Lecture 3: Relational Algebra and SQL Tuesday, March 25, 2008

Lecture 3: Relational Algebra and SQL

Tuesday, March 25, 2008

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Outline

• Relational Algebra: 4.2 (except 4.2.5)

• SQL: 5.2, 5.3, 5.4

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Querying the Database

• Goal: specify what we want from our database – Find all the employees who earn more than $50,000

and pay taxes in New Jersey.

• Could write in C++/Java, but bad idea• Instead use high-level query languages:

– Theoretical: Relational Algebra, Datalog– Practical: SQL

• Relational algebra: a basic set of operations on relations that provide the basic principles.

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Instances of Branch and Staff (part) Relations

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Relational Algebra• Operators: relations as input, new relation as output • Five basic RA operators:

– Set Operators• union, difference • Selection:

– Projection: – Cartesian Product: X

• Derived operators:– Intersection, complement– Joins (natural,equi-join, theta join, semi-join)

• When our relations have attribute names:– Renaming:

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Set Operations: Union

• Union: all tuples in R1 or R2

• Notation: R1 U R2

• R1, R2 must have the same schema

• R1 U R2 has the same schema as R1, R2

• Example: – ActiveEmployees U RetiredEmployees

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Union• R S

– Union of two relations R and S defines a relation that contains all the tuples of R, or S, or both R and S, duplicate tuples being eliminated.

– R and S must be union-compatible.

• If R and S have I and J tuples, respectively, union is obtained by concatenating them into one relation with a maximum of (I + J) tuples.

Example: List all cities where there is either a branch office or a property for rent.

• Pcity(Branch) union Pcity(PropertyForRent) or R = Branch[city] union PropertyForRent[city]

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Intersection• R S

– Defines a relation consisting of the set of all tuples that are in both R and S.


• Expressed using basic operations:

R S = R – (R – S)

Example: List all cities where there is both a branch office and at least one property for rent.

city(Branch) city(PropertyForRent)

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Difference (Minus)

• R – S– Defines a relation consisting of the tuples that are in relation R, but not in S.


• List all cities where there is a branch office but no properties for rent.

city(Branch) – city(PropertyForRent)

Or R = Branch [city] - PropertyForRent [city]

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Set Operations: Difference

• Difference: all tuples in R1 and not in R2

• Notation: R1 – R2

• R1, R2 must have the same schema

• R1 - R2 has the same schema as R1, R2

• Example– AllEmployees - RetiredEmployees

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Relational Algebra Operations

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Set Operations: Selection

• Returns all tuples which satisfy a condition

• Notation: c(R)

• c is a condition: =, <, >, and, or, not

• Output schema: same as input schema

• Find all employees with salary > $40,000:– Salary > 40000 (Employee)

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Selection Example

EmployeeSSN Name DepartmentID Salary999999999 John 1 30,000777777777 Tony 1 32,000888888888 Alice 2 45,000

SSN Name DepartmentID Salary888888888 Alice 2 45,000

Find all employees with salary more than $40,000.Salary > 40000 (Employee)

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Restriction (or Selection)– Works on a single relation R and defines a relation that contains

only those tuples (rows) of R that satisfy the specified condition (predicate).

Example: List all staff with a salary greater than US$10,000.

salary > 10000 (Staff) or R = STAFF Where Salary > 10000

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Projection• Unary operation: returns certain columns• Eliminates duplicate tuples !• Notation: A1,…,An (R)• Input schema R(B1,…,Bm)• Condition: {A1, …, An} {B1, …, Bm}• Output schema S(A1,…,An)• Example: project social-security number and

names:– SSN, Name (Employee)

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Projection col1, . . . , coln(R)

– Works on a single relation R and defines a relation that contains a vertical subset of R, extracting the values of specified attributes and eliminating duplicates.

• Produce a list of salaries for all staff, showing only staffNo, fName, lName, and salary details.

staffNo, fName, lName, salary(Staff) or Staff [staffNo, fName, lName, salary]

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Projection Example

EmployeeSSN Name DepartmentID Salary999999999 John 1 30,000777777777 Tony 1 32,000888888888 Alice 2 45,000

SSN Name999999999 John777777777 Tony888888888 Alice

SSN, Name (Employee)

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Cartesian Product

• Each tuple in R1 with each tuple in R2• Notation: R1 x R2• Input schemas R1(A1,…,An), R2(B1,…,Bm)• Condition: {A1,…,An} ∩ {B1,…Bm} = • Output schema is S(A1, …, An, B1, …, Bm)• Notation: R1 x R2• Example: Employee x Dependents• Very rare in practice; but joins are very often

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Cartesian Product Example Employee Name SSN John 999999999 Tony 777777777 Dependents EmployeeSSN Dname 999999999 Emily 777777777 Joe Employee x Dependents Name SSN EmployeeSSN Dname John 999999999 999999999 Emily John 999999999 777777777 Joe Tony 777777777 999999999 Emily Tony 777777777 777777777 Joe

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Cartesian Product (Multiplication)• R X S

– Defines a relation that is the concatenation of every tuple of relation R with

every tuple of relation S.• List the names and comments of all clients who have viewed a property for

rent.

