ASET

Relational Algebra continues…

ASETRename Operation

• Allows us to name, and therefore to refer to, the results of relational-algebra expressions.

• Allows us to refer to a relation by more than one name.• Example:

x (E)

returns the expression E under the name X• If a relational-algebra expression E has arity n, then

returns the result of expression E under the name X, and with theattributes renamed to A1 , A2 , …., An .

)(),...,,( 21E

nAAAx

ASETComposition of Operations

• Can build expressions using multiple operations• Example: A=C(r x s)

• r x s

• A=C(r x s)

A B

11112222

C D

1010201010102010

E

aabbaabb

A B C D E

122

101020

aab

ASET

Banking Examplebranch (branch_name, branch_city, assets)

customer (customer_name, customer_street, customer_city)

account (account_number, branch_name, balance)

loan (loan_number, branch_name, amount)

depositor (customer_name, account_number)

borrower (customer_name, loan_number)

ASETExample Queries

• Find all loans of over $1200

Find the loan number for each loan of an amount greater than $1200

amount > 1200 (loan)

loan_number (amount > 1200 (loan))

Find the names of all customers who have a loan, an account, or both, from the bank

customer_name (borrower) customer_name (depositor)

ASETExample Queries

• Find the names of all customers who have a loan at the Perryridge branch.

Find the names of all customers who have a loan at the Perryridge branch but do not have an account at any branch of the bank.customer_name (branch_name = “Perryridge”

(borrower.loan_number = loan.loan_number(borrower x loan))) –

customer_name(depositor)

customer_name (branch_name=“Perryridge”

(borrower.loan_number = loan.loan_number(borrower x loan)))

ASETAdditional Operations

• Additional Operations

– Set intersection– Natural join– Aggregation– Outer Join– Division

• All above, other than aggregation, can be expressed using basic operations we have seen earlier

ASETSet-Intersection Operation –

Example• Relation r, s:

• r s

A B

121

A B

23

r s

A B

2

ASETNatural Join Operation – Example• Relations r, s:

A B

12412

C D

aabab

B

13123

D

aaabb

E

r

A B

11112

C D

aaaab

E

s

r s

ASET

Notation: r s

Natural-Join Operation

• Let r and s be relations on schemas R and S respectively. Then, r s is a relation on schema R S obtained as follows:

– Consider each pair of tuples tr from r and ts from s.

– If tr and ts have the same value on each of the attributes in R S, add a tuple t to the result, where

• t has the same value as tr on r

• t has the same value as ts on s

• Example:R = (A, B, C, D)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))

ASETAggregate Functions and Operations• Aggregation function takes a collection of values and returns a single value as

a result.avg: average valuemin: minimum valuemax: maximum valuesum: sum of valuescount: number of values

• Aggregate operation in relational algebra

E is any relational-algebra expression– G1, G2 …, Gn is a list of attributes on which to group

– Each Fi is an aggregate function

– Each Ai is an attribute name

)()(,,(),(,,, 221121E

nnn AFAFAFGGG

ASETAggregate 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

ASET

Aggregate Functions (Cont.)• Result of aggregation does not have a name

– Can use rename operation to give it a name– For convenience, we permit renaming as part of aggregate operation

branch_name g sum(balance) as sum_balance (account)

ASET

Q1. Consider the following relational database, where the primary keys are

underlined. Give an expression in the relational algebra to express each of the

following queries:

employee( person_name, street, city)

Works(person_name, company_name, salary)

company( company_name, city)

manages(person_name, manager_name)