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Database Management Systems Chapter 4 Relational Algebra Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Formal 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. (Nonoperational, declarative.) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 2 Preliminaries 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 SQL Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 3 Example Schema Sailors (sid: integer, sname: string, rating: integer, age: real); Boats (bid: integer, bname: string, color: string); Reserves (sid: integer, bid: integer, day: date); Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 4 Example Instances R1 sid 22 58 We’ll use positional or sid S1 named field notation, 22 assume that names of fields in query results are 31 `inherited’ from names of 58 fields in query input relations. S2 sid 28 31 44 58 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke bid day 101 10/10/96 103 11/12/96 sname rating age dustin 7 45.0 lubber 8 55.5 rusty 10 35.0 sname rating age yuppy 9 35.0 lubber 8 55.5 guppy 5 35.0 rusty 10 35.0 5 Relational Algebra Basic operations: Additional 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. Intersection, join, division, renaming: Not essential, but (very!) useful. Since each operation returns a relation, operations can be composed! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 6 Projection 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?) sname rating yuppy lubber guppy rusty 9 8 5 10 sname,rating(S2) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke age 35.0 55.5 age(S2) 7 Selection 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! (Operator composition.) sid sname rating age 28 yuppy 9 35.0 58 rusty 10 35.0 rating 8(S2) sname rating yuppy 9 rusty 10 sname,rating( rating 8(S2)) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 8 Set Operation - Union R U S returns a relation instance containing all tuples that occur in either relation instance R or relation instance S (or both), and the schema of the result is defined to be identical to the schema of R. R and S are union-compatible S1 S2 They have the same number of the fields, Corresponding fields, taken in order from left to right, have the same domains. sid 22 31 58 sid 28 31 44 58 sname dustin lubber rusty rating 7 8 10 age 45.0 55.5 35.0 sname rating age yuppy 9 35.0 lubber 8 55.5 guppy 5 35.0 rusty 10 35.0 sid sname rating age 22 31 58 44 28 dustin lubber rusty guppy yuppy 7 8 10 5 9 45.0 55.5 35.0 35.0 35.0 S1 S2 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 9 Set Operation - Intersection returns a relation instance containing all tuples that occur in both R and S, and the schema of the result is defined to be identical to the schema of R. R and S must be unioncompatible. S1 RS S2 sid 22 31 58 sid 28 31 44 58 sname dustin lubber rusty rating 7 8 10 age 45.0 55.5 35.0 sname rating age yuppy 9 35.0 lubber 8 55.5 guppy 5 35.0 rusty 10 35.0 sid sname rating age 31 lubber 8 55.5 58 rusty 10 35.0 S1S 2 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 10 Set Operation - Set-Difference sid 22 31 58 sname dustin lubber rusty rating 7 8 10 age 45.0 55.5 35.0 R-S returns a relation S1 instance containing all tuples that occur in R but not in S, and the schema sid sname rating age 28 yuppy 9 35.0 of the result is defined to S2 31 lubber 8 55.5 be identical to the schema 44 guppy 5 35.0 58 rusty 10 35.0 of R. The relations R and S must be unionsid sname rating compatible. 22 dustin 7 age 45.0 S1 S2 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 11 Set Operation – Cross-product R1 sid bid R X S returns a relation instance whose schema contains all the fields of R followed by all the fields of S, this is one tuple <r,s> for each pair of tuples r∈R, s∈S. day 101 10/10/96 103 11/12/96 22 58 sid S1 22 31 58 sname rating age dustin 7 45.0 lubber 8 55.5 rusty 10 35.0 (sid) sname rating age (sid) bid day 22 dustin 7 45.0 22 101 10/10/96 22 dustin 7 45.0 58 103 11/12/96 31 lubber 8 55.5 22 101 10/10/96 31 lubber 8 55.5 58 103 11/12/96 58 rusty 10 35.0 22 101 10/10/96 58 rusty 10 35.0 58 103 11/12/96 S1 X R1 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 12 Renaming Renaming operator ρ takes an relational algebra expression E and returns an instance of a relation R. ( R(1 sid1, 5 sid 2), S1R1) position New name Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 13 Joins One of most useful operations and most commonly used to combine information from two or more relations. A join operation is defined as a cross-product, then a selection, and projection. R S Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 14 Condition Join R c S c ( R S) (sid) 22 31 sname rating age dustin 7 45.0 lubber 8 55.5 S1 (sid) bid 58 103 58 103 S1. sid R1. sid day 11/12/96 11/12/96 R1 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. