1. Murali Mani Relational Algebra

2. Murali Mani What is Relational Algebra? Defines operations (data retrieval) for relational model SQL’s DML (Data Manipulation Language) has data retrieval facilities, which are equivalent to that of relational algebra. SQL and relational algebra are not for complex operations; they support efficient, easy access of large data sets.

3. Murali Mani Basics Relational Algebra is defined on bags, rather than relations (sets). Bag or multiset allows duplicate values; but order is not significant. We can write an expression using relational algebra operators with parentheses Need closure – an operator on bag returns a bag. Relational algebra includes set operators, and other operators specific to relational model.

4. 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)

5. Murali 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

6. Murali 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

7. Murali 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

8. Murali Mani Bag semantics vs Set semantics Union is idempotent for sets: (R1  R2)  R2 = R1  R2 Union is not idempotent for bags. Intersection is idempotent for sets and bags. Difference is idempotent for sets, but not for bags. For sets and bags, R 1  R 2 = R 1 – (R 1 – R 2 ).

9. Murali 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

10. Murali 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

11. Murali 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)

12. Murali 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

13. Murali 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

14. Murali 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.

15. Murali Mani Natural Join example AB 12 12 R1R1 BC 23 45 45 R2R2 ABC 123 123 R 1 ⋈ R 2

16. Murali Mani Theta Join: R ⋈ C S Theta Join is similar to cartesian product, 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

17. 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). ABC 123 456 789 R1R1 BCD 2310 2311 6712 R2R2 R 1 ⋈ o R 2 ABCD 12310 12311 456null 789 6712

18. Murali 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

19. Murali 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

20. Murali Mani Renaming: ρ S(A1, A2, …, An) (R) Rename relation R to S, attributes of R are renamed to A1, A2, …, An ρ S (R) renames relation R to S, keeping the attributes same. BCD 2310 2311 6712 R2R2 XCD 2310 2311 6712 ρ S(X, C, D) (R 2 ) S BCD 2310 2311 6712 ρ S (R 2 ) S

21. Murali Mani Example: Introducing new relations Find 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, and t1 and t2 agree in all attributes common to R and S, then t1 appears in the result. R1 = R ⋈ S R2 = π A R (R1) R ⋉ S = R2 ⋂ R

22. Murali Mani Duplicate Elimination:  (R) Convert a bag to a set. R AB 12 34 12 12  (R) AB 12 34

23. Murali Mani Extended Projection: π L (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. BCD 2310 2311 6712 R π B→A, C+D→X, C, D (R) AXCD 213310 214311 619712

24. Murali Mani Aggregation operators MIN, MAX, COUNT, SUM, AVG AGG B (R) considers only non-null values of R. R AB 12 34 1null 13 MIN B (R) 2 MAX B (R) 4 COUNT B (R) 3 SUM B (R) 9 AVG B (R) 3 COUNT * (R) 4

25. Murali Mani Aggregation Operators MIN, MAX, SUM, AVG must be on any 1 attribute. COUNT can be on any 1 attribute or COUNT * (R) An aggregation operator returns a bag, not a single value ! But SQL allows treatment as a single value. AB 34 σ B=MAX B (R) (R)

26. Murali Mani Grouping Operator:  GL, AL (R)  GL, AL (R) groups all attributes in GL, and performs the aggregation specified in AL. titleyearstarName SW177HF Matrix99KR 6D&7N93HF SW279HF Speed94KR StarsIn  starName, MIN (year)→year, COUNT(title) →num (StarsIn) starNameyearnum HF773 KR942

27. 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. ABC 125 316 127 138 R ABC 125 127 138 316  A,B (R)

28. Murali Mani Relational Algebra Operators Set Operators: Union, Intersection, Difference, Cartesian Product Select, Project Join: Natural Join, Theta Join, (Left/Right) Outer Join Renaming, Duplicate Elimination Aggregation: MIN, MAX, COUNT, SUM, AVG Grouping, Sorting