Relational Algebra

Relational Query Languages

} Theoretical QLs give semantics to Practical QLs

CSCI1270, Lecture 2

Recall: Query = “Retrieval Program”

Theoretical:1. Relational Algebra2. Relational Calculus

a. Tuple Relational Calculus (TRC)b. Domain Relational Calculus (DRC)

Practical:1. SQL (originally: SEQUEL from System R)2. Quel (used in Ingres)3. Datalog (Prolog-like – used in research lab systems)

Language Examples:

Relational Algebra• Basic Operators

1. select ( σ )2. project ( p )3. union ( È )4. set difference ( – )5. cartesian product ( ´ )6. rename ( ρ )

• Closure Property

CSCI1270, Lecture 2

Relational Operator

Relation

Relation

Relation

Relational Operator

Select ( σ )

CSCI1270, Lecture 2

Notation: σpredicate (Relation)Relation: Can be name of table or result of another queryPredicate:

Select rows from a relation based on a predicate

2. Complex • predicate AND predicate• predicate OR predicate• NOT predicate

Idea:

1. Simple• attribute1 = attribute2• attribute = constant value (also: ≠, <, >, ≤, ≥)

Bank Database

CSCI1270, Lecture 2

Account

bname acct_no balanceDowntown

MianusPerryR.H.

BrightonRedwoodBrighton

A-101A-215A-102A-305A-201A-222A-217

500700400350900700750

Depositor

cname acct_noJohnsonSmithHayesTurner

JohnsonJones

Lindsay

A-101A-215A-102A-305A-201A-217A-222

Customer

cname cstreet ccityJonesSmithHayesCurry

LindsayTurner

WilliamsAdamsJohnsonGlennBrooksGreen

MainNorthMainNorthPark

PutnamNassauSpringAlma

Sand HillSenatorWalnut

HarrisonRye

HarrisonRye

PittsfieldStanfordPrincetonPittsfieldPalo AltoWoodsideBrooklynStanford

Branch

bname bcity assetsDowntownRedwood

PerryMianus

R.H.Pownel

N. TownBrighton

BrooklynPalo AltoHorseneckHorseneckHorseneckBennington

RyeBrooklyn

9M2.1M1.7M0.4M8M

0.3M3.7M7.1M

Borrower

cname lnoJonesSmithHayes

JacksonCurrySmith

WilliamsAdams

L-17L-23L-15L-14L-93L-11L-17L-16

Loan

bname lno amtDowntownRedwood

PerryDowntown

MianusR.H.Perry

L-17L-23L-15L-14L-93L-11L-16

1000200015001500500900

1300

Select ( σ )

CSCI1270, Lecture 2

bname bcity assetsDowntownBrighton

BrooklynBrooklyn

9M7.1M

bname bcity assetsDowntown Brooklyn 9M

σ bcity = “Brooklyn” (branch) =

σ assets > $8M (σ bcity = “Brooklyn” (branch)) =

Notation: σpredicate (Relation)

Project ( p )

cstreet ccityMainNorthPark

PutnamNassauSpringAlma

Sand HillSenatorWalnut

HarrisonRye

PittsfieldStanfordPrincetonPittsfieldPalo AltoWoodsideBrooklynStanford

CSCI1270, Lecture 2

Notation: pA1, …, An (Relation)• Relation: name of a table or result of another query• Each Ai is an attribute • Idea: p selects columns (vs. σ which selects rows)

p cstreet, ccity (customer) =

Project ( p )

bcityBrooklyn

Horseneck

CSCI1270, Lecture 2

p bcity (σassets > 5M (branch)) =

Question: Does the result of Project always have the same numberof tuples as its input?

Union ( È )

cnameJohnsonSmithHayesTurnerJones

LindsayJacksonCurry

WilliamsAdams

CSCI1270, Lecture 2

Notation: Relation1 È Relation2

Example:

(p cname (depositor)) È (p cname (borrower)) =

R È S valid only if:

1. R, S have same number of columns (arity)2. R, S corresponding columns have same name and domain

(compatibility)

Depositor

cname acct_noBorrower

cname lno

Schema:

Set Difference ( – )

bname lno amountDowntownRedwood

PerryDowntown

Perry

L-17L-23L-15L-14L-16

100020001500500300

Notation: Relation1 - Relation2

R - S valid only if:

1. R, S have same number of columns (arity)2. R, S corresponding columns have same domain (compatibility)

Example:(p bname (σamount ≥ 1000 (loan))) – (p bname (σ balance < 800 (account))) =

bname acct_no balanceMianus

BrightonRedwoodBrighton

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

700900700850

CSCI1270, Lecture 2

bnameDowntown

Perry

What About Intersection?

