relational algebra...relational algebra basic structure – consider the account table. – it has...

Relational Algebra

Relational

Algebra

7/4/2017 1Md. Golam Moazzam, Dept. of CSE, JU

Relational Algebra

Structure of Relational Databases

– A relational database consists of a collection of tables, each of which is

assigned a unique name.

– A row in a table represents a relationship among a set of values. Since

a table is a collection of such relationships, there is a close

correspondence between the concept of table and the mathematical

concept of relation, from which the relational data model takes its

name.

– Example of a Relation:


Relational Algebra

Basic Structure

– Consider the account table.

– It has three column headers: account-number, branch-name, and

balance.

– For each attribute, there is a set of permitted values, called the domain

of that attribute.

– For the attribute branch-name, for example, the domain is the set of all

branch names.

– Let D1 denote the set of all account numbers,

D2 the set of all branch names, and

D3 the set of all balances.

Any row of account must consist of a 3-tuple (v1, v2, v3), where v1 is an

account number (that is, v1 is in domain D1), v2 is a branch name (that

is, v2 is in domain D2), and v3 is a balance (that is, v3 is in domain D3).


Relational Algebra

Tuple Variable

– A tuple variable is a variable that stands for a tuple.

– In other words, a tuple variable is a variable whose domain is

the set of all tuples.

– In the account relation, there are seven tuples.

– Let the tuple variable t refer to the first tuple of the relation.

– Thus, t[account-number] = “A-101,” and t[branch-name] =

“Downtown”.

– Alternatively, we may write t[1] to denote the value of tuple t

on the first attribute (account-number), t[2] to denote branch-

name, and so on.

– Since a relation is a set of tuples, we use the mathematical

notation of t ∈ r to denote that tuple t is in relation r.


Relational Algebra

Attribute Types

– Each attribute of a relation has a name.

– The set of allowed values for each attribute is called the

domain of the attribute.

– Attribute values are (normally) required to be atomic,

that is, indivisible.

• E.g. multivalued attribute values are not atomic.

• E.g. composite attribute values are not atomic.

– The special value null is a member of every domain.

– The null value causes complications in the definition of

many operations.


Relational Algebra

Query Languages

– A query language is a language in which a user requests

information from the database.

– These languages are usually on a level higher than that of a

standard programming language.

– Query languages can be categorized as either procedural or

nonprocedural.

– In a procedural language, the user instructs the system to

perform a sequence of operations on the database to compute

the desired result.

– In a nonprocedural language, the user describes the desired

information without giving a specific procedure for obtaining

that information.

– Very widely used query language is SQL.

– Some other query languages: QBE and Datalog.


Relational Algebra

Relational Algebra

– Relational algebra is a procedural query language.

– It consists of a set of operations that take one or two relations

as input and produce a new relation as their result.

– The fundamental operations in the relational algebra are select,

project, union, set difference, Cartesian product, and rename.

– select, project, and rename operations are called unary

operations, because they operate on one relation.

– The other three operations operate on pairs of relations and are,

therefore, called binary operations.

– There are several other operations—namely, set intersection,

natural join, division, and assignment.


Relational Algebra

Consider the following database


branch(branch- name, branch-city, assets)

customer(customer-name, customer-street, customer-city)

loan(loan-number, branch-name, amount)

borrower(customer-name, loan-number)

account(account-number, branch-name, balance)

depositor(customer-name, account-number)

Relational Algebra

Select Operation

– The select operation selects tuples that satisfy a given

predicate.

– The lowercase Greek letter sigma (σ) used to denote

selection.

– The predicate appears as a subscript to σ.

– The argument relation is in parentheses after the σ.

– Notation: p(r)

p is called the selection predicate.


Relational Algebra

Select Operation

– Example:

“Select those tuples of the loan relation where the branch is

Perryridge”.

σbranch-name =“Perryridge” (loan)

“Find all tuples in which the amount lent is more than $1200”.

σamount>1200 (loan)

Allow comparisons using =, ≠, <, ≤, >, ≥ in the selection

predicate.

We can combine several predicates by using the connectives:

and (∧), or (∨), and not (￢).


Relational Algebra

Select Operation

– “Find those tuples pertaining to loans of more than $1200

made by the Perryridge branch”.

σbranch-name =“Perryridge”∧ amount>1200 (loan)


Relational Algebra

Project Operation

– Project operation is a unary operation that returns its

argument relation, with certain attributes left out.

