Fundamentals of Database Systems PDF

Fundamentals of Database Systems Seventh Edition Chapter 8 The Relational Algebra and The Relational Calculus...

Fundamentals of Database Systems Seventh Edition Chapter 8 The Relational Algebra and The Relational Calculus (plus QBE- Appendix C) Slides in this presentation contain hyperlinks. JAWS users should be able to get a list of links by using INSERT+F7 Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Chapter Outline 8.1 Relational Algebra – Unary Relational Operations – Relational Algebra Operations From Set Theory – Binary Relational Operations – Additional Relational Operations – Examples of Queries in Relational Algebra 8.2 Relational Calculus – Tuple Relational Calculus – Domain Relational Calculus 8.3 Example Database Application (COMPANY) 8.4 Overview of the QBE language (appendix D). Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Relational Algebra Overview (1 of 4) 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 – This property makes the algebra “closed” (all objects in relational algebra are relations) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Relational Algebra Overview (2 of 4) 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) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Brief History of Origins of Algebra Muhammad ibn Musa al-Khwarizmi (800-847 CE) – from Morocco wrote a book titled al-jabr about arithmetic of variables – Book was translated into Latin. – Its title (al-jabr) gave Algebra its name. Al-Khwarizmi called variables “shay” – “Shay” is Arabic for “thing”. – Spanish transliterated “shay” as “xay” (“x” was “sh” in Spain). – In time this word was abbreviated as x. Where does the word Algorithm come from? – Algorithm originates from “al-Khwarizmi" – Reference: PBS (http://www.pbs.org/empires/islam/innoalgebra.html) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Relational Algebra Overview (3 of 4) Relational Algebra consists of several groups of operations – Unary Relational Operations ▪ SELECT (symbol (  sigma )) ▪ PROJECT (symbol : p ( pi )) ▪ RENAME (symbol : p ( rho )) – Relational Algebra Operations From Set Theory ▪ UNION (È ),INTERSECITON(Ç), DIFFERENCE (or MINUS, − ) ▪ CARTESIAN PRODUCT ( x ) – Binary Relational Operations ▪ JOIN (several variations of JOIN exist) ▪ DIVISIONs Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Relational Algebra Overview (4 of 4) – Additional Relational Operations ▪ OUTER JOINS, OUTER UNION ▪ AGGREGATE FUNCTIONS (These compute summary of information: for example, SUM, COUNT, AVG, MIN, MAX) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Database State for COMPANY All examples discussed below refer to the Company database shown here. Figure 5.7 Referential integrity constraints displayed on the COMPANY relational database schema. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Unary Relational Operations: SELECT (1 of 4) The SELECT operation (denoted by σ (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: σDNO= 4 (EMPLOYEE ) – Select the employee tuples whose salary is greater than $30,000: σSalary>30,000 (EMPLOYEE ) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Unary Relational Operations: SELECT (2 of 4) In general, the select operation is denoted by σ (R ) where – The symbol σ (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 Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Unary Relational Operations: SELECT (3 of 4) SELECT Operation Properties – The SELECT operation σ (R ) produces a relation S that has the same schema (same attributes) as R – SELECT σ is commutative: ▪ σ ( σ (R ) ) = σ ( σ (R ) ) – Because of commutativity property, a cascade (sequence) of SELECT operations may be applied in any order: ▪ σ ( σ ( σ (R ) ) ) = σ ( σ ( σ (R ) ) ) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Unary Relational Operations: SELECT (4 of 4) – A cascade of SELECT operations may be replaced by a single selection with a conjunction of all the conditions: ▪ σ ( σ ( σ (R ) ) = σ ANDAND (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 Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved The Following Query Results Refer to This Database State Figure 5.6 One possible database state for the COMPANY relational database schema. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Unary Relational Operations: Project (1 of 3) PROJECT Operation is denoted by π (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 ) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Unary Relational Operations: Project (2 of 3) The general form of the project operation is:  ( R ) – π (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. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Unary Relational Operations: Project (3 of 3) 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 Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Examples of Applying Select and Project Operations Figure 8.1 Results of SELECT and PROJECT operations. (a) σ (Dno=4 AND Salary>25000) OR (Dno=5 AND Salary>30000) (EMPLOYEE ). (b) Lname,Fname, Salary (EMPLOYEE ). (c) Sex, Salary (EMPLOYEE ). Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved 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 – We can 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. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved 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 ( σDNO=5 (EMPLOYEE) ) OR We can explicitly show the sequence of operations, giving a name to each intermediate relation: – DEP5_EMPS  σDNO=5 (EMPLOYEE) – RESULT  FNAME, LNAME,SALARY (DEP5_EMPS ) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Unary Relational Operations: RENAME (1 of 3) 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) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Unary Relational Operations: RENAME (2 of 3) 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 – S ( R ) changes: ▪ the relation name only to S –  (B1, B2, …,Bn ) ( R ) changes: ▪ the column (attribute) names only to Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Unary Relational Operations: RENAME (3 of 3) 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 Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Example of Applying Multiple Operations and RENAME Figure 8.2 Results of a sequence of operations. (a) Fname,Lname, Salary (σDno=5 (EMPLOYEE ) ). (b) Using intermediate relations and renaming of attributes Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Relational Algebra Operations from Set Theory: UNION (1 of 2) 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) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Relational Algebra Operations from Set Theory: Union (2 of 2) Example: – To retrieve the social security numbers of all employees who either work in department 5 (RESULT1 below) or directly supervise an employee who works in department 5 (RESULT2 below) – We can use the UNION operation as follows: DEP5_EMPS  σDNO=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 Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Figure 8.3 Result of the Union Operation RESULT ← RESULT1  RESULT2 union symbol Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Relational Algebra Operations from Set Theory Type Compatibility of operands is required for the binary set operation UNION , (also for INTERSECTION , and SET DIFFERENCE −, see next slides) are type compatible if: – they have the same number of attributes, and – the domains of corresponding attributes are type compatible The resulting relation for R1  R2 (also for R1  R2, or R1 − R2, see next slides) has the same attribute names as the first operand relation R1 (by convention) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved 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” Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved 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” Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Example to Illustrate the Result of UNION, INTERSECT, and DIFFERENCE Figure 8.4 The set operations UNION, INTERSECTION, and MINUS. (a) Two union-compatible relations. (b) STUDENT  INSTRUCTOR. (c) STUDENT  INSTRUCTOR. (d) STUDENT – INSTRUCTOR. (e) INSTRUCTOR – STUDENT. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved 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 Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Relational Algebra Operations from Set Theory: CARTESIAN PRODUCT (1 of 3) CARTESIAN (or CROSS) PRODUCT Operation – This operation is used to combine tuples from two relations in a combinatorial fashion. – Denoted by – Result is a relation Q with degree n + m attributes: ▪ – 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” Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Relational Algebra Operations from Set Theory: CARTESIAN PRODUCT (2 of 3) Generally, CROSS PRODUCT is not a meaningful operation – Can become meaningful when followed by other operations Example (not meaningful): – FEMALE_EMPS  σ SEX=’F’ (EMPLOYEE ) – EMPNAMES  FNAME, LNAME, SSN (FEMALE_EMPS ) – EMP_DEPENDENTS  EMPNAMES x DEPENDENT EMP_DEPENDENTS will contain every combination of EMPNAMES and DEPENDENT – whether or not they are actually related Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Relational Algebra Operations from Set Theory: CARTESIAN PRODUCT (3 of 3) To keep only combinations where the DEPENDENT is related to the EMPLOYEE, we add a SELECT operation as follows Example (meaningful): – FEMALE_EMPS  σ SEX=’F’ (EMPLOYEE ) – EMPNAMES  FNAME, LNAME, SSN (FEMALE_EMPS ) – EMP_DEPENDENTS  EMPNAMES x DEPENDENT – ACTUAL_DEPS  