Foundations of Information Systems 1 Relational Data Modelpages.iai.uni-bonn.de/behrend_andreas/lehre/FIM/WS16/01 Relational... · The Relation Model Developed by Codd (IBM) in 1970

© Prof. Dr.-Ing. Wolfgang Lehner |

INTELLIGENT DATABASE GROUP

Relational Data Model 1

Foundations of Information Systems

| 2

>

© A. Behrend Foundations of Information Systems |

What is in the Lecture?

1. Database Usage Query Programming Design

© A. Behrend Foundations of Information Systems | 3

>

|

Relational Model

| 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): 377–387.]

| 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 {A1,…,Ak} Attribute Aj: Domain Dj = dom(Aj) Relation: Subset of the cartesian product of the domains

R ⊆ D1 × D2 × ... × Dk, k ≥ 1

Example Domain D1 = {a,b,c}, D2 = {0,1} Cartesian product D1 × D2 = { (a,0), (a,1), (b,0), (b,1), (c,0), (c,1) } Possible relations R1= { (a,0), (b,0), (c,0), (c,1) } or R2 = ∅

| 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 ⊆ D1 × D2 × ... × Dk: 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

R1 A B

a 0

b 0

c 0

c 1

| 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 t1[Name] = Munich, t1[Inhabitants] = 101617 and t1[State] = Bavaria t2[Name] = Dresden, t2[Inhabitants] = 476668 and t2[Germany] = Saxony

In general Database schema = Set of relation schemas Database = Set of relations

Cities Name Inhabitants State Munich 1.353.186 Bavaria Dresden 523.058 Saxony

| 8

>


Primary Key Concept

R PNr Salary

1 1700

2 2172

4 3189

| 9

>


Foreign Key Concept


>

|

Relational Algebra

| 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

| 12

>


Selection

R1 := σC (R2) C is a condition (as in “if”) that refers to attributes of R2 R1 is all those tuples of R2 that satisfy C

Example (relation Sells)

JoeMenu := σbar=“Joe’s”(Sells) bar beer price

Joe’s Bud 2.50

Joe’s Miller 2.75

Sue’s Bud 2.50

Sue’s Miller 3.00

bar beer price

Joe’s Bud 2.50

Joe’s Miller 2.75

| 13

>


Projection

R1 := πL (R2) L is a list of attributes from the schema of R2 R1 is constructed by looking at each tuple of R2, extracting the attributes on list L, in

the order specified, and creating from those components a tuple for R1 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 := πbeer,price(Sells)

beer price

Bud 2.50

Miller 2.75

Miller 3.00

| 14

>


Renaming

R1 := ρR1(A1,…,An)(R2) The ρ operator gives a new schema to a relation Makes R1 be a relation with attributes A1,…,An and the same tuples as R2 Simplified notation: R1(A1,…,An) := R2

R(bar, addr) := Bars name addr

Joe’s Maple St.

Sue’s River Rd

bar addr

Joe’s Maple St.

Sue’s River Rd

Bars R

| 15

>


Extended Projection

For operator πL, we allow L to contain arbitrary expressions Arithmetic on attributes, e.g. A+B C Duplicate occurrences of the same attribute

Example

R2:=πA+BC,AA1, AA2 (R1) A B

1 2

3 4

C A1 A2

3 1 1

7 3 3

R1 R2

| 16

>


Cartesian Product

R3 := R1 × R2

Pair each tuple t1 of R1 with each tuple t2 of R2 Concatenation t1t2 is a tuple of R3 Schema of R3 are the attributes of R1 and R2 (in order) To differentiate attribute A of the same name in R1 and R2 R1.A and R2.A

Example

A B

1 2

3 4

B C

2 6

4 8

9 19

A R1.B R2.B C

1 2 2 6

1 2 4 8

1 2 9 19

3 4 2 6

3 4 4 8

3 4 9 19

R1

R2

R3

R3 := R1 × R2

| 17

>


Natural Join

R3 := R1 ⋈ R2

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.

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.

BarInfo := Sells ⋈ Bars

| 18

>


Theta-Join

R3 := R1 ⋈C R2

Equivalent to Take the product R1 × R2 Then apply σC 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 addr

Joe’s Maple St.

Sue’s River Rd.

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.

BarInfo := Sells ⋈Sells.bar = Bars.name Bars

| 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 := πR-S(R) – πR-S((πR-S(R)×S) – R) or: πR(S×(R÷S)) is subset of R

| 20

>


Example: Division Bars÷S

name beer price

Joe’s Bud 2.50

Joe’s Miller 2.75

Sue’s Bud 2.50

Sue’s Miller 3.00

Bars

beer price

Bud 2.50

Miller 2.75

S

name

Joe’s

Bars÷S

| 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 R3 := R1 ⋈C R2 can be written as R3 := σC (R1×R2)

| 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

Bars

Sells

σaddr = “Maple St.”

σprice<3 AND beer=“Bud”

πname ρR(name)

πbar

∪

| 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

Sells

Sells ρS(bar, beer1, price1)

⋈

πbar

σbeer != beer1 and price=price1

| 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


>

|

Summary

Relational Model Foundations Primary Key/Foreign Key

Relational Algebra Selection Projection Join …

Documents

Foundations of Information Systems 1 Relational Data Modelpages.iai.uni-bonn.de/behrend_andreas/lehre/FIM/WS16/01 Relational... · The Relation Model Developed by Codd (IBM) in 1970