Remember:R ⋂ S = R – (R – S)

SR

R – S

Cartesian Product ( ´ )

depositor.cname

acct_no borrower.cname

lno

JohnsonJohnsonJohnsonJohnsonJohnsonJohnsonJohnsonJohnsonSmith

…

A-101A-101A-101A-101A-101A-101A-101A-101A-215

…

JonesSmithHayes

JacksonCurrySmith

WilliamsAdamsJones

…

L-17L-23L-15L-14L-93L-11L-17L-16L-17…

CSCI1270, Lecture 2

Notation: Relation1 ´ Relation2

Example:

R ´ S like cross product for mathematical relations:• every tuple of R appended to every tuple of S• flattened!!!

depositor ´ borrower =

How many tuples in the result?

A: 56

Rename ( ρ )

CSCI1270, Lecture 2

Notation: r identifier (Relation)renames a relation, or

Notation: r identifier0 (identifier1, …, identifiern) (Relation)renames relation and columns of n-column relation

Use:massage relations to make È, – valid, or ´ more readable

Rename ( ρ )

CSCI1270, Lecture 2

Notation: r identifier0 (identifier1, …, identifiern) (Relation)

Example:

r result (dcname, acctno, bcname, lno) (depositor ´ borrower) =

dccname acctno bcname lno

JohnsonJohnsonJohnsonJohnsonJohnsonJohnsonJohnsonJohnsonSmith

…

A-101A-101A-101A-101A-101A-101A-101A-101A-215

…

JonesSmithHayes

JacksonCurrySmith

WilliamsAdamsJones

…

L-17L-23L-15L-14L-93L-11L-17L-16L-17…

result

Example Query in RA•Determine lno for loans that are for an amount that is larger than the amount of some other loan. (i.e. lno for all non-minimal loans)

CSCI1270, Fall 2008, Lecture 2

Temp1 ß …Temp2 ß … Temp1 ……

Can do in steps:

Example Query in RA

CSCI1270, Lecture 2

lno amtL-17L-23L-15L-14L-93L-11L-16

10002000150015005009001300

1. Find the base data we need

Temp1 ß p lno,amt (loan)

2. Make a copy of (1)

Temp2 ß ρ Temp2 (lno2,amt2) (Temp1) lno2 amt2L-17L-23L-15L-14L-93L-11L-16

10002000150015005009001300

Example Query in RA

CSCI1270, Lecture 2

lno amt lno2 amt2L-17L-17…

L-17L-23L-23…

L-23…

10001000…

100020002000…

2000…

L-17L-23…

L-16L-17L-23…

L-16…

10002000…

130010002000…

1300…

3. Take the cartesian product of 1 and 2

Temp3 ß Temp1 ´ Temp2

Example Query in RA

• p lno (σamt > amt2 (p lno,amt (loan) ´ (ρTemp2 (lno2,amt2) (p lno,amt (loan)))))

CSCI1270, Fall 2008, Lecture 2

4. Select non-minimal loans

Temp4 ß σamt > amt2 (Temp3)

5. Project on lno

Result ß p lno (Temp4)

… or, if you prefer…

ReviewTheoretical Query Languages

CSCI1270, Fall 2008, Lecture 2

1. SELECT ( σ )2. PROJECT ( π )3. UNION ( È )4. SET DIFFERENCE ( – )5. CARTESIAN PRODUCT ( ´ )6. RENAME ( ρ )

Relational Algebra

• Relational algebra gives semantics to practical query languages• Above set: minimal relational algebra

à will now look at some redundant (but useful!) operators

Review

CSCI1270, Lecture 2

Find the names of customers who have both accounts and loans

T1 ß ρT1 (cname2, lno) (borrower)

T2 ß depositor ´ T1

T3 ß σcname = cname2 (T2)Result ß π cname (T3)

Above sequence of operators (ρ, ´, σ) very common.

Express the following query in the RA:

Motivates additional (redundant) RA operators.

