Relational Algebra

Relational Algebra is the mathematical basis for the query language SQL

Introduction.

So now you have learn how to design good conceptual models to store information with the ER-model
And you also know how to turn an ER-model into a Relational model

Let us jump the next step - which is defining the Relational Database and entering the data into the database - and look atdata manipulation

What can we do with the data ????

Relational Algebra: Overview

is amathematical languagefor manipulating relations (ala mathematical definition of "relation" - which is a subset of some cartesian product)

contains:

set operators
relational database specific operators
set functions

is used to specify a result set that satisfies a certain "selection criteria"

the result of an operation in relational algebra is always arelation

Set operations

You should all know whatset unionis:

SupposeA = {i, j, k}andB = {k, x, y}
Then the union ofAandBis{i, j, k, x, y}

Note that tuples in setsAandBmust have the sameFORMAT(same number and type of attributes) to be used in the union operation.

You should also have learned whatset intersectionis:

SupposeA = {i, j, k}andB = {k, x, y}
Then the intersection ofAandBis{k}

Note that tuples in setsAandBmust have the sameFORMAT(same number and type of attributes) to be used in the intersection operation.

You should also have learned whatset differenceis:

SupposeA = {i, j, k}andB = {k, x, y}
Then the difference ofAandBis{i, j}

Note that tuples in setsAandBmust have the sameFORMAT(same number and type of attributes) to be used in the set difference operation.

I have already discussed thecartesian product:click here

Set Functions

Operates on a set of values and produce a single value

To illustrate the various set function, weapplytheset functionson this set of value:

A = {1, 4, 9}

Results:

Relational database specific operations

σ_(Condition)(R) (selection)

Selects tuples from a relationRthat satisfies the conditionCondition
TheConditionexpression containsconstantsand/orattributesof relationR(the condition expressioncannotbe aset!!!)
The result ofσ_(Condition)(R)is asubsetof tuples ofRthat satisfies the boolean conditionCondition

Examples:

Retrieve all employee tuples working for department number 4:

The following diagram shows the result graphically (hopefully, it will illustrates the concept unambiguously):

Retrieve all employees earning more than $30,000:

Retrieve all employees working for department number 4 and eaning more than $30,000:

OR:

π_{(attribute-list)} (projection)

Selects out only theattribute valuesattribute-listfrom all tuples in relationR
Theattribute-listcontains the list of attributes in relationRthat will be selected.
The result ofπ_{(attribute-list)}(R)contains the is asubsetof tuples ofRthat satisfies the boolean conditionCondition

Examples:

Retrieve all SSN from the employees

The following diagram shows the result graphically (hopefully, it will illustrates the concept unambiguously):

Retrieve thesexinformation from all employees:

The followingdiagramshows the result graphically (hopefully, it will illustrates the concept ofprojectionunambiguously):

Notes:

Combining information from 2 relations

Consider the following query:

Observation:

But we must becarefulin combining the information:

   Employee:
   +-----------+--------+---------+-----------+-----+ 
   | ssn       | fname  | lname   | bdate     | dno | .... (other columns omitted)     
   +-----------+--------+---------+-----------+-----+ 
   | 123456789 | John   | Smith   | 09-JAN-55 |   5 | 
   | 333445555 | Frankl | Wong    | 08-DEC-45 |   5 | 
   | 999887777 | Alicia | Zelaya  | 19-JUL-58 |   4 | 
   | 987654321 | Jennif | Wallace | 20-JUN-31 |   4 | 
   | 666884444 | Ramesh | Narayan | 15-SEP-52 |   5 | 
   | 453453453 | Joyce  | English | 31-JUL-62 |   5 | 
   | 987987987 | Ahmad  | Jabbar  | 29-MAR-59 |   4 | 
   | 888665555 | James  | Borg    | 10-NOV-27 |   1 | 
   +-----------+--------+---------+-----------+-----+ 

   Department:
   +----------------+---------+-----------+--------------+
   | dname          | dnumber | mgrssn    | mgrstartdate |
   +----------------+---------+-----------+--------------+
   | Research       |       5 | 333445555 | 22-MAY-78    |
   | Administration |       4 | 987654321 | 01-JAN-85    |
   | Headquarters   |       1 | 888665555 | 19-JUN-71    |
   +----------------+---------+-----------+--------------+

Howto"combine"tuples fromEmployeeandDepartment:

Solution:

First find the department number of the "Research" department:

Graphically:

Then wecombinethedepartmentandemployeethrough a"mindless" cartesian product operation:

The result of this operation is"tagging" an extra column "dnumber"to theemployeerelation:

Nowwe can use aselect operationto select out the"right tuples":

Becausednumberis now anattributeof the (new)relationR₂!!!

Graphically:

Efficiency consideration of queries

Correctnessis themost important propertyof a query

Thesecond most important propertyof aqueryisefficiency

There maybemultiple queriesthat retrieve thecorrect set of tuples(answer)

The difference of the queries may be theirexecution speed

Consider the following solution:

Although thisqueryiscorrect, it will takelonger time to processbecause of thelarger amount ofintermediatedatathat the query produces:

Thecartesian productemployee x departmentwillproducehuge number of tuples- many more than in the previous solution.

