The Relational Model and Relational Algebra

Transcription

1 The Relational Model and Relational Algebra Background Introduced by Ted Codd of IBM Research in Concept of mathematical relation as the underlying basis. The standard database model for most transactional database today. 1

2 Relational Model Concepts A relational database is a collection of relation. A relation resembles a Table or a Flatfile of records. Each row in such a table represents a collection of such related data values ( an instance of the relation depicted by the table) Roll No: Name Adithya Ananth Kumar Barath Kumar Date of Registration Table 1: A example relation called STUDENT Each row is called a tuple and each column is called attribute of the relation Some Definitions A data value is said to be atomic if it cannot be subdivided into smaller data values. For example, the age of a person in years is an atomic value. A Domain is a set of atomic values. Examples are: the set of all Integers, the set of all valid student roll number, the set of all Pakistani cities with population above 6 million etc. A Relation Schema denoted by R(A 1,A 2,A 3,A n ) is made up of relation name R and a list of attributes A 1,A 2,A 3,A n. Each A i is the name of role played some domain D in the schema R. The domain A i is denoted by dom(a i ). 2

3 Some Definitions The Degree of the relation is the number of attributes of its relation schema. A relation r of a relation schema R(A 1,A 2,A 3,A n ), is a set of n-tuples of the form [ t 1,t 2,.,t n ] where each t i dom(a i ). The Relation can be defined as r(r) (dom(a 1 ) x dom(a 2 ) x dom(a n )), where x is the Cartesian product over sets. Characteristics of Relation Ordering of Tuples: Mathematically tuples in relation don t have ordering (although in reality they do have order when stored in files or disk) Ordering of attributes: Attributes within a tuple have to maintain order if they are stored as tuples. However a relation need not necessarily be maintained as a set of ordered tuples as long as a mapping between attribute and value is maintained in each relation instance. Values of Tuples: Each value that makes up a tuple is atomic and cannot be further subdivided. This is also called the first normal form assumption. 3

4 Relational Constraints Domain constraints: domain constraints specify that value of each attribute A must be atomic and a member of dom(a). Key constraints: Given a relational schema R = (A i ), i = 1 n, a subset of attributes K is called a superkey if the values of attributes in K is distinct for every relation instance. This is, for any two tuples t i, t j, (t i t j ) (t i [K] t j [K]). A key k of a relation is a subset of the superkey, k K,such that removal of any attribute in k removes the superkey property for k. They are also called Minimal superkeys. Relational Constraints In STUDENT table (Roll no, Name) is a superkey, while (Roll no) is a minimal key. A relation schema may have more than one key. Each key is called a candidate key. Usually one of the candidate key is chosen to uniquely identify tuples in a relation. such a key is called primary key for a relation. In STUDENT relation, Roll no is a good primary key. 4

5 Relational Constraints Entity Integrity Constraint: The primary key of a tuple may never be null. Referential Integrity Constraint: Informally, a tuple that refers to another tuple from another relation should refer to an existing tuple. In a given relation R 1 a set of attributes FK is said to be a foreign key of R 1 referencing another relation R 2, if the following rules hold: 1. The attribute in FK have the same domain as the primary key of R For every tuple in R 1, the attributes in its Foreign Key FK either reference a tuple in R 2 or is null. Relational Constraints Employee Emp_id Name Works_in Reports_to Department Dept_in Location Managed_by Num_members Foreign Keys are diagrammatically depicted as arrows. Note that foreign key can be from a relation to itself. 5

6 Relational Constraints Semantic Integrity Constraint: Constraints on the values of attributes. Example: The salary of the employee cannot be more than of his/her supervisor. The minimum age of an employee is 18 and maximum is 65. Semantic Integrity constraints are application specific and not part of the relational model, strictly speaking. Such constraints are specified and enforced using general purpose constraint specification languages. Basic Relational Algebra Operations Operations of relational model can be categorized into retrievals and updates. Updates can be either insert of new tuple, update( or modify) an existing tuple or delete an existing tuple. Retrieval is handled mostly by select (denoted by σ) and project (denoted by π) operations. 6

