Nov 18, 2003 Murali Mani
Relational Algebra
B term 2004: lecture 10, 11
Nov 18, 2003 Murali Mani
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.
Nov 18, 2003 Murali Mani
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)
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
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).
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
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)
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
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.
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
Renaming: ρS(A1, A2, …, An) (R) Rename relation R to S, attributes of R are
renamed to A1, A2, …, An ρS (R) renames relation R to S, keeping the
attributes same.
B C D
2 3 10
2 3 11
6 7 12
R2
X C D
2 3 10
2 3 11
6 7 12
ρS(X, C, D) (R2)
SB C D
2 3 10
2 3 11
6 7 12
ρS (R2)
S
Nov 18, 2003 Murali Mani
Example: Introducing new relationsFind the semijoin of 2 relations R, S. Semijoin denoted R ⋉ S is defined as the tuples in R, such that for a tuple t1 in R, if there exists a tuple t2 in S that t1 and t2 agree in all attributes common to R and S, then t1 appears in the result.
R1 = R S⋈R2 = πAR (R1)R ⋉ S = R2 R⋂
Nov 18, 2003 Murali Mani
Duplicate Elimination: (R)
Convert a bag to a set.
R
A B
1 2
3 4
1 2
1 2
(R)
A B
1 2
3 4
Nov 18, 2003 Murali Mani
Extended Projection: πL (R) Here L can be
An attribute (just like simple projection) An expression x → y, where x and y are names of
attributes, this renames attribute x to y. An expression E → z, where E is any expression
involving attributes, constants, and arithmetic and string operators. This has an attribute called z whose values are given by E.
B C D
2 3 10
2 3 11
6 7 12
R πB→A, C+D→X, C, D (R)
A X C D
2 13 3 10
2 14 3 11
6 19 7 12
Nov 18, 2003 Murali Mani
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
Nov 18, 2003 Murali Mani
Grouping Operator: GL, AL (R) Usually we want to perform aggregation not
on all the values of a column, but on a subgroup of the values of a column.
GL, AL (R) groups all attributes in GL, and performs the aggregation specified in AL.
title year starName
SW1 77 HF
Matrix 99 KR
6D&7N 93 HF
SW2 79 HF
Speed 94 KR
StarsIn starName, MIN (year)→year, COUNT(title) →num (StarsIn)
starName year num
HF 77 3
KR 94 2
Nov 18, 2003 Murali Mani
Sorting Operator: L (R)
It sorts the tuples in R. If L is list A1, A2, …, An, it first sorts by A1, then by A2, and so on.
Sort is used as a last operator in an expression.
A B C
1 2 5
3 1 6
1 2 7
1 3 8
RA B C
1 2 5
1 2 7
1 3 8
3 1 6
A,B (R)