Chapter 3. The Relational Model. Database Systems p. 61/569

Transcription

1 Chapter 3 The Relational Model Database Systems p. 61/569

2 Introduction The relational model was developed by E.F. Codd in the 1970s (he received the Turing award for it) One of the most widely-used data models Superseded the hierarchical and network models It was/is being extended with object-oriented and semistructured (XML) concepts Database Systems p. 62/569

3 Basic Definition A relational database contains a set of relations A relation R consists of two parts: An instance R, which is a table with rows and columns Represents the current content of this table A schema R which specifies the name of the relation and the names and data types of the columns Defines the structure of the table Database Systems p. 63/569

4 Example of an Instance Assume we want to keep information on students in the relationstudent: attribute stud_no name date_of_birth 1 Smith, John tuple 2 Miller, Anne Jones, Betty We call the rows tuples We call the columns attributes Database Systems p. 64/569

5 Schema of the Relation Student student has three attributes: stud no,name, and date of birth We associate a domain, or data type, with each attribute D stud_no = integer D name = string D date_of_birth = date So the complete schema looks like this: student(stud_no:integer, name:string, date_of_birth:date) Database Systems p. 65/569

6 Schema Information Schema information is also called metadata It is not the actual data but a description of what the data looks like Database Systems p. 66/569

7 More Definitions A relation (instance) is a subset of the Cartesian product of the domains R D 1 D 2 D n For example, student D stud_no D name D date_of_birth = student integer string date Database Systems p. 67/569

8 Even More Definitions The cardinality of a relation is its size in number of tuples A minimal set of attributes whose values uniquely identify a tuple is called a key Usually, the primary key is underlined student(stud_no, name, date_of_birth) Database Systems p. 68/569

9 Converting ER Relational If you want to implement your database using a relational system and you have an ER diagram you can convert this diagram into a relational schema! This is done in three steps: Converting entities Converting relationships Simplifying the schema Database Systems p. 69/569

10 Converting Entities Every entity is transformed into a separate relation Every attribute of the entity becomes an attribute in the relation Choose the key attribute(s) of the entity as primary key of the relation stud_no name date of birth student student(stud_no, name, date_of_birth) Database Systems p. 70/569

11 Converting Relationships Every relationship is transformed into a separate relation Attributes of this relation are made up of the keys of the participating entity relations + any attributes of the relationship (e.g. grade of the relationshipexamines) What is the key of this relation? Database Systems p. 71/569

12 Converting Relationships(2) The multiplicity of a relationship determines the key N:M the key is a combination of both entity keys 1:N, N:1 key of the entity on the N-side becomes key 1:1 choose one of the entity keys (arbitrarily) Database Systems p. 72/569

13 Examples stud_no name date of birth course_no title credits N M student attends lecture attends(stud_no, course_no) pers_no name office_no course_no title credits professor 1 N gives lecture gives(course_no, pers_no) Database Systems p. 73/569

14 Simplifying the Schema There may be relations with the same key after the second step Relations with the same key are merged into a single relation This happens in the case of 1:N, N:1, and 1:1 multiplicities The relationship relation has the same key as one of the entity relations Database Systems p. 74/569

15 Example lecture andgives have the same key: lecture(course_no, title, credits) gives(course_no, pers_no) lecture(course_no, title, credits, pers_no) Database Systems p. 75/569

16 Example Continued pers no is called a foreign key inlecture It references the key inprofessor (and also contains the same values) Sometimes foreign keys are renamed to make it clear where they come from lecture(course_no, title, credits, prof_pers_no) Database Systems p. 76/569

17 Weak Entities Basically, weak entities are converted like 1:N relationships Major difference: the weak entity has a combined key building no office no 1 N building in office building(building_no,... ) office(building_no, office_no,... ) Database Systems p. 77/569

18 Generalizations Generalizations are converted in three different ways There is no direct equivalent translation into the relational model pers_no name employee is a area assistant professor office no (1) only super-type, (2) only sub-types, (3) all types Database Systems p. 78/569

19 Generalizations(2) Only super-type: employee(pers_no, name, area, office_no) Issue: NULL values Only sub-types: assistant(pers_no, name, area) professor(pers_no, name, office_no) Issue: employees who are neither assistants nor professors All types: employee(pers_no, name) assistant(pers_no, area) professor(pers_no, office_no) Issue: information is distributed among three relations Database Systems p. 79/569

20 Generalizations(3) There is no silver bullet You have to make a decision depending on the requirements of the application Database Systems p. 80/569

21 Query Languages So far we have only looked at how to describe the stored data We haven t covered how to retrieve the data from the database In order to access the data we need a query language The relational model comes with the relational algebra as query language Database Systems p. 81/569

22 Relational Algebra All the operators of the relational algebra are set-oriented and closed: Each operator gets as input one or more sets of tuples and outputs a set of tuples relation relation input... relational algebra operator output relation Let us have a closer look at the operators Database Systems p. 82/569

23 Projection Chooses a subset of attributes (columns) A 1,...,A n from a relation R... and throws away the other attributes: Has the following syntax: π A1,...,A n (R) Database Systems p. 83/569

24 Example student stud_no name date_of_birth Query: Algebra: 1 Smith, John Miller, Anne Jones, Betty Johnson, Boris Output the names of all students π name (student) name Smith, John Miller, Anne Jones, Betty Johnson, Boris Database Systems p. 84/569

25 Projection(2) What happens with duplicates? The (basic) relational model is set-oriented: a relation is a set of tuples That means, duplicates are removed Database Systems p. 85/569

26 Example student stud_no name date_of_birth Query: Algebra: 1 Smith, John Miller, Anne Jones, Betty Johnson, Boris Output the birthdays of all students π date_of_birth (student) date_of_birth Database Systems p. 86/569

27 Selection Chooses a subset of tuples (rows) t 1,...,t n from R... and throws away the other tuples: Has the following syntax: σ p (R) A tuple has to satisfy the predicate p to stay in The usual comparison operators are used in predicates: =,<,,>, Predicates can be combined via,, Database Systems p. 87/569

28 Query: Algebra: Example student stud_no name date_of_birth 1 Smith, John Miller, Anne Jones, Betty Johnson, Boris Get all students with a student number less or equal than 3 and a date of birth after σ stud_no 3 date_of_birth> (student) stud_no name date_of_birth 2 Miller, Anne Jones, Betty Database Systems p. 88/569

29 Cartesian Product Up to now, all operators had one relation as input parameter What do we do if we need information from more than one table? The Cartesian product (or cross product) combines every tuple of one relation with every tuple of another Syntax: R 1 R 2. Database Systems p. 89/569

30 Example lecture course_no... prof_pers_no professor pers_no name... 1 Gamper... 2 Helmer... Query: Algebra: Output all lectures and professors lecture professor Database Systems p. 90/569

31 Result course_no... prof_pers_no pers_no name Gamper Helmer Gamper Helmer Gamper Helmer Gamper Helmer... Database Systems p. 91/569

32 Filtering Out Useless Combinations Many of the tuples on the previous slide merge lectures with professors who have nothing to do with this lecture Useless combinations can be filtered out with a selection For our example, we could rewrite the query to σ prof_pers_no=pers_no (lecture professor) Database Systems p. 92/569

33 Join As cross product/filtering is needed very often, there is a shortcut: a join operator ( ) R 1 R1.A i =R 2.A j R 2 = σ R1.A i =R 2.A j (R 1 R 2 ) Joins can also be implemented more efficiently than Cartesian products Database Systems p. 93/569

34 Example Query: Algebra: Output all lectures together with the responsible professors lecture prof_pers_no=pers_no professor course_no... prof_pers_no pers_no name Gamper Gamper Helmer Helmer... Database Systems p. 94/569

