Operations in the Relational Model

COP 4720Lecture 8

Lecture Notes

Lecture 8COP 4720 2

Outline of Lecture

• Relational Algebra– Set Operations– Projection– Selection– Cartesian Product – Join– Combining Operations

• Completeness of Relational Algebra Operations

• Sec. 3.2

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Database Query Languages

• Given a database, ask questions, get data as answers– Ex: Get all students with GPA > 3.7 who applied to

Berkeley and Stanford and nowhere else

– Ex: Get all humanities departments at campuses in Florida with < 1000 applicants

– Ex: Get the campus with highest average accept rate over the last five years

• Some questions are easy to pose, some are not

• Some questions are easy for DBMS to answer, some are not.

• "Query language" also used to update the database

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Relational Query Languages

• Formal(pure): relational algebra, relational calculus

• Actual: SQL

• In all languages, a query is executed over a set of relations, get a relation as the result

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Relational Algebra• Construct new relations from old ones

– Set of operators– Relations are operands

• Build progressively more complex expressions by applying operators to relations or to rel. algebra expressions (which are relations as well)

• Query is a relational algebra expression– First concrete example of a query language

• Four broad classes of operations– Set operations, selection operations,

operations that combine data from two relations, rename operation

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Sample Relational Schema

Movies(title,year,length,filmType)

MovieStars(name,address,gender,birthdate)

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Relational Operators: Select

Select or Restrict (unary) <predicate> (r) = {t | t r and predicate(t)}

• <predicate> is a conditional expression of the type that we are familiar with from conventional programming languages– <attribute> <op> <attribute>– <attribute> <op> <constant>– attribute in R– op {=,,<,>,, …, AND, OR}

• Ex: length100 (Movies)

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Pictorially

A1 A2 A3 … An

...

i

A1 A2 A3 … An

...

j, i j

title year length filmTypeStar Wars

Mighty Ducks

Wayne’sWorld

1977

1991

1992

124

104

95

color

color

color

Movies

result set

Comparison with null values return a false condition

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Relational Operators: Project

Project (unary) <attr list> (r)

• <attr list> is a list of attributes (columns) from R only

• Ex: title, year, length (Movies) A1 A2 A3 … An

...

i

A1 A2… Ak

...

i

n k, k<=n

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Relational Operators: Rename

Rename (unary) New (Old)

• New is the name and schema of the relation to be renamed (Old)

• Ex: MyMovies (Movies)

• Ex: MyMovies(title,year,length,type) (Movies)

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Set OperationsUnion (binary, commutative, associative)• R SIntersection (binary, commutative, associative)• R SSet Difference (binary)• R - S

– Set of elements in R but not in S– R-S S-R !!

• R(A1,A2,…,An), S(B1,B2,…,Bn) must be union compatible– R and S are of the same degree

– for each i, dom(Ai) = dom(Bi)

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Example

name address gender birthdateCarrie Fisher

Mark Hamil

123 Maple St., Hollywood

456 Oak Rd., Brentwood

F

M

9/9/99

8/8/88

R

name address gender birthdateCarrie Fisher

Harrison Ford

123 Maple St., Hollywood

789 Palm Dr., Beverly Hills

F

M

9/9/99

7/7/77

S

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Sample Operations

name address gender birthdateCarrie Fisher 123 Maple St., Hollywood F 9/9/99

R S

name address gender birthdateCarrie Fisher

Harrison Ford

123 Maple St., Hollywood

789 Palm Dr., Beverly Hills

F

M

9/9/99

7/7/77

R S

Mark Hamil 456 Oak Rd., Brentwood M 8/8/88

name address gender birthdate

R - S

Mark Hamil 456 Oak Rd., Brentwood M 8/8/88

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Cartesian Product

Cartesian Product (binary, commutative, associative)

• R x S• Sets of all pairs that can be formed by

choosing the first element of the pair to be any element of R, the second any element of S

• Resulting schema may be ambiguous– Use R.A or S.A to remove ambiguity for an

attribute that occurs in both schemas– Alternatively, use Rename operator and rename

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Example

A B

1 2

3 4

B C

2 5

4 7

D

6

8

9 10 11

x

A R.BS.B C D

R S

1 2 2 5 6

1 2 4 7 8

1 2 9 10 11

3 4

3 4

3 4

2 5 6

4 7 8

9 10 11

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Join Operations

Natural Join (binary)• R join S R S • Match only those tuples from R and S

that agree in whatever attributes are common to the schemas of R and S– If r and s from r(R) and s(S) are

successfully paired, result is called a joined tuple

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Example

A B

1 2

3 4

B C

2 5

4 7

D

6

8

9 10 11

join

• Resulting schema has attributes from R, either R or S (i.e., joining attribute(s)), and S

• Tuples that fail to pair with any tuple of the other relation are called dangling tuples

A B C D

R S

1 2 5 6

3 4 7 8

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Join Operations

Theta Join (binary)

• R joinC S, where C is an arbitrary join condition

– R S is an arbitrary join condition

• Step 1: take the product of R and S• Step 2: Select from the product only those tuples

that satisfy condition C• As with the product operation, the schema for

the result is the union of the schemas of R and S natural join - join based on common attributes

R S = R x S

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Example

B C

2 3

2 3

D

4

5

7 8 10

joinA<D AND U.BV.B

A B

1 2

6 7

C

3

8

9 7 8

U V

V.BV.C DA U.BU.C

1 2 3 7 8 10

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Combining Operations to Form Queries

• Form expressions of arbitrary complexity by applying operators either to – given relations or – to relations that are the result of applying one

or more relational operators to relations

• Single relational expression, using parenthesis as necessary

• Apply only one operator at a time and create intermediate result relations

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Complete Set of Relational Algebra Expressions

select

project

union

set difference

cartesian product

• Any of the other operators can be expressed as a sequence of operations from this set

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Sample Schema for Exercises

Student(ID, name, address, GPA, SAT)

Campus(location, enrollment, rank)

Apply(ID, location, date, major, decision)

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Sample Queries

PROJECT_{date,decision} (Apply)

Campus X Apply

Find Names and addresses of all students with GPA > 3.7 who applied to CS major and were rejected

List name and address of all students with GPA > 3.7