7 The Select Operation The SELECT operation is used to select (retrieve) a subset of the tuples from a relation that matches the selection condition. σ salary > 3000 (Employee) selects all tuples of relation Employee which matches the criteria Employee.Salary > 3000 The Select Operation The general syntax of SELECT is as follows: σ <Condition> (Relation) where condition is of the form : Condition (Expression) logicaloperator condition Expression Expression attributename operator constant attributename operator attributename Logicaloperator AND OR Operator < > = 7

8 Properties of SELECT operation The SELECT operation is unary. It operates on only one relation. Selection conditions are applied to each tuple individually in the relation. Hence the condition cannot be span more than one tuple. The degree of the relation resulting from σ c (R) is the same as that of R. σ c (R) R The SELECT operation is commutative: σ c1 (σ c2 (R)) σ c2 (σ c1 (R)) Hence, SELECT operation can be cascaded in any fashion. The Project Operation The project operation selects specific columns from the specified relation. π Name, Salary ( Employee) returns a relation comprising of only the Employee.Name and Employee.Salary attributes. The operation has said to have projected Employee relation onto the required relation. 8

9 The Project Operation The general form of PROJECT is as follows : π <attributeslist> (Relation) where attributelist is a comma-separated list of attributes belonging to the specified relation. The result of the PROJECT is in the same order as the specified list of attributes. Properties of Project operation The PROJECT operation removes duplicate in its results. The number of tuples returned by PROJECT is less than or equal to number of tuples in the specified relation. When attrubute list of PROJECT includes the superkey then the number of tuples returned is equal to the number of tuples in the specified relation. PROJECT is not commutative. π l1 π l2 (R) = π l1 (R) if l1 is a substring of l2. Else the operation is an incorrect expression. 9

10 Composition and Assignment Since the input and output of the relational operator is a single relation, they can be composed in any fashion with the output of one operation being the input of the other. π Name, Salary (σ Salary > 3000 (Employee ) ) returns Name and Salary field of all records of Employee, where salary is greater than Results of a query can also be assigned to a name to form a relation by that name. SalaryStatement π Name, Salary (σ Salary > 3000 ( Employee )) create a new relation SalaryStatement out of the specified query. Cartesian Join ( x) PROJECT and SELECT operators expect a single relation. When PROJECT and SELECT need to be run on a multiple relation, a single relation can be generated using the cartesian join ( x ) operator. TABLE : Student TABLE : Lab Roll No Name Lab Name Faculty Dept Arindam Ravi Hema Vasu Speech Lab Open system Lab Distributed computing lab Open system lab Speech Lab Open system Lab Distributed computing lab Prof. Fernandes Prof. Baradwaj Prof. Ramanan EE Consider the Relational Query : σ Student.Lab = Lab.Name (Student x Lab) 10

11 Cartesian Join (x) Student. Roll No Student. Name Arindam Arindam Arindam Student.Lab Speech.Lab Speech.Lab Speech.Lab Lab.Name Speech Lab Open System Lab Distributed computed lab Lab.Faculty Prof.Fernandes Prof. Baradwaj Prof. Ramanan Lab. Dept EE Ravi Ravi Ravi Open system Lab Open system Lab Open system Lab Speech Lab Open system Lab Distributed Computing Lab Prof.Fernandes Prof.Baradwaj Prof.Ramanan EE Hema Hema Hema Distributed Computing Lab Distributed computing Lab Distributed computing lab Speech Lab Open system Lab Distributed computing Lab Prof.Fernandes Prof.Baradwaj Prof.Ramanan EE Vasu Vasu Vasu Open system Lab Open system Lab Open system Lab Speech Lab Open system Lab Distributed Computing Lab Prof.Fernandes Prof.Baradwaj Prof.Ramanan EE Columns matching the query are shown in PINK. Query Result Student. Roll No Student. Name Student.Lab Lab.Name Lab.Faculty Lab. Dept Arindam Speech.Lab Speech.Lab Prof.Fernandes EE Ravi Open system Lab Open system Lab Prof.Baradwaj Hema Distributed computing lab Distributed computing lab Prof.Ramanan Vasu Open system Lab Open system Lab Prof.Baradwaj 11