(clientNo, fName, lName(Client)) X (clientNo, propertyNo, comment (Viewing)) or

Client [clientNo, fName, lName] x Viewing [clientNo, propertyNo, comment ]

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Example - Cartesian product and Selection

• Use selection operation to extract those tuples where Client.clientNo = Viewing.clientNo.Client.clientNo = Viewing.clientNo((clientNo, fName, lName(Client)) (clientNo, propertyNo,

comment(Viewing)))

Cartesian product and Selection can be reduced to a single operation called a Join.

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Renaming

• Does not change the relational instance

• Changes the relational schema only

• Notation: B1,…,Bn (R)

• Input schema: R(A1, …, An)

• Output schema: S(B1, …, Bn)

• Example:

LastName, SocSocNo (Employee)

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Renaming Example

EmployeeName SSNJohn 999999999Tony 777777777

LastName SocSocNoJohn 999999999Tony 777777777

LastName, SocSocNo (Employee)

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Derived Operations

• Intersection can be derived:– R1 ∩ R2 = R1 – (R1 – R2) – There is another way to express it (later)

• Most importantly: joins, in many variants

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Join• Join is a derivative of Cartesian product.

• Equivalent to performing a Selection, using join predicate as selection formula, over Cartesian product of the two operand relations.

• One of the most difficult operations to implement efficiently in an RDBMS and one reason why RDBMSs have intrinsic performance problems.

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Join• Join is a derivative of Cartesian product.

• Equivalent to performing a Selection, using join predicate as selection formula, over Cartesian product of the two operand relations.

• One of the most difficult operations to implement efficiently in an RDBMS and one reason why RDBMSs have intrinsic performance problems.

• Various forms of join operation– Natural join (defined by Codd)– Outer join – Theta join– Equijoin (a particular type of Theta join)– Semijoin

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Theta join (-join)

• R FS– Defines a relation that contains tuples

satisfying the predicate F from the Cartesian product of R and S.

– The predicate F is of the form R.ai S.bi where may be one of the comparison operators (<, , >, , =, ).

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Theta join (-join)• Can rewrite Theta join using basic Selection

and Cartesian product operations.

R FS = F(R S)

Degree of a Theta join is sum of degrees of the operand relations R and S. If predicate F contains only equality (=), the term Equijoin is used.

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Example - Equijoin • List the names and comments of all clients

who have viewed a property for rent.(clientNo, fName, lName(Client)) Client.clientNo = Viewing.clientNo

(clientNo, propertyNo, comment(Viewing))

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EQUIJOIN

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• SELECT table.column, table.column FROM table1, table2 WHERE table1.column1 = table2.column2;

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Natural Join• Notation: R1 R2• Input Schema: R1(A1, …, An), R2(B1, …, Bm)• Output Schema: S(C1,…,Cp)

– Where {C1, …, Cp} = {A1, …, An} U {B1, …, Bm}

• Meaning: combine all pairs of tuples in R1 and R2 that agree on the attributes:– {A1,…,An} ∩ {B1,…, Bm} (called the join attributes)

• Equivalent to a cross product followed by selection• Example Employee Dependents

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Natural Join Example

EmployeeName SSNJohn 999999999Tony 777777777

DependentsSSN Dname999999999 Emily777777777 Joe

Name SSN DnameJohn 999999999 EmilyTony 777777777 Joe

Employee Dependents = Name, SSN, Dname( SSN=SSN2(Employee x SSN2, Dname(Dependents))

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Natural Join

• List the names and comments of all clients who have viewed a property for rent.(clientNo, fName, lName(Client)) Join (clientNo, propertyNo, comment(Viewing))

Or Client [clientNo, fName, lName] Join Viewing [clientNo, propertyNo, comment ]

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Another Natural Join Example

• R= S=

• R S=

A B

X Y

X Z

Y Z

Z V

B C

Z U

V W

Z V

A B C

X Z U

X Z V

Y Z U

Y Z V

Z V W

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Natural Join

• Given the schemas R(A, B, C, D), S(A, C, E), what is the schema of R S ?

• Given R(A, B, C), S(D, E), what is R S ?

• Given R(A, B), S(A, B), what is R S ?

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Theta Join

• A join that involves a predicate

• Notation: R1 R2 where is a condition

• Input schemas: R1(A1,…,An), R2(B1,…,Bm)

• {A1,…An} ∩ {B1,…,Bm} =

• Output schema: S(A1,…,An,B1,…,Bm)

• Derived operator:

R1 R2 = (R1 x R2)

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Semijoin

• R S = A1,…,An (R S)

• Where the schemas are:– Input: R(A1,…An), S(B1,…,Bm)– Output: T(A1,…,An)

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Outer join• To display rows in the result that do not

have matching values in the join column, use Outer join.

• R S– (Left) outer join is join in which tuples from

R that do not have matching values in common columns of S are also included in result relation.

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Outer Join• To display rows in the result that do not have matching values in the join

column, use Outer join.

• R Left Outer Join S– (Left) outer join is join in which tuples from R that do not have matching values in

common columns of S are also included in result relation.

Example:

• Produce a status report on property viewings.propertyNo, street, city(PropertyForRent)

Left Outer Join Viewing

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Types of Joins

• Left Outer Join – keep all of the tuples from the “left” relation

– join with the right relation

– pad the non-matching tuples with nulls

• Right Outer Join– same as the left, but keep tuples from the “right”

relation

• Full Outer Join– same as left, but keep all tuples from both relations

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Left Outer Join

• If we do a left outer join on R and S, and we match on the first column, the result is:

A B

C D

A F

G H

R= S=

A B F

C D -

name phone name email

name phone email

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Example - Left Outer join

• Produce a status report on property viewings.propertyNo, street, city(PropertyForRent)

Viewing

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Right Outer Join

• If we do a right outer join on R and S, and we match on the first column, the result is:

A B

C D

A F

G H

R= S=

A B F

G - H


name phone email

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Full Outer Join

• If we do a full outer join on R and S, and we match on the first column, the result is:

A B

C D

A F

G H

R= S=

A B F

C D -

G - H


name phone email

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OUTER JOIN Example 1

E 5F 4D 3B 2A 1

A 1C 2D 3E 4

R.ColA = S.SColA SR LEFT OUTER JOIN

A 1 A 1D 3E 4

- -- -

D 3E 5

F 4B 2

A 1 A 1D 3E 4

D 3E 5- - C 2

ColA ColB

SColBSColAS

R

R.ColA = S.SColA SR RIGHT OUTER JOIN

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OUTER JOIN Example 2

E 5F 4D 3B 2A 1

A 1C 2D 3E 4

R.ColA = S.SColA SR FULL OUTER JOIN

A 1 A 1D 3E 4

- -- -

D 3E 5

F 4B 2

- - C 2

ColA ColB

SColBSColAS

R

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Division

• Identify all clients who have viewed all properties with three rooms.

(clientNo, propertyNo(Viewing)) (propertyNo(rooms = 3 (PropertyForRent)))

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Natural join• R S

– An Equijoin of the two relations R and S over all common attributes x. One occurrence of each common attribute is eliminated from the result.

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Example - Natural join• List the names and comments of all clients

who have viewed a property for rent.(clientNo, fName, lName(Client))

(clientNo, propertyNo, comment(Viewing))

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Semijoin• R F S

– Defines a relation that contains the tuples of R that participate in the join of R with S.

Can rewrite Semijoin using Projection and Join:

R F S = A(R F S)

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Example - Semijoin• List complete details of all staff who work at

the branch in Glasgow.

Staff Staff.branchNo = Branch.branchNo and Branch.city = ‘Glasgow’ Branch

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Division• R S

– Defines a relation over the attributes C that consists of set of tuples from R that match combination of every tuple in S.

• Expressed using basic operations:T1 C(R)

T2 C((S X T1) – R)

T T1 – T2

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Example - Division• Identify all clients who have viewed all

properties with three rooms.

(clientNo, propertyNo(Viewing)) (propertyNo(rooms = 3 (PropertyForRent)))

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Summary of Relational Algebra

• Five basic operators, many derived

• Combine operators in order to construct queries: relational algebra expressions, usually shown as trees

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RA has Limitations !

• Cannot compute “transitive closure”

• Find all direct and indirect relatives of Fred• Cannot express in RA !!! Need to write C program

Name1 Name2 Relationship

Fred Mary Father

Mary Joe Cousin

Mary Bill Spouse

Nancy Lou Sister

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Equivalences

The same relational algebraic expression can be written in many different ways. The order in which tuples appear in relations is never significant. • A B <=> B A • A B <=> B A • A B <=> B A • (A - B) is not the same as (B - A) • c1 ( c2 (A)) <=> c2 ( c1 (A)) <=> c1 ^ c2 (A) • a1(A) <=> a1( a1,etc(A)) , where etc is any attributes of A. • ...

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Operations on Bags (and why we care)

• Union: {a,b,b,c} U {a,b,b,b,e,f,f} = {a,a,b,b,b,b,b,c,e,f,f}– add the number of occurrences

• Difference: {a,b,b,b,c,c} – {b,c,c,c,d} = {a,b,b,d}– subtract the number of occurrences

• Intersection: {a,b,b,b,c,c}∩{b,b,c,c,c,c,d} = {b,b,c,c}– minimum of the two numbers of occurrences

• Selection: preserve the number of occurrences

• Projection: preserve the number of occurrences (no duplicate elimination)

• Cartesian product, join: no duplicate elimination

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SQL Introduction

• Standard language for querying and manipulating data

Structured Query Language

• Many standards out there: SQL92, SQL2, SQL3, SQL99• Vendors support various subsets of these, but all of what we’ll be

talking about.• Works on bags, rather than sets

• Basic construct:SELECT …FROM …WHERE …

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Selections

Company(sticker, name, country, stockPrice)

Find all US companies whose stock is > 50:

SELECT * FROM Company WHERE country=“USA” AND stockPrice > 50

Output schema: R(sticker, name, country, stockPrice)

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Selections

What you can use in WHERE:

• attribute names of the relation(s) used in the FROM.• comparison operators: =, <>, <, >, <=, >=• apply arithmetic operations: stockprice*2• operations on strings (e.g., “||” for concatenation).• Lexicographic order on strings.• Pattern matching: s LIKE p• Special stuff for comparing dates and times.

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The LIKE operator

• s LIKE p: pattern matching on strings• p may contain two special symbols:

– % = any sequence of characters– _ = any single character

Company(sticker, name, address, country, stockPrice)Find all US companies whose address contains “Mountain”:

SELECT * FROM Company WHERE country=“USA” AND address LIKE “%Mountain%”

• Needed in the 1st assignment !

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Select only a subset of the attributes

SELECT name, stockPrice FROM Company WHERE country=“USA” AND stockPrice > 50

Input schema: Company(sticker, name, country, stockPrice)Output schema: R(name, stock price)

Projections

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Rename the attributes in the resulting table

SELECT name AS company, stockprice AS price FROM Company WHERE country=“USA” AND stockPrice > 50

Input schema: Company(sticker, name, country, stockPrice)Output schema: R(company, price)

Projections with Renamings

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Eliminating Duplicates

SELECT DISTINCT country FROM Company WHERE stockPrice > 50

Without DISTINCT the result is a bag

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Ordering the Results

SELECT name, stockPrice FROM Company WHERE country=“USA” AND stockPrice > 50 ORDERBY country, name

Ordering is ascending, unless you specify the DESC keyword.

Ties are broken by the second attribute on the ORDERBY list, etc.

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Joins

Product ( pname, price, category, maker)Purchase (buyer, seller, store, product)Company (cname, stockPrice, country)Person( per-name, phoneNumber, city)

Find names of people living in Seattle that bought gizmo products, and the names of the stores they bought from

SELECT per-name, store FROM Person, Purchase WHERE per-name=buyer AND city=“Seattle” AND product=“gizmo”

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Disambiguating Attributes

SELECT Person.name FROM Person, Purchase, Product WHERE Person.name=buyer AND product=Product.name AND Product.category=“telephony”

Product (name, price, category, maker)Purchase (buyer, seller, store, product)Person(name, phoneNumber, city)

Find names of people buying telephony products:

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Tuple Variables

SELECT product1.maker, product2.maker FROM Product AS product1, Product AS product2 WHERE product1.category=product2.category AND product1.maker <> product2.maker

Product ( name, price, category, maker)

Find pairs of companies making products in the same category

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Tuple Variables•Tuple variables introduced automatically by the system: Product ( name, price, category, maker)

SELECT name FROM Product WHERE price > 100Becomes: SELECT Product.name FROM Product AS Product WHERE Product.price > 100

Doesn’t work when Product occurs more than once.

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Meaning (Semantics) of SQL Queries

SELECT a1, a2, …, akFROM R1 AS x1, R2 AS x2, …, Rn AS xnWHERE Conditions

1. Nested loops: Answer = {} for x1 in R1 do for x2 in R2 do ….. for xn in Rn do if Conditions then Answer = Answer U {(a1,…,ak)} return Answer

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2. Parallel assignment

Answer = {} for all assignments x1 in R1, …, xn in Rn do if Conditions then Answer = Answer U {(a1,…,ak)} return Answer

Doesn’t impose any order !

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3. Translation to Relational algebra:

a1,,…,ak ( Conditions (R1 x R2 x … x Rn))

Select-From-Where queries are precisely Select-Project-Join

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First Unintuitive SQLismSELECT R.AFROM R, S, TWHERE R.A=S.A OR R.A=T.A

Looking for R ∩ (S U T)

But what happens if T is empty?

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Union, Intersection, Difference(SELECT name FROM Person WHERE City=“Seattle”) UNION(SELECT name FROM Person, