1 Relational Data Model

Transcription

1 Prof. Dr.-Ing. Wolfgang Lehner INTELLIGENT DATABASE GROUP 1 Relational Data Model

2 What is in the Lecture? 1. Database Usage Query Programming Design 2

3 Relational Model 3

4 The Relational Model The Relation Model Developed by Codd (IBM) in 1970 Considered ingenious but impractical in 1970 Conceptually simple Computers lacked power to implement the relational model Today, microcomputers can run sophisticated relational database software [Codd, E.F. (1970). "A Relational Model of Data for Large Shared Data Banks". Communications of the ACM 13 (6): ] 4

5 Relational Model Foundations Foundations Domains: Integer, String[20], Date, Relation R is defined on a relational schema RS Relational Schema RS: Set of attributes {A 1,,A k } Attribute A j : Domain D j = dom(a j ) Relation: Subset of the cartesian product of the domains R D 1 D 2... D k, k 1 Example Domain D 1 = {a,b,c}, D 2 = {0,1} Cartesian product D 1 D 2 = { (a,0), (a,1), (b,0), (b,1), (c,0), (c,1) } Possible relations R 1 = { (a,0), (b,0), (c,0), (c,1) } or R 2 = 5

6 Relational Model Foundations (2) Foundations (cont d) Tuple: Element of a relation Cardinality of a relation: number of tuples in a relation Degree of a relation R D 1 D 2... D k : k 1 Representation of a relation as table Attribut names A and B are column names Order of rows (tuples) and columns (attributes) within a table is important R 1 A B a 0 b 0 c 0 c 1 6

7 Relational Model Foundations (3) Example Relational schema {Name, Inhabitants, State} with dom(name) = String[40], dom(inhabitants) = INTEGER and dom(land) = String[40]} Instances t 1 [Name] = Munich, t 1 [Inhabitants] = and t 1 [State] = Bavaria t 2 [Name] = Dresden, t 2 [Inhabitants] = and t 2 [Germany] = Saxony In general Cities Name Inhabitants State Munich Bavaria Dresden Saxony Database schema = Set of relation schemas Database = Set of relations 7

8 Primary Key Concept R PNr Salary

9 Foreign Key Concept 9

10 Relational Algebra 10

11 Algebra Mathematical system consisting of Operands: variables or values from which new values can be constructed Operators: symbols denoting procedures that construct new values from given values Relational Algebra An algebra whose operands are relations or variables that represent relations Operators are designed to do the most common things that we need to do with relations in a database The result is an algebra that can be used as a query language for relations Core Selection: picking certain rows Projection: picking certain columns Renaming of relations and attributes Products and joins: compositions of relations Union, intersection, and difference Usual set operations, but both operands must have the same relation schema 11

12 Selection R 1 := σ C (R 2 ) C is a condition (as in if ) that refers to attributes of R 2 R 1 is all those tuples of R 2 that satisfy C Example (relation Sells) bar beer price Joe s Bud 2.50 Joe s Miller 2.75 Sue s Bud 2.50 JoeMenu := σ bar= Joe s (Sells) bar beer price Joe s Bud 2.50 Joe s Miller 2.75 Sue s Miller

13 Projection R 1 := π L (R 2 ) L is a list of attributes from the schema of R 2 R 1 is constructed by looking at each tuple of R 2, extracting the attributes on list L, in the order specified, and creating from those components a tuple for R 1 Eliminate duplicate tuples, if any Example (relation Sells) bar beer price Joe s Bud 2.50 Joe s Miller 2.75 Sue s Bud 2.50 Sue s Miller 3.00 Prices := π beer,price (Sells) beer price Bud 2.50 Miller 2.75 Miller

14 Renaming R 1 := ρ R1(A1,,An) (R 2 ) The ρ operator gives a new schema to a relation Makes R 1 be a relation with attributes A 1,,A n and the same tuples as R 2 Simplified notation: R 1 (A 1,,A n ) := R 2 Bars R name addr R(bar, addr) := Bars bar addr Joe s Maple St. Joe s Maple St. Sue s River Rd Sue s River Rd 14

