Relational algebra - ITUitu.dk/~mogel/SIDD2012/lectures/SIDD.2012.05.pdf · Rasmus Ejlers Møgelberg Translating SQL into relational algebra •Expression •Is translated to •Relational

Rasmus Ejlers Møgelberg

Relational algebraIntroduction to Database Design 2012, Lecture 5


Overview

• Use of logic in constructing queries

• Relational algebra

• Translating queries to relational algebra

• Equations expressed in relational algebra

2


Use of logic in constructing queries

• Consider the following problem

• Expressed more formally

• Translate to SQL:

3

Find all students who have taken all courses offered by the biology department

Find all students s such that for all courses c, if c is offered by ‘Biology’ then s has taken c

select * from students where [???]



• The problem is not suitable for SQL, because it uses ‘for all’ and ‘if ... then ...’

• So reformulate

• (using classical logic)

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Find all students s such that for all courses c, if c is offered by ‘Biology’ then s has taken c

Find all students s such that there is no course c such that c is offered by ‘Biology’ and s has not taken c



• This can be formulated in SQL:

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Find all students s such that there is no course c such that c is offered by ‘Biology’ and s has not taken c

select * from student where not exists (select * from course where dept_name = ‘Biology’ and course_id not in (select course_id from takes where takes.id = student.id))

Finds all courses taken by studentFinds all courses offered by Biology not take by student


More logic

• Similar analysis is needed for the challenging exercises this week

• You will see more logic in the course Foundations of Computing

6


Relational algebra


Relational algebra

• A language for expressing basic operations in the relational model

• Two purposes

- Express meaning of queries

- Express execution plans in DBMSs

• SQL is declarative (what)

• Relational algebra is procedual (how)

8


Relational algebra in DBMSs

9

Illustration from book


Projection

• In SQL

• In relational algebra

10

select name, salary from instructor;

Πname, salary(instructor)


Selection

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select * from instructor where salary > 90000;

σsalary>90000(instructor)


Combining selection and projection

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select name, dept_name from instructor where salary > 90000;

Πname, dept name(σsalary>90000(instructor))


Translating SQL into relational algebra

• Expression

• Is translated to

• Relational algebra expression says

- First do selection

- Then do projection

• Relational algebra procedural

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select name, dept_name from instructor where salary > 90000;



Syntax trees

• The syntax tree

• represents the expression (3+4)*5

• Trees grow downwards in computer science!

• Evaluation from bottom up

• Useful graphical way of representing evaluation order (no need for parentheses)

14

*

+ 5

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Syntax trees for relational algebra

• represents

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Πname, dept name

σsalary>90000

instructor



Cartesian products

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mysql> select * from instructor, department;+-------+------------+------------+----------+------------+----------+-----------+| ID | name | dept_name | salary | dept_name | building | budget |+-------+------------+------------+----------+------------+----------+-----------+| 10101 | Srinivasan | Comp. Sci. | 65000.00 | Biology | Watson | 90000.00 | | 10101 | Srinivasan | Comp. Sci. | 65000.00 | Comp. Sci. | Taylor | 100000.00 | | 10101 | Srinivasan | Comp. Sci. | 65000.00 | Elec. Eng. | Taylor | 85000.00 | | 10101 | Srinivasan | Comp. Sci. | 65000.00 | Finance | Painter | 120000.00 | | 10101 | Srinivasan | Comp. Sci. | 65000.00 | History | Painter | 50000.00 | | 10101 | Srinivasan | Comp. Sci. | 65000.00 | Music | Packard | 80000.00 | | 10101 | Srinivasan | Comp. Sci. | 65000.00 | Physics | Watson | 70000.00 | | 12121 | Wu | Finance | 90000.00 | Biology | Watson | 90000.00 | | 12121 | Wu | Finance | 90000.00 | Comp. Sci. | Taylor | 100000.00 | | 12121 | Wu | Finance | 90000.00 | Elec. Eng. | Taylor | 85000.00 | | 12121 | Wu | Finance | 90000.00 | Finance | Painter | 120000.00 | | 12121 | Wu | Finance | 90000.00 | History | Painter | 50000.00 | | 12121 | Wu | Finance | 90000.00 | Music | Packard | 80000.00 | | 12121 | Wu | Finance | 90000.00 | Physics | Watson | 70000.00 | | 15151 | Mozart | Music | 40000.00 | Biology | Watson | 90000.00 | | 15151 | Mozart | Music | 40000.00 | Comp. Sci. | Taylor | 100000.00 | | 15151 | Mozart | Music | 40000.00 | Elec. Eng. | Taylor | 85000.00 | | 15151 | Mozart | Music | 40000.00 | Finance | Painter | 120000.00 | | 15151 | Mozart | Music | 40000.00 | History | Painter | 50000.00 | ... +-------+------------+------------+----------+------------+----------+-----------+84 rows in set (0.01 sec)