35 Combining Relations The names of attributes in a relation need to be unique Important when combining different relations Example: assistant(pers_no, name, area, mgr) professor(pers_no, name, office_no) assistant mgr=pers_no professor We have two different personnel numbers: one for assistants, another for professors Database Systems p. 95/569

36 Renaming In order to keep names unique, we can do renaming: Chooses a subset of attributes and gives them new names Has the following syntax: ρ B1 A 1,...,B n A n (R) Database Systems p. 96/569

37 Example So to avoid clashes between attribute names, we could rename attributes before joining: In relational algebra: ρ a_no pers_no,a_name name (assistant) mgr=p_pers_no ρ p_pers_no pers_no,p_name name (professor) This helps us in distinguishing which attributes come from which relation Database Systems p. 97/569

38 Joins Revisited There are different variants of joins, classified after the join predicate Theta-join (θ-join): the most general join, the predicate may include arbitrary comparison operators That is the one we have looked at so far Equi-join: the join predicate only checks for equality (subset of θ-joins) Natural join: a special version of an equi-join We will now look at this operator in more detail Database Systems p. 98/569

39 Natural Join A natural join does an equi-join on attributes with the same names and projects away the redundant columns does not have a join predicate (it s implicit) Query: Algebra: Output all lectures together with the responsible professors (ρ pers_no prof_pers_no (lecture)) professor course_no... pers_no name Gamper Gamper Helmer Helmer... Database Systems p. 99/569

40 Joins Revisited(2) You can join an arbitrary number of relations in a single relational algebra expression The order does not matter, you ll always get the same result Join operators are commutative and associative Database Systems p. 100/569

41 Example Combine students with the lectures they attend student stud_no name... 1 Smith... 2 Miller lecture course_no title... 1 databases... 2 networks attends stud_no course_no Database Systems p. 101/569

42 Example(2) So we could join the three relations in any of the following ways: student attends lecture student lecture attends attends student lecture attends lecture student lecture student attends lecture attends student You ll notice that the 2nd and 5th permutation joins two relations that do no have a common attribute In that case, the natural join computes a Cartesian product Database Systems p. 102/569

43 Even More Joins If a tuple does not find a join partner in the other relation, it is dropped If you want to keep these tuples, you need to use an outer join Database Systems p. 103/569

44 Full Outer Join In a full outer join ( ) all tuples from both relations are kept L A B C a 1 b 1 c 1 a 2 b 2 c 2 R C D E c 1 d 1 e 1 c 3 d 2 e 2 = result A B C D E a 1 b 1 c 1 d 1 e 1 a 2 b 2 c 2 c 3 d 2 e 2 A natural join would only return the first tuple Database Systems p. 104/569

45 Left Outer Join In a left outer join this is only true for tuples from the relation on the left-hand side L A B C a 1 b 1 c 1 a 2 b 2 c 2 R result C D E = A B C D E c 1 d 1 e 1 a 1 b 1 c 1 d 1 e 1 c 3 d 2 e 2 a 2 b 2 c 2 Database Systems p. 105/569

46 Right Outer Join In a right outer join this is only true for tuples from the relation on the right-hand side L A B C a 1 b 1 c 1 a 2 b 2 c 2 R result C D E = A B C D E c 1 d 1 e 1 a 1 b 1 c 1 d 1 e 1 c 3 d 2 e 2 c 3 d 2 e 2 Database Systems p. 106/569

47 Semi-Join A semi-join does not really join the relations It checks which of the tuples in the relation on the left-hand side satisfies the join predicate... and then keeps those A L B C a 1 b 1 c 1 a 2 b 2 c 2 C R D E = A result B C c 1 d 1 e 1 c 3 d 2 e 2 a 1 b 1 c 1 Database Systems p. 107/569