15 Extended Projection For operator π L, we allow L to contain arbitrary expressions Arithmetic on attributes, e.g. A+B C Duplicate occurrences of the same attribute Example R 1 R 2 A B R 2 :=π A+B C,A A1, A A2 (R 1 ) C A1 A

16 Cartesian Product R 3 := R 1 R 2 Pair each tuple t 1 of R1 with each tuple t 2 of R 2 Concatenation t 1 t 2 is a tuple of R 3 Schema of R 3 are the attributes of R 1 and R 2 (in order) To differentiate attribute A of the same name in R 1 and R 2 R 1.A and R 2.A Example R 1 A B R 2 B C R 3 := R 1 R 2 R 3 A R 1.B R 2.B C

17 Natural Join R 3 := R 1 R 2 Connects two relations Equating attributes of the same name, and Projecting out one copy of each pair of equated attributes Example name beer price Joe s Bud 2.50 Joe s Miller 2.75 Sue s Bud 2.50 Sue s Miller 3.00 name addr Joe s Maple St. Sue s River Rd. BarInfo := Sells Bars name beer price addr Joe s Bud 2.50 Maple St. Joe s Miller 2.75 Maple St. Sue s Bud 2.50 River Rd. Sue s Miller 3.00 River Rd. 17

18 Theta-Join R 3 := R 1 C R 2 Equivalent to Take the product R 1 R 2 Then apply σ C to the result C can be any boolean-valued condition, A θ B, where θ is =, <, >, Example bar beer price Joe s Bud 2.50 Joe s Miller 2.75 Sue s Bud 2.50 Sue s Miller 3.00 name Joe s Sue s addr Maple St. River Rd. BarInfo := Sells Sells.bar = Bars.name Bars bar beer price name addr Joe s Bud 2.50 Joe s Maple St. Joe s Miller 2.75 Joe s Maple St. Sue s Bud 2.50 Sue s River Rd. Sue s Miller 3.00 Sue s River Rd. 18

19 Set Operations Union R S R and S have the same relational schema (except renaming) Attribute domains have to be compatible Difference R/S (or R S) R and S have the same relational schema (except renaming) Attribute domains have to be compatible Derived Operations Intersection R S R S = R (R S) Division R S The result of R S consists of the attributes that are in R but not S and for which it holds that all combinations with tuples in S are present in R R S := π R-S (R) π R-S ((π R-S (R) S) R) or: π R (S (R S)) is subset of R 19

20 Example: Division Bars S Bars name beer price Joe s Bud 2.50 Joe s Miller 2.75 Sue s Bud 2.50 Sue s Miller 3.00 S beer price Bud 2.50 Miller 2.75 Bars S name Joe s 20

21 Building Complex Expressions Combine operators with parentheses and precedence rules Three notations: Sequences of assignment statements, expressions with several operators or expression trees Precedence of relational operators [σ, π, ρ] (highest) [, ] [, ] Example R 3 := R 1 C R 2 can be written as R 3 := σ C (R 1 R 2 ) 21

22 Expression Trees Expression Trees Leaves are operands Interior nodes are operators, applied to their child or children Example Using the relations Bars(name, addr) and Sells(bar, beer, price) Find the names of all the bars that are either on Maple St. or sell Bud for less than $3 π name ρ R(name) σ addr = Maple St. π bar Bars σprice<3 AND beer= Bud Sells 22

23 Expression Trees (2) Example (Self-Join) Using Sells(bar, beer, price), find the bars that sell two different beers at the same price Strategy By renaming, define a copy of Sells, called S(bar, beer1, price1) The natural join of Sells and S consists of tuple (bar, beer, beer1, price, price1) such that the bar sells both beers at this price π bar σ beer!= beer1 and price=price1 ρ S(bar, beer1, price1) Sells Sells 23

24 Result Schemas Union, intersection, and difference: the schemas of the two operands must be the same, so use that schema for the result Selection: schema of the result is the same as the schema of the operand Projection: list of attributes tells us the schema Product: schema is the attributes of both relations Use R.A, etc., to distinguish two attributes named A Theta-join: same as product Natural join: union of the attributes of the two relations Renaming: the operator tells the schema 24

25 Summary Relational Model Foundations Primary Key/Foreign Key Relational Algebra Selection Projection Join 25