Products

• In relational algebra

• Syntax tree

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select * from instructor, department;

×

instructor× department

instructor department


Relational model: natural join

• First cartesian product, then select, then project

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mysql> select * from instructor natural join department;+------------+-------+------------+----------+----------+-----------+| dept_name | ID | name | salary | building | budget |+------------+-------+------------+----------+----------+-----------+| Comp. Sci. | 10101 | Srinivasan | 65000.00 | Taylor | 100000.00 | | Finance | 12121 | Wu | 90000.00 | Painter | 120000.00 | | Music | 15151 | Mozart | 40000.00 | Packard | 80000.00 | | Physics | 22222 | Einstein | 95000.00 | Watson | 70000.00 | | History | 32343 | El Said | 60000.00 | Painter | 50000.00 | | Physics | 33456 | Gold | 87000.00 | Watson | 70000.00 | | Comp. Sci. | 45565 | Katz | 75000.00 | Taylor | 100000.00 | | History | 58583 | Califieri | 62000.00 | Painter | 50000.00 | | Finance | 76543 | Singh | 80000.00 | Painter | 120000.00 | | Biology | 76766 | Crick | 72000.00 | Watson | 90000.00 | | Comp. Sci. | 83821 | Brandt | 92000.00 | Taylor | 100000.00 | | Elec. Eng. | 98345 | Kim | 80000.00 | Taylor | 85000.00 | +------------+-------+------------+----------+----------+-----------+12 rows in set (0.01 sec)


Join in relational algebra

• Join can be defined using other constructors

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×

department instructor

σinstructor.dept name=department.dept name

Πdept name,ID,. . . ,budget


Computation of joins

• In practice joins are not always computed this way

• Consider e.g.

• Can often find relevant entry on right hand side fast without having to construct cartesian product

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Expressing execution plans

• DBMSs use a variant of relational algebra for this

• Still, basic relational algebra good way of understanding meaning of queries

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General joins

• Define

• For example

• Is translated to relational algebra as

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R ��Θ S = σΘ(R× S)

select * from student join advisor on s_ID = ID

student ��(ID=s ID) advisor


Set operations

• Usual set operations in relational algebra

• These only allowed between relations with same set of attributes!

• Warning:

- The book treats relational algebra

- Might have been better to use multiset relational algebra

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R ∪ S

R ∩ S

R \ S


Using left outer join

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mysql> select * from student natural left outer join takes;+-------+----------+------------+----------+-----------+--------+----------+------+-------+| ID | name | dept_name | tot_cred | course_id | sec_id | semester | year | grade |+-------+----------+------------+----------+-----------+--------+----------+------+-------+| 00128 | Zhang | Comp. Sci. | 102 | CS-101 | 1 | Fall | 2009 | A | | 00128 | Zhang | Comp. Sci. | 102 | CS-347 | 1 | Fall | 2009 | A- | | 12345 | Shankar | Comp. Sci. | 32 | CS-101 | 1 | Fall | 2009 | C | | 12345 | Shankar | Comp. Sci. | 32 | CS-190 | 2 | Spring | 2009 | A | | 12345 | Shankar | Comp. Sci. | 32 | CS-315 | 1 | Spring | 2010 | A | | 12345 | Shankar | Comp. Sci. | 32 | CS-347 | 1 | Fall | 2009 | A | | 19991 | Brandt | History | 80 | HIS-351 | 1 | Spring | 2010 | B | | 23121 | Chavez | Finance | 110 | FIN-201 | 1 | Spring | 2010 | C+ | | 44553 | Peltier | Physics | 56 | PHY-101 | 1 | Fall | 2009 | B- | | 45678 | Levy | Physics | 46 | CS-101 | 1 | Fall | 2009 | F | | 45678 | Levy | Physics | 46 | CS-101 | 1 | Spring | 2010 | B+ | | 45678 | Levy | Physics | 46 | CS-319 | 1 | Spring | 2010 | B | | 54321 | Williams | Comp. Sci. | 54 | CS-101 | 1 | Fall | 2009 | A- | | 54321 | Williams | Comp. Sci. | 54 | CS-190 | 2 | Spring | 2009 | B+ | | 55739 | Sanchez | Music | 38 | MU-199 | 1 | Spring | 2010 | A- | | 70557 | Snow | Physics | 0 | NULL | NULL | NULL | NULL | NULL | | 76543 | Brown | Comp. Sci. | 58 | CS-101 | 1 | Fall | 2009 | A | | 76543 | Brown | Comp. Sci. | 58 | CS-319 | 2 | Spring | 2010 | A | | 76653 | Aoi | Elec. Eng. | 60 | EE-181 | 1 | Spring | 2009 | C | | 98765 | Bourikas | Elec. Eng. | 98 | CS-101 | 1 | Fall | 2009 | C- | | 98765 | Bourikas | Elec. Eng. | 98 | CS-315 | 1 | Spring | 2010 | B | | 98988 | Tanaka | Biology | 120 | BIO-101 | 1 | Summer | 2009 | A | | 98988 | Tanaka | Biology | 120 | BIO-301 | 1 | Summer | 2010 | NULL | +-------+----------+------------+----------+-----------+--------+----------+------+-------+23 rows in set (0.00 sec)


Outer join

• Left outer join defined as

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∪

−

student �� takes

student

student �� takes

×

{(null, . . . ,null)}

Πstudent

students who have taken a course

students who have not taken a course


Generalised projections

• Projections can be combined with basic operations on numbers, dates, booleans or strings, e.g.

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Πflight num,capacity−reservations(. . . )


Renaming

• It is often necessary to rename a relation

• The expression

• renames relation S to R and the attributes of S to a, ..., b

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ρR(a,...,b)(S)


Aggregation

• Special symbol for aggregation

• SQL

• Relational algebra

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mysql> select avg(salary), dept_name from instructor -> group by dept_name;

dept nameGaverage(salary)


Aggregation with having

• First group, then select groups

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mysql> select avg(salary), dept_name from instructor -> group by dept_name -> having count(ID)>1;


Having

• Recall that having is just selection on the group level

• Translate

• as

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mysql> select avg(salary), dept_name from instructor -> group by dept_name -> having avg(salary) > 80000;

σavg salary>80000

ρR(avg salary,dept name)

dept nameGaverage(salary)

instructor


Subqueries

• Example

• Insert tree from subquery in tree from outer query

• (details on blackboard)

• Nested queries in where clause are more involved

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mysql> select name from instructor, -> (select max(salary) as max_salary from instructor) as S -> where instructor.salary = S.max_salary;

mysql> select name from instructor -> where salary >= all (select salary from instructor);


Equations


Equations

• Many different relational algebra expressions compute the same thing

• When evaluating queries, DBMS will

- generate many different relational algebra expressions computing the query

- choose the one it thinks is most efficient

• Here we see some basic equalities of expressions

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Relational algebra in DBMSs

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Illustration from book


Equalities (examples)

• Selection is commutative

• Join is commutative

• (only difference is order of attributes)

• Join is associative

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σΘ1(σΘ2(R)) = σΘ2(σΘ1(R))= σΘ1andΘ2(R)

R1 �� R2 = R2 �� R1

R1 �� (R2 �� R3) = (R1 �� R2) �� R3


More equalities

• Suppose only talks about attributes of and similarly only talks about attributes of

• Then

• Right hand side is often much less expensive to compute (DBMS makes such optimizations automatically)

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Θ1 R1

Θ2 R2

σΘ1andΘ2

��

R1 R2

=

��

σΘ1 σΘ2

R1 R2


Summary

• After this lecture you should be able to

- Translate simple queries to relational algebra

- Draw the syntax tree of relational algebra expressions

• Future goal:

- Judge which relational algebra expression represents the most efficient evaluation plan for a query

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Documents

Relational algebra - ITUitu.dk/~mogel/SIDD2012/lectures/SIDD.2012.05.pdf · Rasmus Ejlers Møgelberg Translating SQL into relational algebra •Expression •Is translated to •Relational