– Since a relation is a set, any duplicate rows are

eliminated.

– Projection is denoted by the uppercase Greek letter pi

(Π).

– We list those attributes that we wish to appear in the

result as a subscript to Π.

– The argument relation follows in parentheses.

– Notation:

A1, A2, …, Ak (r)

– where A1, A2 are attribute names and r is a relation name.


Relational Algebra

Project Operation

– Example:

“List all loan numbers and the amount of the loan”.

Πloan-number, amount(loan)

– “Find those customers who live in Harrison.”

Πcustomer-name (σcustomer-city =“Harrison” (customer))


Relational Algebra

Union Operation

– Notation: r s

– Defined as:

r s = {t | t r or t s}

– For r s to be valid:

r, s must have the same number of attributes.

Attribute domains must be compatible.

– Example:

“Find the names of all bank customers who have either an

account or a loan or both”.

Πcustomer-name (borrower ) ∪ Πcustomer-name (depositor)

– Since relations are sets, duplicate values are eliminated.


Relational Algebra

Set Difference Operation

– Set-difference operation, denoted by −, allows us to find tuples

that are in one relation but are not in another.

– Notation: r – s

– Defined as:

r – s = {t | t r and t s}

– Set differences must be taken between compatible relations.

r and s must have the same number of attributes.

Attribute domains of r and s must be compatible.

– Example:

“Find all customers of the bank who have an account but not a

loan”.

Πcustomer-name (depositor) − Πcustomer-name (borrower )


Relational Algebra

Cartesian-Product Operation

– Cartesian-product operation, denoted by a cross (×),

allows us to combine information from any two relations.

– Notation: r x s

– Defined as:

r x s = {t q | t r and q s}

– Assume that attributes of r(R) and s(S) are disjoint.

(That is, R S = ).

– If attributes of r(R) and s(S) are not disjoint, then

renaming must be used.


Relational Algebra


– Example: Relations r, s:

– Thus, r x s:


A B

1

2

C D

10

10

20

10

E

a

a

b

b

r s

A B

1

1

1

1

2

2

2

2

C D

10

10

20

10

10

10

20

10

E

a

a

b

b

a

a

b

b

Relational Algebra


– Example: A=C(r x s)

– r = borrower × loan is

(customer-name, borrower.loan-number, loan.loan-

number, branch-name, amount)

– “Find the names of all customers who have a loan at the

Perryridge branch”.

Πcustomer-name (σborrower .loan-number =loan.loan-number (σbranch-name

=“Perryridge” (borrower × loan)))


A B C D E

1

2

2

10

20

20

a

a

b

Relational Algebra

Rename Operation

– rename operator, denoted by the lowercase Greek letter

rho (ρ).

– Notation: ρx(E)

returns the result of expression E under the name x.

– A second form of the rename operation is as follows.

ρx(A1,A2,...,An) (E)

returns the result of expression E under the name x, and

with the attributes renamed to A1,A2, . . .,An.


Relational Algebra

Rename Operation

– Example:

“Find the largest account balance in the bank.”

Πbalance (account) − Πaccount .balance (σaccount .balance < d.balance

(account × ρd (account)))

“Find the names of all customers who live on the same

street and in the same city as Smith.”

Πcustomer.customer-name(σcustomer.customer-street=smith-addr.street ∧

customer.customer-city=smith-addr.city (customer × ρsmith-addr(street,city)

(Πcustomer-street, customer-city(σcustomer-name= “Smith”(customer)))))


Relational Algebra

Set-Intersection Operation

– set intersection denoted by (∩).

– Notation: r s

– Defined as:

r s ={ t | t r and t s }

– Assume:

r and s must have the same number of attributes.

Attribute domains of r and s must be compatible.


Relational Algebra

Set-Intersection Operation

– Relation r, s:

– Thus, r s :

– Example:

“Find all customers who have both a loan and an

account”.

Πcustomer-name (borrower ) ∩ Πcustomer-name (depositor)


A B

1

2

1

A B

2

3

rs

A B

2

Relational Algebra

Natural-Join Operation

– natural join is a binary operation that allows us to

combine certain selections and a Cartesian product into

one operation.

– It is denoted by the “join” symbol :

– natural-join operation forms a Cartesian product of its

two arguments, performs a selection forcing equality on

those attributes that appear in both relation schemas, and

finally removes duplicate attributes.