Join operations

The join operation (bow tie symbol) is equivalent to:

Example:

Retrieve all employees from the "Research" department

slower (but correct)solution:

It is slower because thejoin operationwill result in more tuples asintermediate results

Thefaster versionis:

The "theta join" operation

The condition in the join operation is of the form:

   Condition

 and Condition

 and ....       
   

Each condition is of the form"x θ y", wherexandyare attributes or constants

Andθis one of the following relational operators:

A join operation with a "general" join condition (whereθcan be any relational operation), is called atheta join

The "equi-join" operation

Often(almost always :-)), we use the equal relational operation in the join operation:

     Condition

 and Condition

 and ....     

whereeach conditionis of the form:

    x == y        

wherexandyare attributes or constants

Equi-join:

The "natural join" (*) operation

The natural join operation is anequi-joinoperation follow by a projection operation that removes one of theduplicateattributes.

Since the equi-join will for tuples when the attribute values are equal, all tuples that produced by an equi-join will have the same attribute value under the join condition attributes.

The natural join operation is denoted by a star (*):

    R1 * attr1,attr2 R2       

Outer-join

Outer-joins:

The3 flavorsofouter-joins:

     A:                 B:

        X     Y            U     V
     +-----+-----+      +-----+-----+                  
     |  1  |  7  |      |  1  |  8  |
     +-----+-----+      +-----+-----+
     |  2  |  5  |      |  2  |  3  |
     +-----+-----+      +-----+-----+
     |  3  |  4  |      |  4  |  9  |
     +-----+-----+      +-----+-----+

  A ?A.X=B.U B:

        X     Y     U     V
     +-----+-----+-----+-----+ 
     |  1  |  7  |  1  |  8  |
     +-----+-----+-----+-----+
     |  2  |  5  |  2  |  3  |
     +-----+-----+-----+-----+
     |  3  |  4  | NULL| NULL|
     +-----+-----+-----+-----+

     A:                 B:

        X     Y            U     V
     +-----+-----+      +-----+-----+                  
     |  1  |  7  |      |  1  |  8  |
     +-----+-----+      +-----+-----+
     |  2  |  5  |      |  2  |  3  |
     +-----+-----+      +-----+-----+
     |  3  |  4  |      |  4  |  9  |
     +-----+-----+      +-----+-----+

  A ?A.X=B.U B:

        X     Y     U     V
     +-----+-----+-----+-----+ 
     |  1  |  7  |  1  |  8  |
     +-----+-----+-----+-----+
     |  2  |  5  |  2  |  3  |
     +-----+-----+-----+-----+
     | NULL| NULL|  4  |  9  |
     +-----+-----+-----+-----+

     A:                 B:

        X     Y            U     V
     +-----+-----+      +-----+-----+                  
     |  1  |  7  |      |  1  |  8  |
     +-----+-----+      +-----+-----+
     |  2  |  5  |      |  2  |  3  |
     +-----+-----+      +-----+-----+
     |  3  |  4  |      |  4  |  9  |
     +-----+-----+      +-----+-----+

  A ?A.X=B.U B:

        X     Y     U     V
     +-----+-----+-----+-----+ 
     |  1  |  7  |  1  |  8  |
     +-----+-----+-----+-----+
     |  2  |  5  |  2  |  3  |
     +-----+-----+-----+-----+
     |  3  |  4  | NULL| NULL|
     +-----+-----+-----+-----+
     | NULL| NULL|  4  |  9  |
     +-----+-----+-----+-----+

Note:

     A:                 B:

        X     Y            X     V
     +-----+-----+      +-----+-----+                  
     |  1  |  7  |      |  1  |  8  |
     +-----+-----+      +-----+-----+
     |  2  |  5  |      |  2  |  3  |
     +-----+-----+      +-----+-----+
     |  3  |  2  |      |  4  |  9  |
     +-----+-----+      +-----+-----+

  A ? B:

        X     Y     V
     +-----+-----+-----+ 
     |  1  |  7  |  8  |
     +-----+-----+-----+
     |  2  |  5  |  3  |
     +-----+-----+-----+
     |  3  |  2  | NULL|
     +-----+-----+-----+
     |  4  | NULL|  9  |
     +-----+-----+-----+

Teaching notes:

More examples....

Find fname and lname of all employees working in the "Research" department that earn more than $50,000

Find fname and lname of John Smith's supervisor

Find fname and lname of all employees that have dependents

Find fname and lname of all employees thatdo nothave dependents

This solution iswrong:

The following diagram shows how the query works on an example database, which illustrateswhythe query is wrong:

The correct solution is:

Help1 = set of SSN of employees with dependents
Help2 = set of SSN of employees without any dependents

Find fname and lname of employees who has more than one dependents of the same sex (i.e., 2 or more boys or 2 or more girls):

NOTE:so join conditions do not always have to be "equal" !!!

A comment...

Clearly, I can easily formulate a gad-sillion different queries in the next day or so, so:

DO NOT

prepare for tests with

memorizing
facts - you gotta understand what's going on and do the queries "from scratch"....

And for those students that have never had me before: I have a reputation to challenge student's grey cells to work overtime.... so you should expect some very interesting queries.... and plenty of opportunity to practice your skill (in the homeworks and SQL projects).

http://www.mathcs.emory.edu/~cheung/Courses/377/Syllabus/4-RelAlg/intro.html

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