1. Oct 28, 2003Murali Mani Relational Algebra B term 2004: lecture 10, 11

2. Oct 28, 2003Murali 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.

3. Oct 28, 2003Murali 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)

4. Oct 28, 2003Murali Mani Union: ∪ Consider two bags R 1 and R 2 that are union- compatible. Suppose a tuple t appears in R 1 m times, and in R 2 n times. Then in the union, t appears m + n times. AB 12 34 12 R1R1 AB 12 34 56 R2R2 AB 12 12 12 34 34 56 R 1 ∪ R 2

5. Oct 28, 2003Murali Mani Intersection: ∩ Consider two bags R 1 and R 2 that are union- compatible. Suppose a tuple t appears in R 1 m times, and in R 2 n times. Then in the intersection, t appears min (m, n) times. AB 12 34 12 R1R1 AB 12 34 56 R2R2 AB 12 34 R 1 ∩ R 2

6. Oct 28, 2003Murali Mani Difference: - Consider two bags R 1 and R 2 that are union- compatible. Suppose a tuple t appears in R 1 m times, and in R 2 n times. Then in R 1 – R 2, t appears max (0, m - n) times. AB 12 34 12 R1R1 AB 12 34 56 R2R2 AB 12 R 1 – R 2

7. Oct 28, 2003Murali 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, R 1  R 2 = R 1 – (R 1 – R 2 ).

8. Oct 28, 2003Murali Mani Cross Product (Cartesian Product): Ⅹ Consider two bags R 1 and R 2. Suppose a tuple t 1 appears in R 1 m times, and a tuple t 2 appears in R 2 n times. Then in R 1 X R 2, t 1 t 2 appears mn times. AB 12 12 R1R1 BC 23 45 45 R2R2 AR 1.BR 2.BC 1223 1223 1245 1245 1245 1245 R 1 X R 2

9. Oct 28, 2003Murali Mani Basic Relational Operations Select, Project, Join Select: denoted σ C (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. ABC 125 346 127 127 R σ (C ≥ 6) (R) ABC 346 127 127

10. Oct 28, 2003Murali Mani Select Select is commutative: σ C2 (σ C1 (R)) = σ C1 (σ C2 (R)) Select is idempotent: σ C (σ C (R)) = σ C (R) We can combine multiple select conditions into one condition. σ C1 (σ C2 (… σ Cn (R)…)) = σ C1 AND C2 AND … Cn (R)

11. Oct 28, 2003Murali Mani Project: π A1, A2, …, An (R) Consider relation (bag) R with set of attributes A R. π A1, A2, …, An (R), where A1, A2, …, An  A R returns the tuples in R, but only with columns A1, A2, …, An. ABC 125 346 127 128 R π A, B (R) AB 12 34 12 12

12. Oct 28, 2003Murali Mani Project: Bag Semantics vs Set Semantics For bags, the cardinality of R = cardinality of π A1, A2, …, An (R). For sets, cardinality of R ≥ cardinality of π A1,A2, …, An (R). For sets and bags project is not commutative project is idempotent

13. Oct 28, 2003Murali Mani Natural Join: R ⋈ S Consider relations (bags) R with attributes A R, and S with attributes A S. Let A = A R ∩ A S. R ⋈ S can be defined as π A R – A, A, A S - A (σ R.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.

14. Oct 28, 2003Murali Mani Natural Join example AB 12 12 R1R1 BC 23 45 45 R2R2 ABC 123 123 R 1 ⋈ R 2

15. Oct 28, 2003Murali 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 = (σ C (R X S)) AB 12 12 R1R1 BC 23 45 45 R2R2 R 1 ⋈ R1.B<R2.B R 2 AR 1.BR 2.BC 1245 1245 1245 1245

16. Oct 28, 2003Murali 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). ABC 123 456 789 R1R1 BCD 2310 2311 6712 R2R2 R 1 ⋈ o R 2 ABCD 12310 12311 456null 789 6712

17. Oct 28, 2003Murali Mani Left Outer Join: R ⋈ o L S 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). ABC 123 456 789 R1R1 BCD 2310 2311 6712 R2R2 R 1 ⋈ o L R 2 ABCD 12310 12311 456null 789

18. Oct 28, 2003Murali Mani Right Outer Join: R ⋈ o R S 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). ABC 123 456 789 R1R1 BCD 2310 2311 6712 R2R2 R 1 ⋈ o R R 2 ABCD 12310 12311 null6712

19. Oct 28, 2003Murali Mani Renaming: ρ S(A1, A2, …, An) (R) Rename relation R to S, attributes of R are renamed to A1, A2, …, An BCD 2310 2311 6712 R2R2 XCD 2310 2311 6712 ρ S(X, C, D) (R 2 ) S

20. Oct 28, 2003Murali Mani Duplicate Elimination: δ (R) Convert a bag to a set. R AB 12 34 12 12 δ (R) AB 12 34

21. Oct 28, 2003Murali Mani Aggregation operators MIN, MAX, COUNT, SUM, AVG The aggregate operators aggregate the values in one column of a relation. R AB 12 34 12 12 MIN (B) = 2 MAX (B) = 4 COUNT (B) = 4 SUM (B) = 10 AVG (B) = 2.5

22. Oct 28, 2003Murali Mani Grouping Operators

23. Oct 28, 2003Murali Mani Sorting Operator

24. Oct 28, 2003Murali Mani Extended Projection