lecture22.pptx relational Algebra basics types
- 1. Lecture 22: Relational Algebra Friday, November 19, 2004
- 2. DBMS Architecture How does a SQL engine work ? • SQL query relational algebra plan • Relational algebra plan Optimized plan • Execute each operator of the plan
- 3. Relational Algebra • Formalism for creating new relations from existing ones • Its place in the big picture: Declartive query language Algebra Implementation SQL, relational calculus Relational algebra Relational bag algebra
- 4. Relational Algebra • Five operators: – Union: – Difference: - – Selection: s – Projection: P – Cartesian Product: • Derived or auxiliary operators: – Intersection, complement – Joins (natural,equi-join, theta join, semi-join) – Renaming: r
- 5. 1. Union and 2. Difference • R1 R2 • Example: – ActiveEmployees RetiredEmployees • R1 – R2 • Example: – AllEmployees -- RetiredEmployees
- 6. What about Intersection ? • It is a derived operator • R1 R2 = R1 – (R1 – R2) • Also expressed as a join (will see later) • Example – UnionizedEmployees RetiredEmployees
- 7. 3. Selection • Returns all tuples which satisfy a condition • Notation: sc(R) • Examples – sSalary > 40000 (Employee) – sname = “Smith” (Employee) • The condition c can be =, <, , >, , <>
- 8. sSalary > 40000 (Employee) SSN Name Salary 1234545 John 200000 5423341 Smith 600000 4352342 Fred 500000 SSN Name Salary 5423341 Smith 600000 4352342 Fred 500000
- 9. 4. Projection • Eliminates columns, then removes duplicates • Notation: P A1,…,An (R) • Example: project social-security number and names: – P SSN, Name (Employee) – Output schema: Answer(SSN, Name)
- 10. P Name,Salary (Employee) SSN Name Salary 1234545 John 200000 5423341 John 600000 4352342 John 200000 Name Salary John 20000 John 60000
- 11. 5. Cartesian Product • Each tuple in R1 with each tuple in R2 • Notation: R1 R2 • Example: – Employee Dependents • Very rare in practice; mainly used to express joins
- 12. Cartesian Product Example Employee Name SSN John 999999999 Tony 777777777 Dependents EmployeeSSN Dname 999999999 Emily 777777777 Joe Employee x Dependents Name SSN EmployeeSSN Dname John 999999999 999999999 Emily John 999999999 777777777 Joe Tony 777777777 999999999 Emily Tony 777777777 777777777 Joe
- 13. Relational Algebra • Five operators: – Union: – Difference: - – Selection: s – Projection: P – Cartesian Product: • Derived or auxiliary operators: – Intersection, complement – Joins (natural,equi-join, theta join, semi-join) – Renaming: r
- 14. Renaming • Changes the schema, not the instance • Notation: r B1,…,Bn (R) • Example: – rLastName, SocSocNo (Employee) – Output schema: Answer(LastName, SocSocNo)
- 15. Renaming Example Employee Name SSN John 999999999 Tony 777777777 LastName SocSocNo John 999999999 Tony 777777777 rLastName, SocSocNo (Employee)
- 16. Natural Join • Notation: R1 || R2 • Meaning: R1 || R2 = PA(sC(R1 R2)) • Where: – The selection sC checks equality of all common attributes – The projection eliminates the duplicate common attributes
- 17. Natural Join Example Employee Name SSN John 999999999 Tony 777777777 Dependents SSN Dname 999999999 Emily 777777777 Joe Name SSN Dname John 999999999 Emily Tony 777777777 Joe Employee Dependents = PName, SSN, Dname(s SSN=SSN2(Employee x rSSN2, Dname(Dependents))
- 18. Natural Join • R= S= • R || S= A B X Y X Z Y Z Z V B C Z U V W Z V A B C X Z U X Z V Y Z U Y Z V Z V W
- 19. Natural Join • Given the schemas R(A, B, C, D), S(A, C, E), what is the schema of R || S ? • Given R(A, B, C), S(D, E), what is R || S ? • Given R(A, B), S(A, B), what is R || S ?
- 20. Theta Join • A join that involves a predicate • R1 || q R2 = s q (R1 R2) • Here q can be any condition
- 21. Eq-join • A theta join where q is an equality • R1 || A=B R2 = s A=B (R1 R2) • Example: – Employee || SSN=SSN Dependents • Most useful join in practice
- 22. Semijoin • R | S = P A1,…,An (R || S) • Where A1, …, An are the attributes in R • Example: – Employee | Dependents
- 23. Semijoins in Distributed Databases • Semijoins are used in distributed databases SSN Name . . . . . . SSN Dname Age . . . . . . Employee Dependents network Employee | ssn=ssn (s age>71 (Dependents)) T = P SSN s age>71 (Dependents) R = Employee | T Answer = R || Dependents
- 24. Complex RA Expressions Person Purchase Person Product sname=fred sname=gizmo P pid P ssn seller-ssn=ssn pid=pid buyer-ssn=ssn P name
- 25. Operations on Bags A bag = a set with repeated elements All operations need to be defined carefully on bags • {a,b,b,c}{a,b,b,b,e,f,f}={a,a,b,b,b,b,b,c,e,f,f} • {a,b,b,b,c,c} – {b,c,c,c,d} = {a,b,b,d} • sC(R): preserve the number of occurrences • PA(R): no duplicate elimination • Cartesian product, join: no duplicate elimination Important ! Relational Engines work on bags, not sets ! Reading assignment: 5.3 – 5.4
- 26. Finally: RA has Limitations ! • Cannot compute “transitive closure” • Find all direct and indirect relatives of Fred • Cannot express in RA !!! Need to write C program Name1 Name2 Relationship Fred Mary Father Mary Joe Cousin Mary Bill Spouse Nancy Lou Sister