Databaser den relationelle model

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1 Databaser den relationelle model

2 Databaser 1. Relationel algebra 2. SQL 3. Objekter i en relationel database 2

3 The relational model From Wikipedia Relational algebra received little attention outside of pure mathematics until the publication of E.F. Codd s relational model of data in Codd proposed such an algebra as a basis for database query languages. Codd s Theorem: relational algebra is essentially equivalent to relational calculus and so first-order logic Relational databases and SQL are based on relational algebra In 1981, Codd received the Turing Award for his contributions 3

4 The relational model The relational model Tuples Cartesian-products Binary relations n-ary relations Relational databases schema database design relational operators (algebra) updates 4

5 Tuple A tuple is a ordered list of values (768, 120) position on the screen (2212XX-XXXX,, Male, Aarhus) (56814,, Postdoc, Computer Science) 5

6 Cartesian Product Given sets A1,...,An, we may form the set of all tuples A 1 A n = {(a 1,...,a n ) a 1 A 1 and and a n A n } Example: (56814,, Postdoc, Datalogi) N Valid names {Student, Postdoc..., Professor} {Physics, Philosophy,..., Computer Science} 6

7 Cartesian Product (Binært) A B = { (a, b) a A b B } Example: A = { Hirtshals, Frederikshavn, Grenå } B = { Kristianssand, Oslo, Göteborg, Varberg } A x B = { (Hi, Kr), (Hi, Os), (Hi, Gö), (Hi, Va), (Fr, Kr), (Fr, Os), (Fr, Gö), (Fr, Va), (Gr, Kr), (Gr, Os), (Gr, Gö), (Gr, Va) } 7

8 N-ary Relations A relation over sets A 1,... A n is a subset of A 1... A n This can be represented as a table with n columns Each row is an n-tuple in the relation and every column is a component of the cartesian product the relation is based upon 8

9 Relations Example: Authors = { Tolkien, Eco, H.C. Andersen } Books = { Lord of the rings, The hobbit, The name of the rose} Years = N Author Book Year Eco Tolkien The Name of the Rose Lord of the Rings Tolkien The Hobbit

10 Relations (Binary) A binary relation between sets A and B is a subset of A B If R A B is a relation with (a, b) R, we often write a R b Z Z is-mother-of Women People Mathematical relations are often defined from predicates {(a, b) a mod b = 0} N N+ {(x, y) y = x 2 } R R We re more interested in arbitrary (finite) subsets hand {A, 2, 3,...,10, J, Q, K} {,,, } employees IDs Names 10

11 Binary relation A binary relation between sets A and B is a subset of A B is-mother-of Women People Example: A = { Hirtshals, Frederikshavn, Grenå } B = { Kristianssand, Oslo, Göteborg, Varberg } R = Færgeforbindelser mellem A og B { (a, b) færgeforbindelse_mellem(a, b) } R = { (Hi, Kr), (Hi, Os), (Hi, Gö), (Hi, Va), (Fr, Kr), (Fr, Os), (Fr, Gö), (Fr, Va), (Gr, Kr), (Gr, Os), (Gr, Gö), (Gr, Va) } 11

12 Schema and instance Relation schema a list of attributes and their corresponding domains similar to the type of a variable in programming languages notation: Book-schema = (author, title, year) Relation instance the current set of tuples, the relation similar to the value of a variable in programming languages notation: books(book-schema) 12

13 Ex: Popular American Broadway musicals (PAPM) Title Month Day Year Theater Perfs My Fair Lady Mark Hellinger 2717 Man of La Mancha ANTA Wash. Sq Oklahoma! St. James 2248 Hair Biltmore 1750 The King and I St. James 1246 Guys and Dolls Forty-Sixth St Cabaret Broadhurst 1166 Damn Yankees Forty-Sixth St Camelot Majestic 878 West Side Story Winter Garden

14 Schema PABM = Title Month Day Year Theater Perfs Title Mængden af titler på Broadway-shows Month Tallene (årets måneder) Day Tallene (månedens dage) Year Positive heltal Theater Mængden af teatre på og omkring Broadway, NY Perfs Mængden af positive heltal 14

15 Example: Musical Creators (MC) Title Book Lyrics Music My Fair Lady Lerner Lerner Loewe Man of La Mancha Wasserman Darion Leigh Oklahoma! Hammerstein Hammerstein Rodgers Hair Ragni & Rado Ragni & Rado MacDermot The King and I Hammerstein Hammerstein Rodgers Guys and Dolls Swerling & Burrows Loesser Loesser Cabaret Masteroff Ebb Kander Damn Yankees Abbott & Wallop Adler & Ross Adler & Ross Camelot Lerner Lerner Loewe West Side Story Laurents Sondheim Bernstein 15

