Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1Relational AlgebraChapter 4, Part ADatabase Management Systems 3ed, R. Ramakrishnan and J. Gehrke 2Relational Query Languages Query languages: Allow manipulation and retrieval of data from a database. Relational model supports simple, powerful QLs: Strong formal foundation based on logic. Allows for much optimization. Query Languages != programming languages! QLs not expected to be “Turing complete”. QLs not intended to be used for complex calculations. QLs support easy, efficient access to large data sets.Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 3Formal Relational Query Languages Two mathematical Query Languages form the basis for “real” languages (e.g. SQL), and for implementation: Relational Algebra: More operational, very useful for representing execution plans. Relational Calculus: Lets users describe what they want, rather than how to compute it. (Non-operational, declarative.)Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 4Preliminaries A query is applied to relation instances, and the result of a query is also a relation instance. Schemas of input relations for a query are fixed (but query will run regardless of instance!) The schema for the result of a given query is also fixed! Determined by definition of query language constructs. Positional vs. named-field notation:  Positional notation easier for formal definitions, named-field notation more readable.  Both used in SQLDatabase Management Systems 3ed, R. Ramakrishnan and J. Gehrke 5Example Instancessid sname rating age22 dustin 7 45.031 lubber 8 55.558 rusty 10 35.0sid sname rating age28 yuppy 9 35.031 lubber 8 55.544 guppy 5 35.058 rusty 10 35.0sidbidday2210110/10/9658 103 11/12/96R1S1S2 “Sailors” and “Reserves” relations for our examples. We’ll use positional or named field notation, assume that names of fields in query results are `inherited’ from names of fields in query input relations.Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 6Relational Algebra Basic operations: Selection ( ) Selects a subset of rows from relation. Projection ( ) Deletes unwanted columns from relation. Cross-product ( ) Allows us to combine two relations. Set-difference ( ) Tuples in reln. 1, but not in reln. 2. Union ( ) Tuples in reln. 1 and in reln. 2. Additional operations: Intersection, join, division, renaming: Not essential, but (very!) useful. Since each operation returns a relation, operationscan be composed! (Algebra is “closed”.)σπ−×Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 7Projectionsnameratingyuppy9lubber8guppy5rusty10πsname ratingS,()2age35.055.5πageS()2 Deletes attributes that are not in projection list. Schema of result contains exactly the fields in the projection list, with the same names that they had in the (only) input relation. Projection operator has to eliminate duplicates! (Why??) Note: real systems typically don’t do duplicate elimination unless the user explicitly asks for it. (Why not?)Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 8SelectionσratingS>82()sidsnameratingage28yuppy935.058rusty1035.0snameratingyuppy9rusty10πσsname ratingratingS,(())>82 Selects rows that satisfy selection condition. No duplicates in result! (Why?) Schema of result identical to schema of (only) input relation. Result relation can be the input for another relational algebra operation! (Operatorcomposition.)Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 9Union, Intersection, Set-Difference All of these operations take two input relations, which must be union-compatible: Same number of fields. `Corresponding’ fields have the same type. What is the schema of result?sidsnameratingage22dustin745.031 lubber 8 55.558rusty1035.044guppy535.028yuppy935.0sidsnameratingage31lubber855.558rusty1035.0SS12∪SS12∩sidsnameratingage22dustin745.0SS12−Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 10Cross-Product Each row of S1 is paired with each row of R1. Result schema has one field per field of S1 and R1, with field names `inherited’ if possible. Conflict: Both S1 and R1 have a field called sid.ρ((,),)CsidsidSR115211→→×(sid) sname rating age (sid) bid day22 dustin 7 45.0 22 101 10/10/9622 dustin 7 45.0 58 103 11/12/9631 lubber 8 55.5 22 101 10/10/9631 lubber 8 55.5 58 103 11/12/9658 rusty 10 35.0 22 101 10/10/9658 rusty 10 35.0 58 103 11/12/96 Renaming operator: Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 11Joins Condition Join: Result schema same as that of cross-product. Fewer tuples than cross-product, might be able to compute more efficiently Sometimes called a theta-join. RcScRS =×σ()(sid)snameratingage(sid)bidday22dustin745.05810311/12/9631lubber855.55810311/12/96SRS sid R sid111 1 . .<Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 12Joins Equi-Join: A special case of condition join where the condition c contains only equalities. Result schema similar to cross-product, but only one copy of fields for which equality is specified. Natural Join: Equijoin on all common fields.sidsnameratingagebidday22dustin745.010110/10/9658rusty1035.010311/12/96SRsid11 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 13Division Not supported as a primitive operator, but useful for expressing queries like: Find sailors who have reserved allboats. Let A have 2 fields, x and y; B have only field y: A/B =  i.e., A/B contains all x tuples (sailors) such that for every ytuple (boat) in B, there is an xy tuple in A. Or: If the set of y values (boats) associated with an x value (sailor) in A contains all y values in B, the x value is in A/B. In general, x and y can be any lists of fields; y is the list of fields in B, and x y is the list of fields of A.{}AyxByx ∈∃∈∀ ,|∪Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 14Examples of Division A/Bsnopnos1p1s1p2s1p3s1p4s2p1s2p2s3p2s4p2s4p4pnop2pnop2p4pnop1p2p4snos1s2s3s4snos1s4snos1AB1B2B3A/B1 A/B2 A/B3Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 15Expressing A/B Using Basic Operators Division is not essential op; just a useful