12 Properties of Cartesian Join ( x ) The Cartesian Join represents a Canonical Join of two or more relations. If relation R having R tuples and relation S having S tuples are joined using x, then R x S = R S Cartesian join is too inefficient for most queries involving multiple relations. Relational Algebra (contd) Join operators and additional relational operators 12

13 Relational Algebra Summary Based on the concept of mathematical relation Building block: a relation comprising of attributes within domains Tuples + Schema = Relation Properties: Ordering, Duplicates Constraints: Key constraints, entity constraints, referential integrity constraints Basic retrieval operators: Select, Project Composability of relational operators and Cartesian join Theta Join Operator ( θ ) Theta join is used to combine related tuples from two or more relations ( specified by the condition theta ), to form a single tuple. Consider the earlier relation Student and Lab. The relation : Student Student.Lab = Lab.Name Lab returns the same result as : σ Student.Lab = Lab.Name (Student x Lab) However, the number of tuples generated by Join is 4 rather than 12 as in the Cartesian join. Note that Student.lab is a foreign key into Lab (and Lab.Name is a primary key of Lab) and referential Integrity has to be maintained to perform this join. 13

14 Theta Join Operator The general form of theta join is as follows : R join condition S where R and S are relation and join condition is a logical expression over attributes of R and S. Tuples whose join attributes are null do not appear in the final result. Theta Join Operator Theta join where the only comparison operator used is the equals (=) sign, are called equijoins. A natural join denoted by * is an equijoin,where the attribute names at both ends of the equality sign are the same. In this case, the attribute is not repeated in the final result. Consider the Student and the Lab example. Let the Student relation schema be modified as Student(RollNo,Name,Labname) and Lab relation schema be modified as Lab(LabName,Faculty,Dept). The natural join Student * Lab has a scheme (Roll No, Name, LabName, Faculty, Dept) 14

15 Renaming TA(id,LabName) π Roll No,Lab (Student) The above relation projects RollNo and Lab attributes from Student and renames them as id and LabName respectively in the output relation. It also names the output relation as TA. The above can also be achieved by rename (ρ) operator as : ρ TA(id,LabName) (π Roll No,Lab ( Student )) Renaming The general form of the rename operator is as follows : ρ S(B1, B2,.Bn) Or (R) ρ S (R) Or ρ (B1, B2,.Bn) ( R) The first form renames both relation name and attribute names, the second form renames only the relation name and the third form renames only attribute names. 15

16 Set Theoretic Operations Set theoretic operation like UNION ( U ), INTERSECTION ( ) and SET DIFFERENCE can be applied to compatible relations. Compatibility : Two relations R ( A 1,A 2 A n ) and S ( B 1,B 2,.B n ) are compatible if: they have the same number of attributes and for all The Union operator R U S returns the set of all tuples that are present in either R or S or both. Duplicate tuples will be eliminated. The Intersection operator R S returns the set of all tuples are present in both R and S. The Set Difference operator R S returns the set of all tuples in R but not in S. Note that, but The Division Operator The Division operator ( ) is used to denote the conditions where a given relation R is to be split based on whether its association with every tuple in an other relation S. Let the set of attributes of R be Z and set of attributes of S be X. Let Y = Z X, that is the set of all attributes of R that are not in S. The result of the operation is as follows. For every tuple, R S A a1 a2 a2 a1 a3 a1 a2 B b1 b1 b1 b2 b2 b3 b3 A a1 a2 B b1 b3 16

