Relational Algebra (CB Chapter 5.1)

CPSC 356 Database

Ellen Walker

Hiram College

Relational Algebra

• Mathematical language for operating on relations

• Each operator takes one or more relations as input, and produces one relation as output

• Relational algebra operators can be implemented as functions in a programming language

• A relational algebra expression indicates which operators to use, in which order

Example Relation (PhoneBook)

First Last Dept Email Phone Title

Louis Oliphant CPSC Louis 5275 Asst

Ellen Walker CPSC WalkerEL 5250 Prof

Laura VanWormer PHYS VanwormerLA 5249 Prof

Cathy Mansor WEC MansorCN 5957 Dean

Brad Goodner BIOL GoodnerBW 5260 Prof

Another Relation (Dept)

Name Building

CPSC Colton

WEC Hinsdale

PHYS Gerstacker

BIOL Gerstacker

MGMT Hinsdale

Selection (condition )

• The output relation has all rows of the input relation that satisfy the condition

(Title=“prof”) Phonebook

First Last Dept Email Phone Title

Laura VanWormer PHYS VanwormerLA 5249 Prof

Ellen Walker CPSC WalkerEL 5250 Prof

Brad Goodner BIOL GoodnerBW 5260 Prof

Valid Conditions

• Basic condition– Dept = ‘CS’ attribute compare to const– First < Last attribute compare to attribute

• Combination of conditions using AND, OR, or NOT

• Note: All attributes must come from the relation in the selection

Pseudocode for Selection

Select (input relation, condition, &output rel)

Do for every tuple in the input relation

If the tuple satisfies the condition

Copy the tuple to the output relation

Time = O(number of tuples in input relation)

Projection (attributes)

• Create a new relation with only the listed attributes in it.

last, phone PhoneBook Last Phone

Oliphant 5275

Walker 5250

VanWormer 5249

Mansor 5957

Goodner 5260

Relations Have No Duplicates

dept PhoneBook

• Only 4 rows, even though PhoneBook had 5!

Dept

CPSC

PHYS

WEC

BIOL

Pseudocode for Projection

Project (input relation, attribute-list, &output rel)

Do for every tuple in the input relation

Do for every attribute in the input relation

If the attribute is in the attribute-list

copy the value to the output relation

Remove duplicates in the output relation

Time = O(number of tuples in relation + time for duplicate-removal)

Combining Select & Project

first, last ( title=prof PhoneBook )

First Last

Laura VanWormer

Ellen Walker

Brad Goodner

Remember: A Relation is a Set

• A set is an unordered collection of unique elementsSet S = {1,2,3} “1 is an element of S”

{a,b,c} = {a,c,b}

• A subset of a set is another set whose elements all come from the original set.{a,b} is a subset of {a,c,b}

{1,2,3} is a subset of {1,2,3}

{1,2,4} is not a subset of {1,2,3}

{} (the empty set) is a subset of every set!

Basic Set Operations

• Union: the set of all elements in either or both original sets– {1,2} union {2,3} = {1,2,3}

• Intersection: the set of all elements in both original sets (only)– {1,2} intersect {2,3} = {2}

• Set Difference: the set of all elements in the first but not the second set– {1,2} – {2,3} = {1}

Applying to Relations

• Relations must be “comparable”– Same set of attributes in each relation!

• Union = all tuples• Intersection = all matching tuples• Set Difference = all tuples from first but not

second

Basic Operation Examples

• R1 = dept PhoneBook

• R2 = name as “dept” Dept– Rename attribute to be same

• R1 R2 = R2 (in this case)• R1 R2 = R1 (in this case)• R1 – R2 = { } (empty set)• R2 – R1 =

Dept

CPSC

PHYS

WEC

BIOL

Dept

CPSC

PHYS

WEC

BIOL

MGMT

R1 R2

Dept

MGMT

Another Set Operation

• Cartesian product: a set of ordered pairs, where each contains one element from each original set{1,2,3} x {a, b} =

{(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)}

• For Relations: create a new relation with every combination of tuples

Cartesian Product (X)

dept, last PhoneBook X Dept

Last Dept Name Bldg

Oliphant CPSC CPSC Colton

Walker CPSC CPSC Colton

VanWormer

PHYS CPSC Colton

Mansor WEC CPSC Colton

Goodner BIOL CPSC Colton

Oliphant CPSC WEC Hinsdale

Walker CPSC WEC Hinsdale

VanWormer

PHYS WEC Hinsdale

Mansor WEC WEC Hinsdale

Goodner BIOL WEC Hinsdale

Oliphant CPSC PHYS Colton

Walker CPSC PHYS ColtonEtc….