48 Semi-Join(2) There is also the right-handed version: A L B C a 1 b 1 c 1 a 2 b 2 c 2 C R D E = C result D E c 1 d 1 e 1 c 3 d 2 e 2 c 1 d 1 e 1 Database Systems p. 108/569

49 Set Operations Set operations such as union, intersection, and difference can also be applied to relations... provided that both relations have the same schema, meaning: same number of attributes (with the same names) corresponding attributes have the same data type Database Systems p. 109/569

50 Union prof1 pers_no name 1 Gamper 2 Helmer prof2 pers_no name 2 Helmer 3 Wood Query: Algebra: Create the union of the relation prof1 and prof2 prof1 prof2 pers_no name 1 Gamper 2 Helmer 3 Wood Database Systems p. 110/569

51 Intersection prof1 pers_no name 1 Gamper 2 Helmer prof2 pers_no name 2 Helmer 3 Wood Query: Algebra: Which professors are in both relations? prof1 prof2 pers_no name 2 Helmer Database Systems p. 111/569

52 Set Difference prof1 pers_no name 1 Gamper 2 Helmer prof2 pers_no name 2 Helmer 3 Wood Query: Algebra: Which professors are in the first but not the second relation? prof1 \ prof2 pers_no name 1 Gamper Database Systems p. 112/569

53 Relational Division Relational division is the most complicated operator found in relational algebra Let s start nice and simple: In arithmetic, division is the opposite of multiplication The Cartesian product can be seen as multiplying two relations So relational division can be interpreted as the opposite of the Cartesian product Database Systems p. 113/569

54 Relational Division(2) Let s look at a concrete example: R A a b c S B 1 2 R S A B a 1 a 2 b 1 b 2 c 1 c 2 Dividing R S by R or S gives us the following results: (R S) R = S (R S) S = R Database Systems p. 114/569

55 Relational Division(3) What happens if we apply relational division T R to an arbitrary relation T? That means, T is not the immediate result of a Cartesian product R X Can t be that arbitrary: R T has to hold Roughly similar to dividing 8 by 3 (integer division) 8 is not divisible by 3, but we can say that 8 3 = 2 remainder 2 When computing T R, we try to find a relation R such that R R covers as much as possible of T Database Systems p. 115/569

56 Relational Division(4) Let s look at a concrete example again: T R A B S A B a 2 (R R ) a (S S) b 1 1 b (S S) b 2 (R R ) 2 c (S S) c 1 (S S) c 2 (R R ) T S = S T R = R S A b c R B 2 Database Systems p. 116/569

57 Relational Division(5) S does not include a, as this would produce [a,1], which is not in T S S creates four tuples and we have the remainder [a,2] R does not include 1, as this would again produce [a,1], which is not in T R R creates three tuples and we have the remainder [b,1], [c,1] We have some leftover tuples not covered by the Cartesian product Database Systems p. 117/569

58 What Do We Need It For? Relational division is used to answer queries involving universal quantification: attends stud_no course_no Query: lect1 course_no 1 2 lect2 course_no 1 3 lect3 course_no List thestud no of all students who attend all lectures inlect1/lect2/lect Database Systems p. 118/569

59 Result attends lect1 stud_no 1 3 attends lect2 stud_no 2 3 attends lect3 stud_no 3 Database Systems p. 119/569

60 Formal Definition There is also a formal definition using other operators: { }} { 1 {}}{ T R = π (T \R) (T)\π (T \R) (( π (T \R) (T) R)\T) }{{} } 2 {{ } 4 1. Take the values in T of attributes not shared with R 2. Combine them with R in a Cartesian product 3. Subtract all the stuff that is in T (Now we are left with what s missing in T from the full Cartesian product) 4. Subtract this from 1. and we are now left with the values participating in a product 3 Database Systems p. 120/569

61 Summary The relational model and its variants are the most common data models used in DBMSs In this chapter we covered the (more theoretical) basics of the model: Definition of the model Transforming an ER diagram into a relational schema A first basic query language Database Systems p. 121/569