– Notation: r s

– Let R = (A, B, C, D) and S = (E, B, D)

Result schema = (A, B, C, D, E)

– r s is defined as:

– r.A, r.B, r.C, r.D, s.E (r.B = s.B r.D = s.D (r x s))


Relational Algebra


– Example:

“Find the names of all customers who have a loan at the

bank, along with the loan number and the loan amount.”

– Using Cartesian product:

Πcustomer-name, loan.loan-number, amount

(σborrower .loan-number =loan.loan-number (borrower × loan))

– Using Natural-Join operation:

Πcustomer-name, loan-number, amount (borrower loan)


Relational Algebra


– “Find the names of all branches with customers who

have an account in the bank and who live in Harrison.”

Πbranch-name (σcustomer-city =“Harrison”

(customer account depositor))


Relational Algebra

Division Operation

– division operation, denoted by ÷, is suited to queries that

include the phrase “for all.”

– Example:

“Find all customers who have an account at all the branches

located in Brooklyn.”

r1 = Πbranch-name (σbranch-city =“Brooklyn” (branch))

r2 = Πcustomer-name, branch-name (depositor account)

– Now, find customers who appear in r2 with every branch name

in r1.

Πcustomer-name, branch-name (depositor account)

÷ Πbranch-name (σbranch-city =“Brooklyn” (branch))


Relational Algebra

Assignment Operation

– Assignment operation () provides a convenient way to

express complex queries.

Aggregate Functions

– Aggregate functions take a collection of values as input

and return a single value as a result.

• avg: average value

• min: minimum value

• max: maximum value

• sum: sum of values

• count: number of values


Relational Algebra

Aggregate Functions

– The aggregate function avg returns the average of the

values.

– The aggregate function min returns the minimum value

in a collection.

– The aggregate function max returns the maximum value

in a collection.

– The aggregate function sum returns the sum of the

values.

– The aggregate function count returns the number of the

elements in the collection.


Relational Algebra

Aggregate Functions

– Aggregate operation in relational algebra:

G1, G2, …, Gn g F1( A1), F2( A2),…, Fn( An) (E)

– E is any relational-algebra expression.

– G1, G2 …, Gn is a list of attributes on which to group

(group by).

– Each Fi is an aggregate function.

– Each Ai is an attribute name.


Relational Algebra

Aggregate Functions

– “Find out the total sum of salaries of all the part-time

employees in the bank.”

gsum(salary)(pt-works)

; pt-works(employee-name, branch-name, salary)

– The symbol g is the letter G in calligraphic font; read it

as “calligraphic G.”

– Its subscript specifies the aggregate operation to be

applied.


Relational Algebra

Aggregate Functions

– “Find the number of branches appearing in the pt-works

relation.”

gcount - distinct(branch-name)(pt-works)

– “Find the total salary sum of all part-time employees at

each branch of the bank.”

branch-name gsum(salary)(pt-works)

– The attribute branch-name in the left-hand subscript of G

indicates that the input relation pt-works must be divided

into groups based on the value of branch-name.


Relational Algebra

Outer Join

– outer-join operation is an extension of the join operation

to deal with missing information.

– It avoids loss of information.

– Consider loan and borrower relations:


3000

4000

1700

loan-number amount

L-170

L-230

L-260

branch-name

Downtown

Redwood

Perryridge

customer-name loan-number

Jones

Smith

Hayes

L-170

L-230

L-155

Relational Algebra

Outer Join

– Inner join operation (natural join)

– loan borrower

– We have lost some information. We can use the outer-join

operation to avoid this loss of information.

– There are actually three forms of the outer-join operation:

• left outer join, denoted by ;

• right outer join, denoted by and

• full outer join, denoted by .


loan-number amount

L-170

L-230

3000

4000

customer-name

Jones

Smith

branch-name

Downtown

Redwood

Relational Algebra

Outer Join

– Left-outer join

– loan borrower

– left outer join takes all tuples in the left relation that did

not match with any tuple in the right relation, pads the

tuples with null values for all other attributes from the

right relation, and adds them to the result of the natural

join.