Pseudocode for Cartesian Product

Product (relation1, relation2, &output rel)Do for every tuple in relation1 Do for every tuple in relation2 Build a row with all attributes from both relations

Add it to the output relation

Time = O(number of tuples in relation1 * number of tuples in relation 2)

This is the most expensive operation in relational algebra!

Join Combines X and Select

• Theta Join: any condition R1 X R2

• Equijoin: equality condition R1 X R2

• Natural Join: equality condition R1 X R2

– Project to remove one copy of each equal attribute

• Left or Right Outer Join: – Include all tuples from (left or right) side, even if they don’t

have a match

Naïve Pseudocode for Join

• Join (rel1, rel2, condition, output rel)• product (rel1, rel2, tmp)• select (tmp, condition, output rel)

• Time: Same as Cartesian Product• To keep time down, keep the size of the

relations down -- we’ll look at this later!

Let’s Try Some Examples:

• What are the first and last names of all professors who don’t work in Hinsdale?

• What are the telephone extensions of people who work in Hinsdale?

• Which buildings contain people whose phone numbers are between 5000 and 5200?

• List the Dept. Name, Building Name, and phone numbers for All departments (even those without phone numbers).

First and last names of all professors who don’t work in Hinsdale

• Select “all professors” Title=“Prof” (Phonebook)

• Select “Departments not in Hinsdale” Bldg != “Hinsdale” (Dept)

• Connect these relations where depts match– (Title=“Prof” (Phonebook)) |X| Dept=Name (Bldg != “Hinsdale” (Dept))

• One project to get the final result first,last ((Title=“Prof” (Phonebook)) |X| Dept=Name (Bldg != “Hinsdale”

(Dept)))

Telephone extensions of people who work in Hinsdale

• Select “Departments in Hinsdale” and project to just Dept to make the table smaller Name (Bldg = “Hinsdale” (Dept))

• Join with PhoneBook to get only those in the right departments Name (Bldg = “Hinsdale” (Dept)) |X| Dept=Name PhoneBook)

• Project to get just the extensions Phone ( Name (Bldg = “Hinsdale” (Dept)) |X| Dept=Name PhoneBook))

Buildings with phone numbers between 5000 and 5300

• Select to get phone numbers from 5000 to 5300 and Project to have only the dept attribute dept (5000<=Phone && 5300>=Phone (PhoneBooks))

• Join with Dept to associate department names with buildings– (dept (5000<=Phone && 5300>=Phone (PhoneBooks)) |X| dept=name (Dept))

• Project to get just the building names bldg ((dept (5000<=Phone && 5300>=Phone (PhoneBooks)) |X| dept=name

(Dept)))

Dept. Name, Building, and phone numbers for all departments

• Join to include info from *ALL* departments. This requires an outer join– Dept X| name=dept PhoneBook

• Project to get the right attributes name, bldg,phone Dept X| name=dept PhoneBook

Aggregation

• Operations that allow you to combine all the values in a table (column) in some way:

• COUNT• SUM• AVG• MIN• MAX

– Examples:• How many CPSC majors are there?• What is the average GPA of CPSC majors?

Grouping

• Aggregate all elements in one column based on values in another column

• Aggregation operators (COUNT, SUM, AVG, MIN, MAX)• Format:

– group-by-attribute, F OPER attribute (table)

• Note: F is a backward cursive F in the book.

Grouping Examples

• List average GPAs by major

major F AVG GPA(Student)

• What is the average GPA of CPSC students?σmajor=CPSC (major F AVG GPA(Student))

• List number of faculty in each department • dept F COUNT id(Faculty)