DB_Lec09.pdf Introduction to Database System PDF

CSAI 202  Outline Unary Relational Operations Relational Algebra Operations From Set Theory Binary Relational Operations Additional Relational Operations Examples of Queries in Relational Algebra Query Optimization: Using Heuristics in Query Optimization...

CSAI 202  Outline Unary Relational Operations Relational Algebra Operations From Set Theory Binary Relational Operations Additional Relational Operations Examples of Queries in Relational Algebra Query Optimization: Using Heuristics in Query Optimization 12/18/2023 Introduction to Database Systems 2 Relational Algebra Overview Relational algebra is the basic set of operations for the relational model These operations enable a user to specify basic retrieval requests (or queries) The result of an operation is a new relation, which may have been formed from one or more input relations 12/18/2023 Introduction to Database Systems 3 Relational Algebra Overview The algebra operations thus produce new relations These can be further manipulated using operations of the same algebra A sequence of relational algebra operations forms a relational algebra expression The result of a relational algebra expression is also a relation that represents the result of a database query (or retrieval request) 12/18/2023 Introduction to Database Systems 4 Relational Algebra operations Unary Relational Operations SELECT (symbol:  (sigma)) PROJECT (symbol:  (pi)) RENAME (symbol:  (rho)) Relational Algebra Operations From Set Theory UNION (  ), INTERSECTION (  ), DIFFERENCE (or MINUS, – ) CARTESIAN PRODUCT ( × ) Binary Relational Operations JOIN (several variations of JOIN exist) DIVISION Additional Relational Operations OUTER JOINS, OUTER UNION AGGREGATE FUNCTIONS (These compute summary of information: for example, SUM, COUNT, AVG, MIN, MAX) 12/18/2023 Introduction to Database Systems 5 Database State for COMPANY 12/18/2023 Introduction to Database Systems 6 Unary Relational Operations: SELECT The SELECT operation (denoted by  (sigma)) is used to select a subset of the tuples from a relation based on a selection condition. The selection condition acts as a filter Keeps only those tuples that satisfy the qualifying condition Tuples satisfying the condition are selected whereas the other tuples are discarded (filtered out) Examples: Select the EMPLOYEE tuples whose department number is 4: ▪ DNO = 4 (EMPLOYEE) Select the employee tuples whose salary is greater than $30,000: ▪ SALARY > 30,000 (EMPLOYEE) 12/18/2023 Introduction to Database Systems 7 Unary Relational Operations: SELECT In general, the select operation is denoted by   (R) where the symbol  (sigma) is used to denote the select operator the selection condition is a Boolean (conditional) expression specified on the attributes of relation R tuples that make the condition true are selected ▪ appear in the result of the operation tuples that make the condition false are filtered out ▪ discarded from the result of the operation 12/18/2023 Introduction to Database Systems 8  SELECT Operation Properties The SELECT operation (R) produces a relation S that has the same schema (same attributes) as R SELECT is commutative: ((R)) = ((R)) Because of commutative property, a cascade (sequence) of SELECT operations may be applied in any order: (((R)) =  (((R))) A cascade of SELECT operations may be replaced by a single selection with a conjunction of all the conditions: (((R)) =  AND AND (R))) The number of tuples in the result of a SELECT is less than (or equal to) the number of tuples in the input relation R 12/18/2023 Introduction to Database Systems 9  Unary Relational Operations: PROJECT PROJECT Operation is denoted by  (pi) This operation keeps certain columns (attributes) from a relation and discards the other columns. PROJECT creates a vertical partitioning ▪ The list of specified columns (attributes) is kept in each tuple ▪ The other attributes in each tuple are discarded Example: To list each employee’s first and last name and salary, the following is used:  LNAME, FNAME,SALARY(EMPLOYEE) 12/18/2023 Introduction to Database Systems 10 Unary Relational Operations: PROJECT The general form of the project operation is: (R)   (pi) is the symbol used to represent the project operation  is the desired list of attributes from relation R. The project operation removes any duplicate tuples This is because the result of the project operation must be a set of tuples ▪ Mathematical sets do not allow duplicate elements. 12/18/2023 Introduction to Database Systems 11 PROJECT Operation Properties The number of tuples in the result of projection (R) is always less or equal to the number of tuples in R If the list of attributes includes a key of R, then the number of tuples in the result of PROJECT is equal to the number of tuples in R PROJECT is not commutative ((R) ) = (R) as long as contains the attributes in 12/18/2023 Introduction to Database Systems 12  Examples 12/18/2023 Introduction to Database Systems 14  Relational Algebra Expressions We may want to apply several relational algebra operations one after the other Either we can ▪ write the operations as a single relational algebra expression by nesting the operations, Or ▪ apply one operation at a time and create intermediate result relations. In the latter case, we must give names to the relations that hold the intermediate results. 12/18/2023 Introduction to Database Systems 15  Single expression versus sequence of relational operations (Example) To retrieve the first name, last name, and salary of all employees who work in department number 5, we must apply a select and a project operation We can write a single relational algebra expression as follows: FNAME, LNAME, SALARY(DNO=5(EMPLOYEE)) OR We can explicitly show the sequence of operations, giving a name to each intermediate relation: DEP5_EMPS  DNO=5(EMPLOYEE) RESULT  FNAME, LNAME, SALARY(DEP5_EMPS) 12/18/2023 Introduction to Database Systems 16  Unary Relational Operations: RENAME The RENAME operator is denoted by  (rho) In some cases, we may want to rename the attributes of a relation or the relation name or both Useful when a query requires multiple operations Necessary in some cases (see JOIN operation later) 12/18/2023 Introduction to Database Systems 17  Unary Relational Operations: RENAME The general RENAME operation can be expressed by any of the following forms: S(B1, B2, …, Bn)(R) ▪ changes both the relation name to S, and the column (attribute) names to B1, B1, …Bn S(R) ▪ changes the relation name only to S (B1, B2, …, Bn)(R) ▪ changes the column (attribute) names only to B1, B1, …Bn 12/18/2023 Introduction to Database Systems 18  Unary Relational Operations: RENAME For convenience, we also use a shorthand for renaming attributes in an intermediate relation: If we write: ▪ RESULT  FNAME, LNAME, SALARY(DEP5_EMPS) ▪ RESULT will have the same attribute names as DEP5_EMPS (same attributes as EMPLOYEE) If we write: ▪ RESULT(F,M, L, S, B, A, SX, SAL, SU, DNO)  RESULT (F,M,L,S,B,A,SX,SAL,SU, DNO)(DEP5_EMPS) ▪ The 10 attributes of DEP5_EMPS are renamed to F, M, L, S, B, A, SX, SAL, SU, DNO, respectively ▪ Note: the  symbol is an assignment operator 12/18/2023 Introduction to Database Systems 19  Examples 12/18/2023 Introduction to Database Systems 21  Relational Algebra Operations from Set Theory: UNION UNION Operation Binary operation, denoted by  The result of R  S, is a relation that includes all tuples that are either in R or in S or in both R and S Duplicate tuples are eliminated The two operand relations R and S must be “type compatible” (or UNION compatible) ▪ R and S must have same number of attributes ▪ Each pair of corresponding attributes must be type compatible (have same or compatible domains) 12/18/2023 Introduction to Database Systems 22  UNION: Example To retrieve the social security numbers of all employees who either work in department 5 or directly supervise an employee who works in department 5 We can use the UNION operation as follows: DEP5_EMPS  DNO=5(EMPLOYEE) RESULT1  SSN(DEP5_EMPS) RESULT2(SSN)  SUPERSSN(DEP5_EMPS) RESULT  RESULT1  RESULT2 The union operation produces the tuples that are in either RESULT1 or RESULT2 or both 12/18/2023 Introduction to Database Systems 23  UNION: Example 12/18/2023 Introduction to Database Systems 24  Type Compatibility Type Compatibility of operands is required for the binary set operation UNION  , (also for INTERSECTION  , and SET DIFFERENCE –) R1(A1, A2,..., An) and R2(B1, B2,..., Bn) are type compatible if: they have the same number of attributes, and the domains of corresponding attributes are type compatible (i.e. dom(Ai)=dom(Bi) for i=1, 2,..., n). The resulting relation for (R1R2, R1R2, or R1–R2) has the same attribute names as the first operand relation R1 (by convention) 12/18/2023 Introduction to Database Systems 25  Relational Algebra Operations from Set Theory: INTERSECTION INTERSECTION is denoted by  The result of the operation R  S, is a relation that includes all tuples that are in both R and S The attribute names in the result will be the same as the attribute names in R The two operand relations R and S must be “type compatible” 12/18/2023 Introduction to Database Systems 26  Relational Algebra Operations from Set Theory: SET DIFFERENCE SET DIFFERENCE (also called MINUS or EXCEPT) is denoted by – The result of R – S, is a relation that includes all