17 Other Relational Operators Outer Join : A normal join operation requires referential integrity. Every attribute in R that refers S should refer to an existing tuple in S. If either the referencing attribute of S is null, the tuple is not returned. Outer join is used when all tuples are required even when referential integrity is not met. Left outer join includes tuples even when the referencing attribute is null and Right outer join includes tuples even when the referenced attribute is null or referenced tuple does not exist. An outer join which is both left and right is called a Full outer join. Outer Union : Outer Union computes the union of partially compatible relations. Two relations R(X) and S(Z) are said to be partially compatible if there exists W X and Y Z such that for every A i W and B i Y, dom(a i ) = dom(b i ). Outer Union of R and S creates a new relation that includes all attributes from both relation. Null values fill up all undefined attributes. The Complete Set of the Operators The following set of relational operators : {,, U,--, x } is called the complete set of the relational operators because all other operators can be mathematically expressed as a sequence of the above operators. For Example : Similarly : 17

18 Relational Operators on Bags A bag or a multi-set is a set that may have multiple occurrences of a given element Bags are necessary when aggregation (sum, average, ) has to be performed over query results Tolerating bags while answering queries reduces unnecessary overheads Set-theoretic operations on bags Consider two bags: R where tuple t occurs n times and S, where tuple t occurs m times. The union of bags or disjoint union of sets denoted by R + S is a bag that contains R + S tuples; or m+n occurrences of tuple t. In the intersection of bags R S, tuple t occurs min(n,m) times. In the set difference of bags R S, tuple t occurs max(0, n-m) times. 18

19 Other operators on Bags The select operator on bags is applied to each tuple in the bag independently and duplicates are not removed from the result The project operator on bags is applied to each tuple independently and duplicates are not removed from the result. Faster than project operator on sets. Cartesian product on bags: If tuple t occurs m times in relation R and tuple s occurs n times in relation S, then tuple ts occurs mn times in RxS. Algebraic expressions on bags The following algebraic expressions are valid when R and S are sets, but invalid when they are bags: (R S) T = (R T) (S T) [1-1-1 principle] R (S T) = (R S) (R T) [2-1-2 principle] σ C OR D (R) = σ C (R) σ D (R) 19

20 Relational Operators Example Consider the following schema: Employee (FNAME,MINIT,LNAME,PAN,DOB,ADDR GENDER,SALARY,SUPERVISOR,DNO) Department (DNAME, DNUMBER, MGRPAN, MGRSTARTDATE) Dept_Locations (DNUMBER, DLOCATION) Project (PNAME, PNUMBER, PLOCATION, DNUM) Works_on (EPAN, PNO, HOURS) Relational Operators Example Query 1: Retrieve the name and address of all employees who work for Research department: Research_dept σ DNAME= Research (Department) Research_emps (Research_dept DNUMBER=DNO EMPLOYEE) Result π FNAME,LNAME,ADDRESS (Research_emps) 20

21 Relational Operators Example Query 2: Find the names of employees who work on all projects controlled by department no. 5: Dpt5_proj π PNUMBER (σ DNUM=5 (Project)) Emp_Prj(PAN,PNUMBER) π EPAN,PNO (Works_on) Result_Pans Emp_Prj Dpt5_proj Result π FNAME,LNAME (Result_Pans * Employee) Relational Operators Example Query 3: List the names of all employees who are not working on any projects: All_Pans π PAN (Employee) Prj_Pans π EPAN (Works_on) NoPrj_Pans All_Pans Prj_Pans Result π FNAME,LNAME (NoPrj_Pans * Employee) 21

22 Summary Definition of relation schema, relation, domain, attributes. Difference between the mathematical concept of relation and their implementation as tables or files. Constraints in relational model : Domain Constraints, Key constraints (Super key, Candidate key, Primary key definition), Entity Integrity constraint, Referential integrity constraint. Basic relational algebra operation :, Relational Algebra operators : outer join, outer union. The complete set of relational operators. 22