σ SSN=ESSN (EMP_DEPENDENTS ) – RESULT  FNAME, LNAME, DEPENDENT_NAME ( ACTUAL_DEPS ) RESULT will now contain the name of female employees and their dependents Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Figure 8.5 The Cartesian Product (CROSS PRODUCT) Operation (1 of 3) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Figure 8.5 The Cartesian Product (CROSS PRODUCT) Operation (2 of 3) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Figure 8.5 The Cartesian Product (CROSS PRODUCT) Operation (3 of 3) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Binary Relational Operations: JOIN (1 of 2) JOIN Operation – 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 – where R and S can be any relations that result from general relational algebra expressions. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Binary Relational Operations: JOIN (2 of 2) 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 Mgr_ssn=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 Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Figure 8.6 Result of the JOIN Operation D E P T dash M G R left arrow DEPARTMENT super absolute value of x sub start expression m g r dash s s n = S s n end expression EMPLOYEE X ← DEPT_MGR DEPARTMENT Mgr_ssn=Ssn EMPLOYEE Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Some Properties of JOIN (1 of 2) Consider the following JOIN operation: – R(A1, A 2,..., A n ) R.A i =S.B j 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—r from R and s from S, but only if they satisfy the join condition – 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 Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Some Properties of JOIN (2 of 2) 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: – Most join conditions involve one or more equality conditions “AND”ed together; for example: – Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Binary Relational Operations: EQUIJOIN EQUIJOIN Operation The most common use of join involves join conditions with equality comparisons only Such a join, where the only comparison operator used is =, is called an EQUIJOIN. – In the result of an EQUIJOIN we always have one or more pairs of attributes (whose names need not be identical) that have identical values in every tuple. – The JOIN seen in the previous example was an EQUIJOIN. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Binary Relational Operations: NATURAL JOIN Operation (1 of 2) NATURAL JOIN Operation – Another variation of JOIN called NATURAL JOIN — denoted by * — was created to get rid of the second (superfluous) attribute in an EQUIJOIN condition. ▪ because one of each pair of attributes with identical values is superfluous – The standard definition of natural join requires that the two join attributes, or each pair of corresponding join attributes, have the same name in both relations – If this is not the case, a renaming operation is applied first. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Binary Relational Operations: NATURAL JOIN Operation (2 of 2) Example: To apply a natural join on the DNUMBER attributes of DEPARTMENT and DEPT_LOCATIONS, it is sufficient to write: – DEPT_LOCS  DEPARTMENT * DEPT_LOCATIONS Only attribute with the same name is DNUMBER An implicit join condition is created based on this attribute: DEPARTMENT.DNUMBER=DEPT_LOCATIONS.DNUMBER Another example: Q  R ( A,B,C,D ) * S ( C,D,E ) – The implicit join condition includes each pair of attributes with the same name, “AND”ed together: ▪ R.C=S.C AND R.D.S.D – Result keeps only one attribute of each such pair: ▪ Q(A,B,C,D,E) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Example of NATURAL JOIN Operation Figure 8.7 Results of two natural join operations. (a) proj_dept  project * dept. (b) dept_locs  department *dept_locations. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Complete Set of Relational Operations The set of operations including SELECT σ, PROJECT π , UNION∪, DIFFERENCE −, RENAME  , and CARTESIAN PRODUCT X is called a complete set because any other relational algebra expression can be expressed by a combination of these five operations. For example: – R  S = (R  S ) – ((R − S )  (S − R )) – R S = σ ( R  S ) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Binary Relational Operations: DIVISION DIVISION Operation – The division operation is applied to two relations – R(Z) ÷ S(X), where X subset Z. Let Y = Z − X (and hence Z = X ∪Y); that is, let Y be the set of attributes of R that are not attributes of S. – The result of DIVISION is a relation T(Y) that includes a tuple t if tuples tR appear in R with tR Y  = t, and with ▪ tR  X  = tS for every tuple tS in S. – For a tuple t to appear in the result T of the DIVISION, the values in t must appear in R in combination with every tuple in S. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Example of DIVISION Figure 8.8 The DIVISION operation. (a) Dividing SSN_PNOS by SMITH_PNOS. (b) T ← R ÷ S. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Table 8.1 Operations of Relational Algebra (1 of 2) Operation Purpose Notation (R ) sigma sub start expression less than sign selection SELECT Selects all tuples that satisfy the selection condition from a σ condition greater than sign left parenthesis R right parenthesis relation R. (R ) pi sub start expression less than sign attribute list PROJECT Produces a new relation with only some of the attributes of R,  greater than sign end expression left parenthesis R right parenthesis and removes duplicate tuples. R sub 1 join start expression less than sign join THETA JOIN Produces all combinations of R1 R2 condition greater than sign end expression R sub 2 tuples from R1 and R2 that satisfy the join condition. R sub 1 join start expression less than sign join EQUIJOIN Produces all the combinations R condition 1 greater R , OR than sign end 2 expression R sub 2 OR of tuples from R1 and R2 that R sub 1 join sub start expression left parenthesis less R than sign join attributes 1 greater than sign right 1 (), ( ) 2 satisfy a join condition with only parenthesis comma left parenthesis less than sign join attributes 2 greater than sign right parenthesis end equality comparisons. expression R sub 2 R sub 1 asterisk start expression less than sign join NATURAL JOIN Same as EQUIJOIN except that R* condition greater 1 2 expression R sub 2 the join attributes of R2 are not comma OR R sub 1 asterisk sub start expression left R* parenthesis less that sign join attributes 1 greater than 1 ( ), ( ) included in the resulting relation; sign right parenthesis comma left parenthesis less than sign join attributes 2 greater than sign right parenthesis if the join attributes have the R OR R * R comma 2 R sub 2 OR 1 R sub2 1 asterisk R sub 2. same names, they do not have to be specified at all. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Table 8.1 Operations of Relational Algebra (2 of 2) Operation Purpose Notation UNION Produces a relation that includes all R sub 1 union R sub 2 the tuples in R1 or R2 or both R1 and R1  R2 R2; R1 and R2 must be union compatible. INTERSECTION Produces a relation that includes all R1  R2 R sub 1 inter section R sub 2 the tuples in both R1 and R2; R1 and R2 must be union compatible. DIFFERENCE Produces a relation that includes all R1 − R2 R sub 1 minus R sub 2 the tuples in R1 that are not in R2; R1 and R2 must be union compatible. CARTESIAN PRODUCT Produces a relation that has the R1  R2 R sub 1 times R sub 2 attributes of R1 and R2 and includes as tuples all possible combinations of tuples from R1 and R2. ( ) ( ) DIVISION Produces a relation R(X) that includes R sub1 left parenthesis Z right parenthesis all tuples t[X] in R1(Z) that appear in R1 RZsub2 divided R Y left2 parenthesis Y right parenthesis R1 in combination with every tuple from R2(Y), where Z = X ∪ Y. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Query Tree Notation Query Tree – An internal data structure to represent a query – Standard technique for estimating the work involved in executing the query, the generation of intermediate results, and the optimization of execution – Nodes stand for operations like selection, projection, join, renaming, division, …. – Leaf nodes represent base relations – A tree gives a good visual feel of the complexity of the query and the operations involved – Algebraic Query Optimization consists of rewriting the query or modifying the query tree into an equivalent tree. (see Chapter 15) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Example of Query Tree Figure 8.9 Query tree corresponding to the relational algebra expression for Q2. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Additional Relational Operations: Aggregate Functions and Grouping A type of request that cannot be expressed in the basic relational algebra is to specify mathematical aggregate functions on collections of values from the database. Examples of such functions include retrieving the average or total salary of all employees or the total number of employee tuples. – These functions are used in simple statistical queries that summarize information from the database tuples. Common functions applied to collections of numeric values include – SUM, AVERAGE, MAXIMUM, and MINIMUM. The COUNT function is used for counting tuples or values. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Aggregate Function Operation Use of the Aggregate Functional operation Ƒ – F MAX Salary (EMPLOYEE ) retrieves the maximum salary value from the EMPLOYEE relation – F MIN Salary (EMPLOYEE ) retrieves the minimum Salary value from the EMPLOYEE relation – F SUM Salary (EMPLOYEE ) retrieves the sum of the Salary from the EMPLOYEE relation – F COUNT Ssn,AVERAGE Salary (EMPLOYEE ) computes the count (number) of employees and their average salary ▪ Note: count just counts the number of rows, without removing duplicates Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Using Grouping With Aggregation The previous examples all summarized one or more attributes for a set of tuples – Maximum Salary or Count (number of) Ssn Grouping can be combined with Aggregate Functions Example: For each department, retrieve the DNO, COUNT SSN, and AVERAGE SALARY A variation of aggregate operation Ƒ allows this: – Grouping attribute placed to left of symbol – Aggregate functions to right of symbol – Dno F COUNT Ssn, AVERAGE Salary ( EMPLOYEE ) Above operation groups employees by DNO (department number) and computes the count of employees and average salary per department Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Figure 8.10 the Aggregate Function Operation a. b. c. (a) Dno No_of_employees Average_sal (b) Dno Count_ssn Average_salary 5 4 33250 5 4 33250 4 3 31000 4 3 31000 1 1 55000 1 1 55000 (c) Count_ssn Average_salary 8 35125 Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Figure 7.1a Results of GROUP by and HAVING (in SQL). Q24 Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Additional Relational Operations (1 of 6) Recursive Closure Operations – Another type of operation that, in general, cannot be specified in the basic original relational algebra is recursive closure. ▪ This operation is applied to a recursive relationship. – An example of a recursive operation is to retrieve all SUPERVISEES of an EMPLOYEE e at all levels — that is, all EMPLOYEE e’ directly supervised by e; all employees e’’ directly supervised by each employee e’; all employees e’’’ directly supervised by each employee e’’; and so on. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Additional Relational Operations (2 of 6) Although it is possible to retrieve employees at each level and then take their union, we cannot, in general, specify a query such as “retrieve the supervisees of ‘James Borg’ at all levels” without utilizing a looping mechanism. – The SQL3 standard includes syntax for recursive closure. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Figure 8.11 A Two-Level Recursive Query Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Additional Relational Operations (3 of 6) The OUTER JOIN Operation – In NATURAL JOIN and EQUIJOIN, tuples without a matching (or related) tuple are eliminated from the join result ▪ Tuples with null in the join attributes are also eliminated ▪ This amounts to loss of information. – A set of operations, called OUTER joins, can be used when we want to keep all the tuples in R, or all those in S, or all those in both relations in the result of the join, regardless of whether or not they have matching tuples in the other relation. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Additional Relational Operations (4 of 6) The left outer join operation keeps every tuple in the first or left relation R in ; if no matching tuple is found in S, then the attributes of S in the join result are filled or “padded” with null values. A similar operation, right outer join, keeps every tuple in the second or right relation S in the result of A third operation, full outer join, denoted by keeps all tuples in both the left and the right relations when no matching tuples are found, padding them with null values as needed. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Figure 8.12 The Result of a LEFT OUTER JOIN Operation Result Fname Minit Lname Dname John B Smith NULL Franklin T Wong Research Alicia J Zelaya NULL Jennifer S Wallace Administration Ramesh K Narayan NULL Joyce A English NULL Ahmad V Jabbar NULL James E Borg Headquarters Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Additional Relational Operations (5 of 6) OUTER UNION Operations – The outer union operation was developed to take the union of tuples from two relations if the relations are not type compatible. – This operation will take the union of tuples in two relations R(X, Y) and S(X, Z) that are partially compatible, meaning that only some of their attributes, say X, are type compatible. – The attributes that are type compatible are represented only once in the result, and those attributes that are not type compatible from either relation are also kept in the result relation T(X, Y, Z). Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Additional Relational Operations (6 of 6) Example: An outer union can be applied to two relations whose schemas are STUDENT(Name, SSN, Department, Advisor) and INSTRUCTOR(Name, SSN, Department, Rank). – Tuples from the two relations are matched based on having the same combination of values of the shared attributes— Name, SSN, Department. – If a student is also an instructor, both Advisor and Rank will have a value; otherwise, one of these two attributes will be null. – The result relation STUDENT_OR_INSTRUCTOR will have the following attributes: STUDENT_OR_INSTRUCTOR (Name, SSN, Department, Advisor, Rank) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Examples of Queries in Relational Algebra: Procedural Form Q1: Retrieve the name and address of all employees who work for the ‘Research’ department. RESEARCH_DEPT   DNAME=’Research’ (DEPARTMENT ) RESEARCH_EMPS  (RESEARCH_DEPT DNUMBER= DNOEMPLOYEEEMPLOYEE) RESULT  FNAME, LNAME, ADDRESS (RESEARCH_EMPS ) Q6: Retrieve the names of employees who have no dependents. ALL_EMPS  SSN (EMPLOYEE) EMPS_WITH_DEPS(SSN)  ESSN (DEPENDENT) EMPS_WITHOUT_DEPS  (ALL_EMPS − EMPS_WITH_DEPS) RESULT  LNAME, FNAME (EMPS _ WITHOUT _ DEPS * EMPLOYEE ) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Examples of Queries in Relational Algebra – Single Expressions As a single expression, these queries become: Q1: Retrieve the name and address of all employees who work for the ‘Research’ department.  Fname,Lname, Address( Dname= ‘Research’ (DEPARTMENT Dnumber=Dno (EMPLOYEE )) Q6: Retrieve the names of employees who have no dependents. Lname,Fname ((Ssn (EMPLOYEE) − Ssn(Essn (DEPENDENT))) * EMPLOYEE) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Relational Calculus (1 of 2) A relational calculus expression creates a new relation, which is specified in terms of variables that range over rows of the stored database relations (in tuple calculus) or over columns of the stored relations (in domain calculus). In a calculus expression, there is no order of operations to specify how to retrieve the query result—a calculus expression specifies only what information the result should contain. – This is the main distinguishing feature between relational algebra and relational calculus. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Relational Calculus (2 of 2) Relational calculus is considered to be a nonprocedural or declarative language. This differs from relational algebra, where we must write a sequence of operations to specify a retrieval request; hence relational algebra can be considered as a procedural way of stating a query. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Tuple Relational Calculus (1 of 2) The tuple relational calculus is based on specifying a number of tuple variables. Each tuple variable usually ranges over a particular database relation, meaning that the variable may take as its value any individual tuple from that relation. A simple tuple relational calculus query is of the form t | COND (t ) – where t is a tuple variable and COND (t) is a conditional expression involving t. – The result of such a query is the set of all tuples t that satisfy COND (t). Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Tuple Relational Calculus (2 of 2) Example: To find the first and last names of all employees whose salary is above $50,000, we can write the following tuple calculus expression: t.FName, t.LName |EMPLOYEE (t ) AND t.Salary > 50000 The condition EMPLOYEE(t) specifies that the range relation of tuple variable t is EMPLOYEE. The first and last name (PROJECTIONFNAME, LNAME ) of each EMPLOYEE tuple t that satisfies the condition t.SALARY>50000 (SELECTION σ SALARY >50000 ) will be retrieved. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved The Existential and Universal Quantifiers (1 of 2) Two special symbols called quantifiers can appear in formulas; these are the universal quantifier and the existential quantifier Informally, a tuple variable t is bound if it is quantified, meaning that it appears in an clause; otherwise, it is free. If F is a formula, then so are where t is a tuple variable. – The formula is true if the formula F evaluates to true for some (at least one) tuple assigned to free occurrences of t in F; otherwise is false. – The formula is true if the formula F evaluates to true for every tuple (in the universe) assigned to free occurrences of t in F; otherwise is false. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved The Existential and Universal Quantifiers (2 of 2) is called the universal or “for all” quantifier because every tuple in “the universe of” tuples must make F true to make the quantified formula true. is called the existential or “there exists” quantifier because any tuple that exists in “the universe of” tuples may make F true to make the quantified formula true. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved  Example Query Using Existential Quantifier (1 of 2) Retrieve the name and address of all employees who work for the ‘Research’ department. The query can be expressed as : {t.FNAME, t.LNAME, t.ADDRESS | EMPLOYEE ( t ) and(d) (DEPARTMENT ( d ) andd.DNAME = ‘Research’ and d.DNUMBER = t.DNO) } The only free tuple variables in a relational calculus expression should be those that appear to the left of the bar ( | ). – In above query, t is the only free variable; it is then bound successively to each tuple. If a tuple satisfies the conditions specified in the query, the attributes F NAME, LNAME, and ADDRESS are retrieved for each such tuple. – The conditions EMPLOYEE (t) and DEPARTMENT(d) specify the range relations for t and d. – The condition d.DNAME = ‘Research’ is a selection condition and corresponds to a SELECT operation in the relational algebra, whereas the condition d.DNUMBER = t.DNO is a JOIN condition. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Example Query Using Existential Quantifier (2 of 2) Find the names of employees who work on all the projects controlled by department number 5. The query can be: Exclude from the universal quantification all tuples that we are not interested in by making the condition true for all such tuples. – The first tuples to exclude (by making them evaluate automatically to true) are those that are not in the relation R of interest. In query above, using the expression not(PROJECT(x)) inside the universally quantified formula evaluates to true all tuples x that are not in the PROJECT relation. – Then we exclude the tuples we are not interested in from R itself. The expression not (x.DNUM=5) evaluates to true all tuples x that are in the project relation but are not controlled by department 5. Finally, we specify a condition that must hold on all the remaining tuples in R. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Languages Based on Tuple Relational Calculus (1 of 2) The language SQL is based on tuple calculus. It uses the basic block structure to express the queries in tuple calculus: – SELECT – FROM – WHERE SELECT clause mentions the attributes being projected, the FROM clause mentions the relations needed in the query, and the WHERE clause mentions the selection as well as the join conditions. – SQL syntax is expanded further to accommodate other operations. (See Chapter 8). Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Languages Based on Tuple Relational Calculus (2 of 2) Another language which is based on tuple calculus is QUEL which actually uses the range variables as in tuple calculus. Its syntax includes: – RANGE OF IS Then it uses – RETRIEVE – WHERE This language was proposed in the relational DBMS INGRES. (system is currently still supported by Computer Associates – but the QUEL language is no longer there). Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved The Domain Relational Calculus (1 of 2) Another variation of relational calculus called the domain relational calculus, or simply, domain calculus is equivalent to tuple calculus and to relational algebra. The language called QBE (Query-By-Example) that is related to domain calculus was developed almost concurrently to SQL at IBM Research, Yorktown Heights, New York. – Domain calculus was thought of as a way to explain what QBE does. Domain calculus differs from tuple calculus in the type of variables used in formulas: – Rather than having variables range over tuples, the variables range over single values from domains of attributes. To form a relation of degree n for a query result, we must have n of these domain variables— one for each attribute. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved The Domain Relational Calculus (2 of 2) An expression of the domain calculus is of the form { x1, x2,...,xn | COND(x1, x 2 ,...,xn , x n+1, x n+2 ,...,x n+m )} – where x1, x 2 ,...,x n , x n+1, x n+2 ,...,x n+m are domain variables that range over domains (of attributes) – and COND is a condition or formula of the domain relational calculus. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Example Query Using Domain Calculus (1 of 2) Retrieve the birthdate and address of the employee whose name is ‘John B. Smith’. Query : {uv | (q) (r) (s) (t) (w) (x) (y) (z) (EMPLOYEE(qrstuvwxyz) and q =’John’ and r =’B’ and s =’Smith’)} Abbreviated notation EMPLOYEE(qrstuvwxyz) uses the variables without the separating commas: EMPLOYEE(q,r,s,t,u,v,w,x,y,z) Ten variables for the employee relation are needed, one to range over the domain of each attribute in order. – Of the ten variables only u and v are free. Specify the requested attributes, BDATE and ADDRESS, by the free domain variables u for BDATE and v for ADDRESS. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Example Query Using Domain Calculus (2 of 2) Specify the condition for selecting a tuple following the bar ( | )— – namely, that the sequence of values assigned to the variables qrstuvwx yz be a tuple of the employee relation and that the values for q (F NAME), r (MINIT), and s (LNAME) be ‘John’, ‘B’, and ‘Smith’, respectively. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved QBE: A Query Language Based on Domain Calculus (Appendix C) (1 of 2) This language is based on the idea of giving an example of a query using “example elements” which are nothing but domain variables. Notation: An example element stands for a domain variable and is specified as an example value preceded by the underscore character. P. (called P dot) operator (for “print”) is placed in those columns which are requested for the result of the query. A user may initially start giving actual values as examples, but later can get used to providing a minimum number of variables as example elements. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved QBE: A Query Language Based on Domain Calculus (Appendix C) (2 of 2) The language is very user-friendly, because it uses minimal syntax. QBE was fully developed further with facilities for grouping, aggregation, updating etc. and is shown to be equivalent to SQL. The language is available under QMF (Query Management Facility) of DB2 of IBM and has been used in various ways by other products like ACCESS of Microsoft, and PARADOX. For details, see Appendix C in the text. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved QBE Examples (1 of 3) QBE initially presents a relational schema as a “blank schema” in which the user fills in the query as an example: Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Example Schema as a QBE Query Interface Figure C.1 The relational schema of Figure 3.5 as it may be displayed by QBE. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved QBE Examples (2 of 3) The following domain calculus query can be successively minimized by the user as shown: Query : {uv | (q) (r) (s) (t) (w) (x) (y) (z) (EMPLOYEE(qrstuvwxyz) and q =’John’ and r =’B’ and s =’Smith’)} Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Four Successive Ways to Specify a QBE Query Figure C.2 Four ways to specify the query Q0 in QBE. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved QBE Examples (3 of 3) Specifying complex conditions in QBE : A technique called the “condition box” is used in QBE to state more involved Boolean expressions as conditions. The C.4(a) gives employees who work on either project 1 or 2, whereas the query in C.4(b) gives those who work on both the projects. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Complex Conditions with and Without a Condition Box as a Part of QBE Query Figure C.3 Specifying complex conditions in QBE. (a) The query Q0A. (b) The query Q0B with a condition box. (c) The query Q0B without a condition box. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Handling AND conditions in a QBE Query Figure C.4 Specifying EMPLOYEES who work on both projects. (a) Incorrect specification of an AND condition. (b) Correct specification. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Join in QBE : Examples The join is simply accomplished by using the same example element (variable with underscore) in the columns being joined from different (or same as in C.5 (b)) relation. Note that the Result is set us as an independent table to show variables from multiple relations placed in the result. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Performing Join with Common Example Elements and Use of a Result Relation Figure C.5 Illustrating JOIN and result relations in QBE. (a) The query Q1. (b) The query Q8. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Aggregation in QBE Aggregation is accomplished by using.CNT for count,.MAX,.MIN,.AVG for the corresponding aggregation functions Grouping is accomplished by.G operator. Condition Box may use conditions on groups (similar to HAVING clause in SQL – see Section 8.5.8) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Aggregation in QBE : Examples Figure C.6 Functions and grouping in QBE. (a) The query Q23. (b) The query Q23A. (c) The query Q24. (d) The query Q26. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Negation in QBE : Example Figure C.7 Illustrating negation by the query Q6. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Updating in QBE : Example Figure C.8 Modifying the database in QBE. (a) Insertion. (b) Deletion. (c) Update in QBE. Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved Chapter Summary Relational Algebra – Unary Relational Operations – Relational Algebra Operations From Set Theory – Binary Relational Operations – Additional Relational Operations – Examples of Queries in Relational Algebra Relational Calculus – Tuple Relational Calculus – Domain Relational Calculus Overview of the QBE language (appendix C) Copyright © 2016, 2011, 2007 Pearson Education, Inc. All Rights Reserved