16 Database design Schema PABM-scheme = (Title, Month, Day, Year, Theater, Perfs) MC-scheme = (Title, Book, Lyrics, Music) Alternative 1 All-scheme = (Title, Month, Day, Year, Theater, Perfs, Book, Lyrics, Music) Alternative 2 Where-scheme = (Title, Theater) When-scheme = (Title, Month, Day, Year) Author-scheme = (Title, Book) Run-scheme = (Title, Perfs) Lyricist-scheme = (Title, Lyrics) Composer-scheme = (Title, Music) 16

17 Queries The relational algebra is a query language Consists of operations that take one or two relations as an input and produce a new relation as a result queries may be composed Example: Which shows had opening night in March? Title Month Day Year Theater Perfs My Fair Lady Mark Hellinger 2717 Oklahoma! St. James 2248 The King and I St. James

18 Operators Select (σ) σ P (R) = The tuples form relation R that satisfies predicate P σ cuts horisontally Project (π) π A1, A2,..., An (R) = R but only with the attributes A 1, A 2,..., A n π cuts vertically Cartesian-product ( ) R 1 R 2 = The tuples that comes from combining all elements in R 1 with all elements of R 2 18

19 Operators Rename (ρ) ρ X (E) returns the result of the expression E under the name X ρ X(A1, A 2,..., A n )(E) returns E under the name X with renamed attributes Natural join ( ) R 1 R 2 = R 1 R 2 with only those tuples matching on attributes with the same name are included 19

20 Example: Select σ σ Perfs > 2000 (PABM) Title Month Day Year Theater Perfs My Fair Lady Mark Hellinger 2717 Man of La Mancha ANTA Wash. Sq Oklahoma! St. James 2248 σ Theater = Forty-Sixth St. (PABM) Title Month Day Year Theater Perfs Guys and Dolls Forty-Sixth St Damn Yankees Forty-Sixth St

21 Example: Projection (π) and Composition π Title, Perfs (σ Theater = Forty-Sixth St. (PABM)) Title Perfs Guys and Dolls 1200 Damn Yankees

22 Example: Natural Join ( ) Where Author Title Theater Title Book My Fair Lady Mark Hellinger My Fair Lady Lerner Oklahoma! St. James Oklahoma! Hammerstein Hair Biltmore The King and I Hammerstein Where Author Title Theater Book My Fair Lady Mark Hellinger Lerner Oklahoma! St. James Hammerstein Øvelse: Hvordan ser Where Author ud? Hvad er skemaet? 22

23 Expressions in the relational algebra Fundamental operations: Union Difference Cross Product Select Project Rename Extra operations: Intersection Natural join Division Expressions E are built up according to r, a relation in the database or a relation variable constant relations like {(The Kid,1921), (UTurn,1997)} E 1 E 2, E 1 E 2, E 1 E 2 σ P (E), π A1,...,An (E) ρ x (E) or ρ x(a1,...,an) (E) Statements assignment: r E insertion: r r E deletion: r r E 23

24 Examples E 1 E 2 E 2 E 1 - E 2 E 1? Hvordan kan vi finde E 1 E 2 ved hjælp af de basale operationer? Hvad skal gælde om skemaerne for E 1 og E 2? 24

25 Tricky example! How can we express natural join with the fundamental operations? 25

26 Find the musical with most performances! Title Month Day Year Theater Perfs My Fair Lady Mark Hellinger 2717 Man of La Mancha ANTA Wash. Sq Oklahoma! St. James 2248 Hair Biltmore 1750 The King and I St. James 1246 Guys and Dolls Forty-Sixth St Cabaret Broadhurst 1166 Damn Yankees Forty-Sixth St Camelot Majestic 878 West Side Story Winter Garden

27 Find the musical with most performances! Compare all with all r σ PABM.perfs < x.perfs ( PABM x ρ x (PABM)) What do we get? Now we project r π name, perfs (r) What do we get? Finally the difference π name, perfs (PABM) - r 27

Relational Algebra. Module 3, Lecture 1. Database Management Systems, R. Ramakrishnan 1

Relational Algebra. Module 3, Lecture 1. Database Management Systems, R. Ramakrishnan 1 Relational Algebra Module 3, Lecture 1 Database Management Systems, R. Ramakrishnan 1 Relational Query Languages Query languages: Allow manipulation and retrieval of data from a database. Relational model