tuples that are in R but not in S The attribute names in the result will be the same as the attribute names in R The two operand relations R and S must be “type compatible” 12/18/2023 Introduction to Database Systems 27  Example 12/18/2023 Introduction to Database Systems 29  Some properties of UNION, INTERSECT, and DIFFERENCE Notice that both union and intersection are commutative operations; that is R  S = S  R, and R  S = S  R Both union and intersection can be treated as n-ary operations applicable to any number of relations as both are associative operations; that is R  (S  T) = (R  S)  T (R  S)  T = R  (S  T) The minus operation is not commutative; that is, in general R – S ≠ S – R 12/18/2023 Introduction to Database Systems 30  Relational Algebra Operations from Set Theory: CARTESIAN PRODUCT CARTESIAN (or CROSS) PRODUCT Operation This operation is used to combine tuples from two relations in a combinatorial fashion. Denoted by R(A1, A2,..., An) × S(B1, B2,..., Bm) Result is a relation Q with degree n + m attributes: ▪ Q(A1, A2,..., An, B1, B2,..., Bm), in that order. The resulting relation state has one tuple for each combination of tuples—one from R and one from S. Hence, if R has nR tuples (denoted as |R| = nR ), and S has nS tuples, then R × S will have nR × nS tuples. The two operands do NOT have to be "type compatible” 12/18/2023 Introduction to Database Systems 31  Relational Algebra Operations from Set Theory: CARTESIAN PRODUCT Generally, CROSS PRODUCT is not a meaningful operation Can become meaningful when followed by other operations Example (not meaningful): FEMALE_EMPS  SEX=’F’(EMPLOYEE) EMPNAMES  FNAME, LNAME, SSN (FEMALE_EMPS) EMP_DEPENDENTS  EMPNAMES × DEPENDENT EMP_DEPENDENTS will contain every combination of EMPNAMES and DEPENDENT whether or not they are actually related 12/18/2023 Introduction to Database Systems 32  Relational Algebra Operations from Set Theory: CARTESIAN PRODUCT To keep only combinations where the DEPENDENT is related to the EMPLOYEE, we add a SELECT operation as follows Example (meaningful): FEMALE_EMPS  SEX=’F’(EMPLOYEE) EMPNAMES  FNAME, LNAME, SSN(FEMALE_EMPS) EMP_DEPENDENTS  EMPNAMES × DEPENDENT ACTUAL_DEPS  SSN=ESSN(EMP_DEPENDENTS) RESULT  FNAME, LNAME, DEPENDENT_NAME(ACTUAL_DEPS) RESULT will now contain the name of female employees and their dependents 12/18/2023 Introduction to Database Systems 33  Example of applying CARTESIAN PRODUCT 12/18/2023 34  Binary Relational Operations: JOIN JOIN Operation (denoted by ⨝ ) The sequence of CARTESIAN PRODECT followed by SELECT is used quite commonly to identify and select related tuples from two relations A special operation, called JOIN combines this sequence into a single operation This operation is very important for any relational database with more than a single relation, because it allows us combine related tuples from various relations The general form of a join operation on two relations R(A1, A2,..., An) and S(B1, B2,..., Bm) is: R⨝S where R and S can be any relations that result from general relational algebra expressions. 12/18/2023 Introduction to Database Systems 35  Binary Relational Operations: JOIN Example: Suppose that we want to retrieve the name of the manager of each department. To get the manager’s name, we need to combine each DEPARTMENT tuple with the EMPLOYEE tuple whose SSN value matches the MGRSSN value in the department tuple. We do this by using the join ⨝ operation. DEPT_MGR  DEPARTMENT ⨝ MGRSSN=SSN EMPLOYEE MGRSSN=SSN is the join condition Combines each department record with the employee who manages the department The join condition can also be specified as DEPARTMENT.MGRSSN= EMPLOYEE.SSN 12/18/2023 Introduction to Database Systems 36 12/18/2023 Introduction to Database Systems 37 Applying the JOIN operation DEPT_MGR  DEPARTMENT ⨝MGRSSN=SSNEMPLOYEE 12/18/2023 Introduction to Database Systems 38  Some properties of JOIN Consider the following JOIN operation: R(A1, A2,..., An) ⨝R.Ai=S.BjS(B1, B2,..., Bm) Result is a relation Q with degree n + m attributes: ▪ Q(A1, A2,..., An, B1, B2,..., Bm), in that order. The resulting relation state has one tuple for each combination of tuples—r from R and s from S, but only if they satisfy the join condition r[Ai]=s[Bj] Hence, if R has nR tuples, and S has nS tuples, then the join result will generally have less than nR * nS tuples. Only related tuples (based on the join condition) will appear in the result 12/18/2023 Introduction to Database Systems 39  Some properties of JOIN The general case of JOIN operation is called a Theta-join: R ⨝theta S The join condition is called theta Theta can be any general Boolean expression on the attributes of R and S; for example: R.Ai