Relational AlgebraAdditional Operators

6. Update ( ß ) (we’ve already been using this)

CSCI1270, Lecture 2

÷2. Division ( )

3. Generalized Projection (π)4. Aggregation

!"1. Natural Join ( )

!" !" !"5. Outer Joins ( )

• 1&2 Redundant: Can be expressed in terms of minimal RAe.g. depositor borrower =

π …(σ…(depositor ´ ρ…(borrower)))

• 3 – 6 Added for extra power

!"

Natural Join

CSCI1270, Lecture 2

Idea: combines ρ, ´, σ

A B C D1223

αααβ

+--+

10102010

E B D‘a’‘a’‘b’‘c’

ααββ

10201010

r s

A B C D E12233

αααΒβ

+--++

1010201010

‘a’‘a’‘a’‘b’‘c’

=

Relation1 Relation2Notation:

!"

πcname,acct_no,lno (σcname=cname2 (depositor ´ ρt(cname2,lno) (borrower)))≡

!"depositor borrower

!"

Division

CSCI1270, Lecture 2

A Bαααβγγγγδδ

1231134612

B12

r

s Aαδ

=÷

Query: Find values for A in r which have corresponding Bvalues for all B values in s

Relation1 Relation2Notation:

Idea: expresses “for all” queries

÷

Division

CSCI1270, Lecture 2

A Bαααβγγγγδδ

1231134612

B12

r

s Aαδ

=

17 ÷ 3 = 5

÷

The largest value of i such that: i ´ 3 ≤ 17

t

Relational Division

The largest value of t such that:( t ´ s Í r )

Another way to look at it: and ÷ ´

Division

CSCI1270, Lecture 2

A B C D Eαααββγγγ

aaaaaaaa

αγγγγγγβ

aabababb

11113111

D E

ab

11

r

s A B Cαγ

aa

γγ

=÷ t

r ÷ s = t such that s X t Í r

A More Complex Example(a,1) and (b,1), i.e., relation Sare paired with same prefix

which thus goes in the answer.

Division Adds No Power(power = set of functions that can be expressed)

Definition in terms of the basic algebra operationLet r(R) and s(S) be relations, and let S Í R

r ÷ s = ÕR-S (r) – ÕR-S ( (ÕR-S (r) x s) – ÕR-S,S(r))

To see why– ÕR-S,S(r) simply reorders attributes of r

– ÕR-S(ÕR-S (r) x s) – ÕR-S,S(r)) gives those tuples t in

ÕR-S (r) such that for some tuple u Î s, tu Ï r.

Generalized Projection

CSCI1270, Lecture 2

p e1,…,en(Relation)

e1,…,en can include arithmetic expressions – not just attributes

cname limit balanceJonesTurner

50003000

20002500

credit =

π cname, limit - balance (credit) = cname limit-balanceJonesTurner

3000500

Notation:

Example

Then…

Aggregate Functions and Operations} An aggregate 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 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

(can be empty)◦ Each Fi is an aggregate function◦ Each Ai is an attribute name

CSCI1270, Lecture 2

Aggregate Operation – Example} Relation r:

CSCI1270, Lecture 2

A B

aabb

abbb

C

77310

g sum(c) (r)sum-C

27No grouping

Aggregate Operation – Example

} Relation account grouped by branch-name:

CSCI1270, Lecture 2

branch-name g sum(balance) (account)

branch-name account-number balancePerryridgePerryridgeBrightonBrightonRedwood

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

400900750750700

branch-name sum(balance)PerryridgeBrightonRedwood

13001500700

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

CSCI1270, Lecture 2

branch-name g sum(balance) as sum-balance (account)

Outer Joins

CSCI1270, Lecture 2

bname lno amtDowntownRedwood

Perry

L-170L-230L-260

300040001700

loan =cname lnoJonesSmithHayes

L-170L-230L-155

borrower =

=bname lno amt cname

DowntownRedwood

L-170L-230

30004000

JonesSmith

Join result loses…à any record of Perryà any record of Hayes

Motivation:

!"loan borrower =

Outer Joins

CSCI1270, Lecture 2

bname lno amtDowntownRedwood

Perry

L-170L-230L-260

300040001700

loan =cname lnoJonesSmithHayes

L-170L-230L-155

borrower =

bname lno amt cnameDowntownRedwood

Perry

L-170L-230L-260

300040001700

JonesSmith

┴

• preserves all tuples in left relation1. Left Outer Join ( )!"