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 15 Equi-Join A special case of condition join where the condition c contains only equalities. sid sname rating age bid day 22 dustin 7 45.0 101 10/10/96 58 rusty 10 35.0 103 11/12/96 S1 R1 S1.sid R1.sid Result schema similar to cross-product, but only one copy of fields for which equality is specified. Natural Join: Equijoin on all common fields. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 16 Division Not supported as a primitive operator, but useful for expressing queries like: Find sailors who have reserved all boats. Let A have 2 fields, x and y; B have only 1 field y: A/B contains all x tuples such that for every y tuple in B, there is an x y tuple in A. Or: If the set of y values associated with an x value 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. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 17 Examples of Division A/B sno s1 s1 s1 s1 s2 s2 s3 s4 s4 pno p1 p2 p3 p4 p1 p2 p2 p2 p4 pno p2 sno s1 s2 s3 s4 sno s1 s4 sno s1 A A/B1 A/B2 A/B3 pno p2 p4 B1 B2 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke pno p1 p2 p4 B3 18 Expressing A/B Using Basic Operators Division is not essential op; just a useful shorthand. (Also true of joins, but joins are so common that systems implement joins specially.) Idea: For A/B, compute all x values that are not `disqualified’ by some y value in B. x value is disqualified if by attaching y value from B, we obtain an xy tuple that is not in A. Disqualified x values: A/B: x ( A) x (( x ( A) B) A) all disqualified tuples Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 19 Q1: Find names of sailors who’ve reserved boat #103 Solution 1: Solution 2: sname(( bid 103 (Temp1, Reserves) Sailors) C bid 103 Re serves) ( Temp2, Temp1 Sailors) C sname (Temp2) Solution 3: sname ( bid 103 (Re serves Sailors)) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke C 20 Q2: Find names of sailors who’ve reserved a red boat Information about boat color only available in Boats; so need an extra join: sname (( Boats) Re serves Sailors) color ' red ' A more efficient solution: sname ( (( Boats) Re s) Sailors) sid bid color ' red ' A query optimizer can find this, given the first solution! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 21 Q5: Find sailors who’ve reserved a red or a green boat Can identify all red or green boats, then find sailors who’ve reserved one of these boats: (Tempboats, ( color ' red ' color ' green ' Boats)) sname(Tempboats Re serves Sailors) Can also define Tempboats using union! (How?) What happens if is replaced by in this query? Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 22 Q6: Find sailors who’ve reserved a red and a green boat Previous approach won’t work! Must identify sailors who’ve reserved red boats, sailors who’ve reserved green boats, then find the intersection (note that sid is a key for Sailors): (Tempred, sid (Tempgreen, (( sid color ' red ' (( Boats) Re serves)) color ' green' Boats) Re serves)) sname((Tempred Tempgreen) Sailors) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 23 Q9: Find the names of sailors who’ve reserved all boats Uses division; schemas of the input relations to / must be carefully chosen: (Tempsids, ( sid, bid Re serves) / ( bid Boats)) sname (Tempsids Sailors) To find sailors who’ve reserved all ‘Interlake’ boats: ..... / bid ( bname ' Interlake' Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke Boats) 24 Summary The relational model has rigorously defined query languages that are simple and powerful. Relational algebra is more operational; useful as internal representation for query evaluation plans. Several ways of expressing a given query; a query optimizer should choose the most efficient version. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 25