Jones

Smith

null

loan-number amount

L-170

L-230

L-260

3000

4000

1700

customer-namebranch-name

Downtown

Redwood

Perryridge

Relational Algebra

Outer Join

– Right-outer join

– loan borrower

– right outer join is symmetric with the left outer join: It

pads tuples from the right relation that did not match any

from the left relation with nulls and adds them to the

result of the natural join.


loan-number amount

L-170

L-230

L-155

3000

4000

null

customer-name

Jones

Smith

Hayes

branch-name

Downtown

Redwood

null

Relational Algebra

Outer Join

– Full-outer join

– loan borrower

– full outer join does both of those operations, padding

tuples from the left relation that did not match any from

the right relation, as well as tuples from the right relation

that did not match any from the left relation, and adding

them to the result of the join.


loan-number amount

L-170

L-230

L-260

L-155

3000

4000

1700

null

customer-name

Jones

Smith

null

Hayes

branch-name

Downtown

Redwood

Perryridge

null

Relational Algebra

Null Values

– It is possible for tuples to have a null value, denoted by

null, for some of their attributes.

– null signifies an unknown value or that a value does not

exist.

– The result of any arithmetic expression involving null is

null.

– Aggregate functions simply ignore null values.

– For duplicate elimination and grouping, null is treated

like any other value, and two nulls are assumed to be the

same


Relational Algebra

Modification of the Database

– The content of the database may be modified using the

following operations:

• Deletion

• Insertion

• Updating

– All these operations are expressed using the assignment

operator.


Relational Algebra

Deletion

– A delete request is expressed similarly to a query, except

instead of displaying tuples to the user, the selected

tuples are removed from the database.

– We can delete only whole tuples; we cannot delete values

on only particular attributes.

– In relational algebra a deletion is expressed by:

r ← r − E

where r is a relation and E is a relational-algebra query.


Relational Algebra

Deletion

Example:

“Delete all of Smith’s account records.”

depositor ← depositor − σcustomer-name =“Smith” (depositor)

“Delete all loans with amount in the range 0 to 50.”

loan ← loan − σamount≥0 and amount≤50 (loan)

“Delete all accounts at branches located in Needham.”

r1 ← σbranch-city =“Needham” (account branch)

r2 ← Πbranch-name, account -number, balance (r1)

account ← account − r2


Relational Algebra

Insertion

– To insert data into a relation, we either specify a tuple to be

inserted or write a query whose result is a set of tuples to be

inserted.

– The relational algebra expresses an insertion by:

r ← r ∪ E

where r is a relation and E is a relational-algebra expression.

– Example:

“Insert information in the database specifying that Smith has

$1200 in account A-973 at the Perryridge branch.”

account←account ∪ {(A-973, “Perryridge”, 1200)}

depositor←depositor ∪ {(“Smith”, A-973)}


Relational Algebra

Updating

– A mechanism to change a value in a tuple without

changing all values in the tuple.

– Use the generalized-projection operator to do this task:

r ← ΠF1,F2,...,Fn(r)

where each Fi is either the ith attribute of r, if the ith

attribute is not updated, or, if the attribute is to be

updated, Fi is an expression, involving only constants

and the attributes of r, that gives the new value for the

attribute.


Relational Algebra

Updating

– Example:

“Make interest payments by increasing all balances by 5

percent.”

account ← Πaccount-number, branch-name, balance *1.05 (account)

“Pay all accounts with balances over $10,000 6 percent

interest and pay all others 5 percent.”

account ← ΠAN,BN, balance *1.06 (σbalance>10000 (account))

∪ ΠAN, BN balance *1.05 (σbalance≤10000 (account))

where the abbreviations AN and BN stand for account-

number and branch-name, respectively.


Relational Algebra

Views

– In some cases, it is not desirable for all users to see the

entire logical model (i.e., all the actual relations stored in

the database.)

– Consider a person who needs to know a customer’s loan

number but has no need to see the loan amount. This

person should see a relation described, in the relational

algebra, by

customer-name, loan-number (borrower loan)

Any relation that is not of the conceptual model but is

made visible to a user as a “virtual relation” is called a

view.


Relational Algebra

View definition

– A view is defined using the create view statement which

has the form:

create view v as <query expression>

where <query expression> is any legal relational algebra

query expression. The view name is represented by v.

– Once a view is defined, the view name can be used to

refer to the virtual relation that the view generates.

– View definition is not the same as creating a new relation

by evaluating the query expression.


Relational Algebra

View definition

– Example:

Create a view (named all-customer) consisting of

branches and their customers.


create view all-customer as

branch-name, customer-name (depositor account)

branch-name, customer-name (borrower loan)

relational algebra...relational algebra basic structure – consider the account table. – it has...

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