

5 6

R1 ∪ R2


Intersection: ∩ Consider two bags R1 and R2 that are union-

compatible. Suppose a tuple t appears in R1 m times, and in R2 n times. Then in the intersection, t appears min (m, n) times.

A B

1 2

3 4

1 2

R1

A B

1 2

3 4

5 6

R2

A B

1 2

3 4

R1 ∩ R2


Difference: - Consider two bags R1 and R2 that are union-

compatible. Suppose a tuple t appears in R1 m times, and in R2 n times. Then in R1 – R2, t appears max (0, m - n) times.

A B

1 2

3 4

1 2

R1

A B

1 2

3 4

5 6

R2

A B

1 2

R1 – R2


Bag semantics vs Set semantics Union is idempotent for sets – (R1 ∪ R2) ∪

R2 = R1 ∪ R2 Union is not idempotent for bags. Intersection and difference are idempotent for

sets and bags. For sets and bags, R1 R2 = R1 – (R1 – R2).


Cross Product (Cartesian Product): Ⅹ Consider two bags R1 and R2. Suppose a tuple

t1 appears in R1 m times, and a tuple t2 appears in R2 n times. Then in R1 X R2, t1t2 appears mn times.

A B

1 2

1 2

R1

B C

2 3

4 5

4 5

R2 A R1.B R2.B C

1 2 2 3

1 2 2 3

1 2 4 5

1 2 4 5

1 2 4 5

1 2 4 5

R1 X R2


Basic Relational Operations Select, Project, Join Select: denoted σC (R): selects the subset of

tuples of R that satisfies selection condition C. C can be any boolean expression, its clauses can be combined with AND, OR, NOT.

A B C

1 2 5

3 4 6

1 2 7

1 2 7

R σ(C ≥ 6) (R)

A B C

3 4 6

1 2 7

1 2 7


Select

Select is commutative: σC2 (σC1 (R)) = σC1 (σC2

(R)) Select is idempotent: σC (σC (R)) = σC (R) We can combine multiple select conditions

into one condition. σC1 (σC2 (… σCn (R)…)) = σC1 AND C2 AND … Cn (R)


Project: πA1, A2, …, An (R)

Consider relation (bag) R with set of attributes AR. πA1, A2, …, An (R), where A1, A2, …, An AR returns the tuples in R, but only with columns A1, A2, …, An.

A B C

1 2 5

3 4 6

1 2 7

1 2 8

R πA, B (R)

A B

1 2

3 4

1 2

1 2


Project: Bag Semantics vs Set Semantics

For bags, the cardinality of R = cardinality of πA1, A2, …, An (R).

For sets, cardinality of R ≥ cardinality of πA1,A2,

…, An (R). For sets and bags

project is not commutative project is idempotent


Natural Join: R ⋈ S Consider relations (bags) R with attributes

AR, and S with attributes AS. Let A = AR ∩ AS. R ⋈ S can be defined as

πAR – A, A, AS - A (σR.A1 = S.A1 AND R.A2 =S.A2 AND … R.An=S.An (R X S))where A = {A1, A2, …, An}The above expression says: select those tuples in R X S that agree in values for each of the A attributes, and project the resulting tuples such that we have only one value for each A attribute.


Natural Join example

A B

1 2

1 2

R1

B C

2 3

4 5

4 5

R2

A B C

1 2 3

1 2 3

R1 ⋈ R2


Theta Join: R ⋈C S Theta Join is similar to natural join, except that

we can specify any condition C. It is defined as

R ⋈

C S = (σC (R X S))

A B

1 2

1 2

R1

B C

2 3

4 5

4 5

R2

R1 ⋈

R1.B<R2.BR2

A R1.B R2.B C

1 2 4 5

1 2 4 5

1 2 4 5

1 2 4 5


Outer Join: R ⋈o S Similar to natural join, however, if there is a

tuple in R, that has no “matching” tuple in S, or a tuple in S that has no matching tuple in R, then that tuple also appears, with null values for attributes in S (or R).

A B C

1 2 3

4 5 6

7 8 9

R1

B C D

2 3 10

2 3 11

6 7 12

R2

R1 ⋈o R2

A B C D

1 2 3 10

1 2 3 11

4 5 6 null

7 8 9 null

null 6 7 12


Left Outer Join: R ⋈oLS

Similar to natural join, however, if there is a tuple in R, that has no “matching” tuple in S, then that tuple also appears, with null values for attributes in S (note: a tuple in S that has no matching tuple in R does not appear).

A B C

1 2 3

4 5 6

7 8 9

R1

B C D

2 3 10

2 3 11

6 7 12

R2

R1 ⋈o

L R2

A B C D

1 2 3 10

1 2 3 11

4 5 6 null

7 8 9 null


Right Outer Join: R ⋈oRS

Similar to natural join, however, if there is a tuple in S, that has no “matching” tuple in R, then that tuple also appears, with null values for attributes in R (note: a tuple in R that has no matching tuple in S does not appear).

A B C

1 2 3

4 5 6

7 8 9

R1

B C D

2 3 10

2 3 11

6 7 12

R2

R1 ⋈o

R R2

A B C D

1 2 3 10

1 2 3 11

null 6 7 12


Renaming: ρS(A1, A2, …, An) (R)

Rename relation R to S, attributes of R are renamed to A1, A2, …, An

B C D

2 3 10

2 3 11

6 7 12

R2X C D

2 3 10

2 3 11

6 7 12

ρS(X, C, D) (R2)

S


Duplicate Elimination: δ (R)

Convert a bag to a set.

R

A B

1 2

3 4

1 2

1 2

δ (R)

A B

1 2

3 4


Aggregation operators

MIN, MAX, COUNT, SUM, AVG The aggregate operators aggregate the

values in one column of a relation.

R

A B

1 2

3 4

1 2

1 2

MIN (B) = 2MAX (B) = 4COUNT (B) = 4SUM (B) = 10AVG (B) = 2.5


Grouping Operators


Sorting Operator


Extended Projection

oct 28, 2003murali mani relational algebra b term 2004: lecture 10, 11

Documents

bags r

cn r slide

b r ab

c2 c1 r

c1 c2 r

denoted c r

relation bag r

c1 c2 cn r