Relational Algebra – Basis for Relational Query

LanguagesBased on presentation by Juliana Freire

Formal relational query languages

Is this the Algebra you know?Algebra -> operators and atomic operandsExpressions -> applying operators to atomic operands and/or other

expressions

Algebra of arithmetic: operands are variables and constants, and operators are the usual arithmetic operators

E.g., (x+y)*2 or ((x+7)/(y-3)) + x

Relational algebra: operands are variables that stand for relations and relations (sets of tuples), and operations include union, intersection, selection, projection, Cartesian product, etc

– E.g., (π c-ownerChecking-account) ∩ (π s-ownerSavings-account)

What is a query?A query is applied to relation instances, and the result ofa query is also a relation instance. (view, query)– Schemas of input and output fixed, but instances not. • Operators refer to relation attributes by position or name:– E.g., Account(number, owner, balance, type)

– Positional notation easier for formal definitions, named-fieldnotation more readable.– Both used in SQL

Relational Algebra OperationsThe usual set operations: union, intersection, difference

•Operations that remove parts of relations: selection, projection

•Operations that combine tuples from two relations: Cartesian product, join

•Since each operation returns a relation, operations can be composed!

Removing Parts of Relations

• Selection – rows

• Projection - columns

Selection: Example

Another selection

Example of Projection

Projection removes duplicates

Set Operations

• Union

• Intersection

• Difference

What happens when sets unite?

Union Operation – Example• Relations r, s:

r s:

A B

1

2

1

A B

2

3

rs

A B

1

2

1

3

Union Example

Union Compatibility

Intersection

Set Difference Operation – Example

• Relations r, s:

r – s:

A B

1

2

1

A B

2

3

rs

A B

1

1

Difference

Another way to show intersection?

Summary so far:

• E1 U E2 : union

• E1 - E2 : difference

• E1 x E2 : cartesian product

c(E1) : select rows, c = condition (book has p for predicate)

• IIs(E1) : project columns : s =selected columns

x(c1,c2) (E1) : rename, x is new name of E1, c1 is new name of

column

Combining Tuples of Two Relations

• Cross product (Cartesian product)

• Joins

Cartesian-Product Operation – Example

Relations r, s:

r x s:

A B

1

2

A B

11112222

C D

1010201010102010

E

aabbaabb

C D

10102010

E

aabbr

s

Cross Product Example

Cross Product

Renaming operator:

Rename whole relation: Teacher X secondteacher(Teacher)

Teacher.t-num, Teacher.t-name, secondteacher.t-num, secondteacher.t-name

OR rename attribute before combining:

Teacher X secondteacher(t-num2, t-name2)(Teacher)

t-num, t-name, t-num2, t-name2

OR rename after combining

c(t-num1, t-name1, t-num2, t-name2)(Teacher X Teacher)

t-num1, t-name1, t-num2, t-name2

• How to resolve????

Join: Example

Join : Example

Condition Join

Equi and Natural Join

Divide operator

Divide Operation

Divide Definition

When to divide?

Division Example

Dividing without division sign

Working out an example

Assignment operation

Why would we use Relational Algebra?????

Equivalencies help

ER vs RA

• Both ER and the Relational Model can be used to model the structure of a database.

• Why is it the case that there are only Relational Databases and no ER databases?

RA vs Full Programming Language

• Relational Algebra is not Turing complete. There are operations that cannot be expressed in relational algebra.

• What is the advantage of using this language to query a database?

Summary of Operators updated

• Summary so far:• E1 U E2 : union• E1 - E2 : difference• E1 x E2 : cartesian product c(E1) : select rows, c = condition (book has p for predicate)

• IIs(E1) : project columns : s =selected columns

x(c1,c2) (E1) : rename, x is new name of E1, c1 is new name of column

• E1 E2 : division

• E1 E2 : join, c = match condition

Practice• Find names of stars who’ve appeared in a 1994

movie

• Information about movie year available in Movies; so need an extra join:σyear=1994(πname(Stars AppearIn Movies))⋈ ⋈

• An even more efficient solution:πname(Stars ⋈πname(AppearIn (π⋈ movieIdσyear=1994(Movies)))A query optimizer can find this, given the first solution!

• A more efficient solution:πname(Stars AppearIn (σ⋈ ⋈ year=1994( Movies))

Extended Relational Algebra Operations

• Generalized projection

• Outer join

• Aggregate functions

Generalized projection – calculate fields

Aggregate Operation – Example• Relation r:

A B

C

7

7

3

10

g sum(c) (r) sum(c )

27

Aggregate

• Functions on more than one tuple

• Samples:– Sum– Count-distinct– Max– Min– Count– Avg

• Use “as” to renamebranchname g sum(balance) as totalbalance (account)

Aggregate Operation – Example

• Relation account grouped by branch-name:

branch_name g sum(balance) (account)

branch_name account_number balance

PerryridgePerryridgeBrightonBrightonRedwood

A-102A-201A-217A-215A-222

400900750750700

branch_name sum(balance)

PerryridgeBrightonRedwood

13001500700

Outer Join

• Keep the outer side even if no join

• Fill in missing fields with nulls

Outer Join – Example• Relation loan

Relation borrower

customer_name loan_number

JonesSmithHayes

L-170L-230L-155

300040001700

loan_number amount

L-170L-230L-260

branch_name

DowntownRedwoodPerryridge

Outer Join – Example• Inner Join

loan Borrower

loan_number amount

L-170L-230

30004000

customer_name

JonesSmith

branch_name

DowntownRedwood

JonesSmithnull

loan_number amount

L-170L-230L-260

300040001700

customer_namebranch_name

DowntownRedwoodPerryridge

Left Outer Join

loan Borrower

Outer Join – Example

loan_number amount

L-170L-230L-155

30004000null

customer_name

JonesSmithHayes

branch_name

DowntownRedwoodnull

loan_number amount

L-170L-230L-260L-155

300040001700null

customer_name

JonesSmithnullHayes

branch_name

DowntownRedwoodPerryridgenull

Full Outer Join

loan borrower

Right Outer Join

loan borrower

Summary of Operators - Full• E1 U E2 : union

• E1 - E2 : difference

• E1 x E2 : cartesian product

c(E1) : select rows, c = condition (book has p for predicate)

• IIs(E1) : project columns : s =selected columns separated by commas,

can have calculations included x(c1,c2) (E1) : rename, x is new name of E1, c1 is new name of column

• E1 E2 : division

• E1 E2 : join, c = match condition

• E1 E2 : outer join, c = match condition, keep the side with the arrows

: assignment – give a new name to an expression to make it easy to read

• as : rename a calculated column

• attribute1g function (attribute2) (E1) : perform function on attribute2 whenever

attribute1 changes