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Lecture 10: Relational Algebra

Lecture 10: Relational Algebra. Relational Algebra. A formalism for creating new relations from existing ones Its place in the big picture:. Declarative query language. Algebra. Implementation. Relational algebra Relational bag algebra. SQL, relational calculus. Relational Algebra.

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Lecture 10: Relational Algebra

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  1. Lecture 10: Relational Algebra

  2. Relational Algebra • A formalism for creating new relations from existing ones • Its place in the big picture: Declarativequerylanguage Algebra Implementation Relational algebraRelational bag algebra SQL,relational calculus

  3. 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

  4. 1. Union and 2. Difference • R1  R2 • Example: • ActiveEmployees  RetiredEmployees • R1 – R2 • Example: • AllEmployees – RetiredEmployees

  5. What about Intersection ? • It is a derived operator • R1  R2 = R1 – (R1 – R2) • Also expressed as a join (will see later) • Example • UnionizedEmployees  RetiredEmployees

  6. 3. Selection • Returns all tuples which satisfy a condition • Notation: sc(R) • Examples • sSalary > 40000 (Employee) • sname = “Smith” (Employee) • The condition c can be =, <, , >,, <>

  7. Find all employees with salary more than $40,000. sSalary > 40000(Employee)

  8. 4. Projection • Eliminates columns, then removes duplicates • Notation: PA1,…,An(R) • Example: project social-security number and names: • PSSN, Name (Employee) • Output schema: Answer(SSN, Name)

  9. PSSN, Name (Employee)

  10. 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

  11. 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

  12. Renaming • Changes the schema, not the instance • Notation: rB1,…,Bn (R) • Example: • rGivenName, SocSecNo(Employee) • Output schema: Answer(GivenName, SocSecNo)

  13. Renaming Example Employee Name SSN John 999999999 Tony 777777777 • GivenName, SocSecNo (Employee) GivenName SocSecNo John 999999999 Tony 777777777

  14. 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

  15. Natural Join Example Employee Name SSN John 999999999 Tony 777777777 Dependents SSN Dname 999999999 Emily 777777777 Joe Employee ⋈ Dependents = PName, SSN, Dname(sSSN=SSN2(Employee rSSN2, Dname(Dependents))) Name SSN Dname John 999999999 Emily Tony 777777777 Joe

  16. Natural Join • R= S= • R ⋈ S=

  17. 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 ?

  18. Theta Join • A join that involves a predicate • R1 ⋈qR2 = sq(R1  R2) • Here q can be any condition

  19. Equi-join • A theta join where q is an equality • R1 ⋈A=BR2 = sA=B(R1  R2) • Example: • Employee ⋈SSN=SSN Dependents • Most useful join in practice

  20. Semijoin • R ⋉ S = PA1,…,An (R ⋈ S) • Where A1, …, An are the attributes in R • Example: • Employee ⋉ Dependents

  21. Semijoins in Distributed Databases • Semijoins are used in distributed databases Dependents Employee network Employee ⋈ssn=ssn (sage>71 (Dependents)) T = PSSNs age>71 (Dependents) R = Employee ⋉ T Answer = R ⋈ Dependents

  22. Complex RA Expressions Pname ⋈buyer-ssn=ssn ⋈pid=pid ⋈seller-ssn=ssn Pssn Ppid sname=`Fred’ sname=gizmo Person Purchase Person Product

  23. 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} • sC(R): preserve the number of occurrences • PA(R): no duplicate elimination • Cartesian product, join: no duplicate elimination • New operator for Duplicate Elimination Relational Engines work on bags, not sets ! Reading assignment: 5.1-5.2

  24. Finally: RA has Limitations • Cannot compute “transitive closure” • Find all direct and indirect relatives of Fred • Cannot be expressed in RA!

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