┴ = NULL

loan borrower =!"

Outer Joins

CSCI1270, Fall 2008, Lecture 2

bname lno amt cnameDowntownRedwood

┴

L-170L-230L-155

30004000

┴

JonesSmithHayes

!"

bname lno amtDowntownRedwood

Perry

L-170L-230L-260

300040001700

loan =cname lnoJonesSmithHayes

L-170L-230L-155

borrower =

• preserves all tuples in right relation2. Right Outer Join ( )

┴ = NULL

loan borrower =!"

Outer Joins

CSCI1270, Fall 2008, Lecture 2

bname lno amtDowntownRedwood

Perry

L-170L-230L-260

300040001700

loan =cname lnoJonesSmithHayes

L-170L-230L-155

borrower =

• preserves all tuples in both relations3. Full Outer Join ( )

┴ = NULL

!"

bname lno amt cnameDowntownRedwood

Perry┴

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

300040001700

┴

JonesSmith

┴Hayes

loan borrower =!"

Update

CSCI1270, Lecture 2

1. Deletion: r ß r – s e.g., account ß account – σbname=Perry (account)(deletes all Perry accounts)

2. Insertion: r ß r È se.g., branch ß branch È {(Waltham, Boston, 7M)}(inserts new branch with bname = Waltham, bcity = Boston, assets = 7M)

3. Update: r ß πe1,…,en (r)

e.g., depositor ß depositor È (ρtemp (cname,acct_no) (borrower))(adds all borrowers to depositors, treating lno’s as acct_no’s)

e.g., account ß πbname,acct_no,bal*1.05 (account)(adds 5% interest to account balances)

Identifier ß QueryNotation: Common Uses:

Views• Limited access to DB.• Tailored schema

• 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 describedas:

Õcustomer-name, loan-number (borrower loan)

• A relation that is made visible to a user as a “virtual relation” is called a view.

View Definition• A view is defined using the create view statement which has the

formcreate view v as <query expression>

where <query expression> is any legal relational algebra query expression. The view name given as 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 Rather, a view definition causes the saving of an expression to be substituted into queries using the view.

View Examples

• Consider the view (named all-customer) consisting of branches and their customers.

create view all-customer asÕbranch-name, customer-name (depositor account)ÈÕbranch-name, customer-name (borrower loan)

• We can find all customers of the Perryridgebranch by writing:Õcustomer-name

(sbranch-name = “Perryridge” (all-customer))

Updates Through View• Database modifications expressed as views must be translated to

modifications of the actual relations in the database.• Consider the person who needs to see all loan data in the loan

relation except amount. The view given to the person, branch-loan, is defined as:

create view branch-loan asÕbranch-name, loan-number (loan)

• Since we allow a view name to appear wherever a relation name is allowed, the person may write:

branch-loan ¬ branch-loan È {(“Perryridge”, L-37)}

Updates Through Views (Cont.)• The previous insertion must be represented

by an insertion into the actual relation loanfrom which the view branch-loan is constructed.

• An insertion into loan requires a value for amount. The insertion can be dealt with by either.– rejecting the insertion and returning an error

message to the user.– inserting a tuple (“L-37”, “Perryridge”, null) into

the loan relation

Updates Through Views (Cont.)

• Some updates through views are impossible to translate.create view v as sbranch-name = “Perryridge” (account))

v ¬ v È (L-99, Downtown, 23)• Others cannot be translated uniquely

all-customer ¬ all-customer È (Perryridge, John)• Have to choose loan or account, and

create a new loan/account number!

Views Defined Using Other Views

• One view may be used in the expression defining another view

• A view relation v1 is said to depend directly on a view relation v2 if v2 is used in the expression defining v1

• A view relation v1 is said to depend on view relation v2 if either v1 depends directly on v2 orthere is a path of dependencies from v1 to v2

View Expansion

• Let view v1 be defined by an expression e1 that may itself contain uses of view relations.

• View expansion of an expression repeats the following replacement step:repeat

Find any view relation vi in e1

Replace the view relation vi by the expression defining vi

until no more view relations are present in e1

• As long as the view definitions are not recursive, this loop will terminate