Relational algebra. Bence Molnár

Transcription

1 Relational algebra Bence Molnár

2 Introduction

3 About relational algebra Basic properties of relational algebra: Well defined mathematical background Based on set theory Can be introduced through Armstrong axioms In practice: Query language relational algebra Database design of RDBMS Makes SQL language much easier based on

4 Memo Column=Attribute Address Peter Rod Phone Education Company Budapest Mechanical engineer Szerszámgyártó Zrt. Stephan Artamis Cegléd Civil engineer Út kivitelező Nyrt. Frank Torem Budapest Economist Elszámolok Kft. Stephane Boren Budapest Secondary school Út kivitelező Nyrt Row=Record=Tuple Cell=Field

5 Relational scheme

6 Relational scheme Following table is given: s ID 1 Peter Smith 4 Jack Daniels 4 1 Relational scheme: s(id,, ) Relation: Two dimensional data set in table format

7 Relational scheme - terms Table is the realization of a relation s relation ID 1 4 Peter Smith Jack Daniels Relation scheme: s(id,, ) of relation Attributes set of attributes 4 1

8 Relational scheme - properties Properties of relation Attribute (column) order is indifferent, free to swap (some operation requires matching order). Record (row) order is indifferent, free to swap One attribute and a given record contains one and only one component In our case: one record can be repeated with same attribute values (set vs. multiset)

9 Scheme examples Profile(ID,, Profile, Moment of inertia, Price) Sphere(ID, Road(ID, X, Y, Z, R), Class) RealEstate(ID, Area, Price) Create Parcel_number, a realization (create the table) Owner,

10 Attribute types

11 Attribute types All attributes must have an attribute type, which defines the set of valid values for a cell. Basic types (examples): Numeric Integer Real Text Logical AutoNumber

12 Attribute types Complex types (examples): Mask: , XXXXX Polyline, sphere, geometric entities Binary Large Object (BLOB) Image, MP, etc... Attribute types are often included to the scheme: s(id : AutoNumber, : Text, : Integer)

13 Selecting proper types From storage point of view Like on a paper sheet: wide predefined column with waste a lot of paper, while strait won't fit Min and max of possible values (1-9, a-z, a-z...) From operation point of view Mathematical operations Ordering (eg. Chinese characters) Possibility to compare cell values (eg. color vs. colour) Referencing Code table Other relation attribute values

14 Other examples Profiles(ID: AutoNumber, : Text, Profile: Real, Moment of inertia : Real, Price: Integer) Sphere(ID: AutoNumber, Real, Z: Real, R: Real) X: Road(ID: : AutoNumber, Class: Integer) Real, Realty(ID: Y: Text, AutoNumber, Parcel_number: Text, Owner: Text, Area: Real, Price: Integer) Multiple perfect solution is possible!

15 Candidate key, Superkey

16 Superkey Superkey: set of attributes which holds that there are no two distinct tuples (rows) that have the same values for the attributes in this set SzK1={{ID}, {ID, ID Age }, {ID, Age}, 1 18 {ID,, Age}} SzK=SzK1 U {{Age}, {Age, }} ID Age

17 Candidate key Key: a minimal set of attributes necessary to identify a tuple ID Age K1={{ID}} ID Age K={{ID}, {Age}}

18 Candidate key example ID Course Math Biology Math SK={{ID, Course}, {, ID, Course}, } K={ID, Course} Compound key

19 Keys in scheme Up to now super and candidate key were created based on one or more realization. However it can be specified to ensure an attribute to be a key. It's marked in the scheme as follows: s(id,, )

20 Examples Profile(: Moment of Integer) Sphere(X: Text, Profile: Real, inertia: Real, Price: Real, Y: Real, Z: Real, R: Real) Road(: Text, Class: Integer) Realty(Parcel_number: Text, Owner: Text, Area: Real, Price: Integer) Sometimes candidate key is missing!

21 Operations

22 Operations To perform a two-variable set operation, the following have to be met on both relations (R and S): R and S should store the same set of attributes Attributes need to be ordered in the same way

23 Set operation 1 - Union Mark: S R Pete Great Erica Euro Bob Davis U John Little Multiset = Pete Great Erica Euro Bob Davis John Little

24 Set operation - Intersection Mark:S R Pete Great Erica Euro Bob Davis John Little S R \ =

25 Set operation - Difference Mark: Also S\R called: complement Peter Great Erica Euro Bob Davis \ John Little = Peter Great Erica Euro Bob Davis Operation order!!!

26 Projection Mark: π name ( π attr1, attr,... (S ) Stephan Smith Peter Great Erica Euro Erica Euro Bob Davis Bob Davis π name, grade ( )= Stephan Smith Peter Great Attendance Stephan Smith Stephan Smith Peter Great 14 Peter Great Erica Euro Erica Euro 1 Bob Davis Bob Davis )=

27 Selection Mark: σ attr1 R value R attr R value R... (S ) R ('=',' <',' >',' ',' ',' ',' ',' ') σ grade= ( Stephan Smith Peter Great Erica Euro Stephan Smith σ grade>1 attendance>10 ( ) = Peter Great Stephan Smith Attendance Stephan Smith 1 14 Peter Great 14 Erica Euro 1 Bob Davis 10 )= On Board

28 Cartesian Product Mark: A B Stephan Smith Peter Great Attendance Stephan Smith 10 Peter Great 14 A B = A. B. Attendance Stephan Smith Stephan Smith 10 Stephan Smith Peter Great 14 Peter Great Stephan Smith 10 Peter Great Peter Great 14

29 Natural Join Mark: A B Attendance Stephan Smith Stephan Smith 10 Peter Great Peter Great 14 Luis Great Attendance Stephan Smith 10 Peter Great 14 = Only available if attribute names are exactly the same

30 Theta-Join Mark: A B attr1 R attr R... R ('=','<',' >',' ',' ',' ',' ',' ') Stephan Smith Peter Great Stephan A. = B. Attendance >10 Smith Attendance Peter Great 14 Attendance 10 Peter Great 14 Luis Great = As it based on Cartesian product, attribute names must be specified as base of join.

31 Example 1 S Attendance Stephan Smith 8 Stev Alan 14 Ann Moro 10 Peter Great 1 14 Specify with relation algebra the list of students who managed the course. π name (σ grade >1 attendance >10 (S ))

32 Example A Course Score Stephan Smith Math 0 8 Stephan Smith Graphics Stev Alan Statics 4 10 Peter Great Math 1 14 B Year Stephan Smith 1 Ann Moro Stev Alan 1 Peter Great 1 Attendance C Course MinScore Math 40 Graphics 60 Statics 0

33 Example 1) List students who visit graph! ) List students who visit first year! ) List courses where more than 4 point is required. 4) List students and their year who visit math. ) List students and their successfully managed courses. 6) List managed courses of students and corresponding student names who are visiting first year.

34 Example 1) π (σ Course=' Graph ' ( A)) ) π (σ Year =1 ( B)) ) π Course (σ MinScore >4 (C )) 4) π, Year (σ A. Course=' Math ' ( B) ) π, Course ( A 6) A) A. Score>C. MinScore A. Course=C.Course π, Course (σ Year=1 ( B) (A Syntax!!! A. Score>C. MinScore A.Course=C.Course C) C ))

35 Thank you!