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Relational Algebra

Relational Algebra. 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.

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Relational Algebra

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  1. Relational Algebra Murali Mani

  2. 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. Murali Mani

  3. 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. Murali Mani

  4. 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) Murali Mani

  5. Union:  • Consider two bags R1 and R2 that are union-compatible. Suppose a tuple t appears in R1m times, and in R2n times. Then in the union, t appears m + n times. R1  R2 R1 R2 Murali Mani

  6. Intersection: ∩ • Consider two bags R1 and R2 that are union-compatible. Suppose a tuple t appears in R1m times, and in R2n times. Then in the intersection, t appears min (m, n) times. R1 R2 R1 ∩R2 Murali Mani

  7. Difference: - • Consider two bags R1 and R2 that are union-compatible. Suppose a tuple t appears in R1m times, and in R2n times. Then in R1 – R2, t appears max (0, m - n) times. R1 R2 R1 – R2 Murali Mani

  8. 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, R1R2 = R1 – (R1 – R2). Murali Mani

  9. Cross Product (Cartesian Product): X • Consider two bags R1 and R2. Suppose a tuple t1 appears in R1m times, and a tuple t2 appears in R2n times. Then in R1 X R2, t1t2 appears mn times. R1 XR2 R1 R2 Murali Mani

  10. 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. σ(C ≥ 6) (R) R Murali Mani

  11. 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) Murali Mani

  12. 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. πA, B (R) R Murali Mani

  13. 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 Murali Mani

  14. 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. Murali Mani

  15. Natural Join example R1 R2 R1 ⋈R2 Murali Mani

  16. 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 = (σC (R X S)) R1 ⋈ R1.B<R2.BR2 R1 R2 Murali Mani

  17. 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). R1 ⋈o R2 R1 R2 Murali Mani

  18. 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). R1 ⋈oLR2 R1 R2 Murali Mani

  19. 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). R1 ⋈oRR2 R1 R2 Murali Mani

  20. 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. ρS(X, C, D) (R2) ρS (R2) R2 S S Murali Mani

  21. 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 = πAR (R1) R ⋉ S = R2 ⋂ R Murali Mani

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

  23. 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. πB→A, C+D→X, C, D (R) R Murali Mani

  24. Aggregation operators • MIN, MAX, COUNT, SUM, AVG • AGGB(R) considers only non-null values of R. SUMB (R) COUNTB (R) MINB (R) R AVGB (R) COUNT* (R) MAXB (R) Murali Mani

  25. 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. σB=MAXB (R) (R) Murali Mani

  26. Grouping Operator: GL, AL (R) • GL, AL (R) groups all attributes in GL, and performs the aggregation specified in AL. starName, MIN (year)→year, COUNT(title) →num (StarsIn) StarsIn Murali Mani

  27. 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 (R) R Murali Mani

  28. 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 Murali Mani

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