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Relational Algebra Ch. 7.4 – 7.6

Relational Algebra Ch. 7.4 – 7.6. John Ortiz. Relational Query Languages. Query languages: allow manipulation and retrieval of data from a database. Relational QLs are simple & powerful. Strong formal foundation based on logic. Allows for much optimization.

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Relational Algebra Ch. 7.4 – 7.6

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  1. Relational AlgebraCh. 7.4 – 7.6 John Ortiz

  2. Relational Query Languages • Query languages: allow manipulation and retrieval of data from a database. • Relational QLs are simple & powerful. • Strong formal foundation based on logic. • Allows for much optimization. • Query languages != programming languages! • Not intended for complex calculations. • Support easy, efficient access to large data sets. Relational Algebra

  3. Preliminaries • A query is applied to relation instances, and the result of a query is also a relation instance. • Schemas of input & result relations are fixed (determined by relations & query language constructs). • A query is specified against schemas (regardless of instances). • Attributes may be referenced either by names or by positions (two notation systems). Relational Algebra

  4. Relational Algebra • Basic Operations: • Selection (): choose a subset of rows. • Projection (): choose a subset of columns. • Cross Product (): Combine two tables. • Union (): unique tuples from either table. • Set difference (): tuples in R1 not in R2. • Renaming (): change names of tables & columns • Additional Operations (for convenience): • Intersection, joins (very useful), division, outer joins, aggregate functions, etc. Relational Algebra

  5. Result Students Selection • Format: selection-condition(R). Choose tuples that satisfy the selection condition. • Result has identical schema as the input. Major = ‘CS’(Students) • Selection condition is a Boolean expression including =, , <, , >, , and, or, not. Relational Algebra

  6. Students Result Projection • Format: attribute-list(R). Retain only those columns in the attribute-list. • Result must eliminate duplicates. Major(Students) • Operations can be composed. Name, GPA(Major = ‘CS’(Students)) Relational Algebra

  7. Cross Product • Format: R1  R2. Each row of R1 is paired with each row of R2. • Result schema consists of all attributes of R1 followed by all attributes of R2. • Problem: Columns may have identical names. Use notation R.A, or renaming attributes. • Only some rows make sense. Often need a selection to follow. Relational Algebra

  8. Students  Awards Students Awards Example of Cross Product Relational Algebra

  9. Students CS_Students Renaming • Format: S(R) or S(A1, A2, …)(R): change the name of relation R, and names of attributes of R CS_Students(Major = ‘CS’(Students)) Relational Algebra

  10. Union, Intersection, Set Difference • Format: R1 R2 (R1  R2, R1  R2). Return all tuples that belong to either R1 or R2 (to both R1 and R2; to R1 but not to R2). • Requirement: R1 and R2 are union compatible. • With same number of attributes. • Corresponding attributes have same domains. • Schema of result is identical to that of R1. May need renaming. • Duplicates are eliminated. Relational Algebra

  11. TAs  RAs TAs  RAs TAs RAs TAs  RAs Examples of Set Operations Relational Algebra

  12. Joins • Theta Join. • Format: R1 join-condition R2. • Returns tuples in join-condition(R1 R2) • Equijoin. • Same as Theta Join except the join-condition contains only equalities. • Natural Join. • Same as Equijoin except that equality conditions are on common attributes and duplicate columns are eliminated. Relational Algebra

  13. Students Profs Result Examples of Joins • Theta Join. Students Students.Age<=Profs.Age Profs Relational Algebra

  14. Result Result Examples of Joins (cont.) • Equijoin. Students Prof=PID AND Name=Pname Profs • Natural Join. Students Profs Relational Algebra

  15. Some Questions About Joins * • What is the result of R1 R2 if they do not have a common attribute? • What is the result of R R? • Consider relations Students(SSN, Name, GPA, Major, Age, PSSN) Profs(PSSN, Name, Office, Age, Dept) • Which type of join should be used to find pairs of names of students and their advisors? • Can a natural join be used? How? Relational Algebra

  16. Division • Format: R1  R2. Restriction: Every attribute in R2 is in R1. • For R1(A1, ..., An, B1, ..., Bm) R2(B1, ..., Bm) and T = A1, ..., An (R1), Return the subset of T, say W, such that every tuple in W  R2 is in R1. • W is the largest subset of T, such that, (W  R2)  R1 Relational Algebra

  17. Takes CS_Req Result An Example of Division Takes  CS_Req • What is the meaning of this expression? Relational Algebra

  18. Students Result Grouping & Aggregate Functions • Format: group_attributes F aggregate_functions ( r ) • Partition a relation into groups • Apply aggregate function to each group • Output grouping and aggregation values, one tuple per group • Ex: Major F count(SID), avg(GPA) (Students) Relational Algebra

  19. Dangling Tuples in Join • Usually, only a subset of tuples of each relation will actually participate in a join. • Tuples of a relation not participating in a join are dangling tuples. • How do we keep dangling tuples in the result of a join? (Why do we want to do that?) • Use null values to indicate a “no-join” situation. Relational Algebra

  20. Outer Joins • Left Outer Join. • Format: R1 R2. Similar to a natural join but keep all dangling tuples of R1. • Right Outer Join. • Format: R1 R2. Similar to a natural join but keep all dangling tuples of R2. • (Full) Outer Join. • Format: R1 R2. Similar to a natural join but keep all dangling tuples of both R1 & R2. • Can also have Theta Outer Joins. Relational Algebra

  21. Students Awards Result Examples of Outer Joins • Left Outer Join. Students Awards Relational Algebra

  22. R S Relational Algebra Exercises • Find the result of these expressions. • R S • R R.C=S.C S • B,E((B,C R) (E<7 S)) • (A,BR) - S(A,B) (D,C S) Relational Algebra

  23. Queries In Relational Algebra Consider the following database schema: Students(SSN, Name, GPA, Age, MajorDept) Enrollment(SSN, CourseNo, Grade) Courses(CourseNo, Title, DName) Departments(DName, Location, Phone) • Two methods: • Use temporary relations. • One expression per query. Relational Algebra

  24. Queries In Relational Algebra • List student name and course title such that the student has an A in the course and the course is not offered by the student’s major department. • Find those students who got an A in any course. • Find the department of the students and the courses. • Find the final answer. Relational Algebra

  25. Summary • Relational model provides simple yet powerful formal query languages. • Relational algebra is procedural and used for internal representation of queries. • Several ways to express a given query. DBMS should choose the most efficient plan. • Any language able to express all relational algebra queries is relational complete. Relational Algebra

  26. Summary (cont.) Lots useful properties. • C1(C2(R)) = C2(C1(R)) = C1 and C2(R) • L1(L2(R)) = L1(R) , if L1  L2 • R1 R2 = R2  R1 • R1  (R2  R3) = (R1 R2)  R3 • R1 R2 = R2 R1 • R1 (R2 R3) = (R1 R2) R3 Relational Algebra

  27. Look Ahead • Next topic: Translation form ER/EER to relational model • Read from the textbook: • Chapter 14.1 – 14.2 Relational Algebra

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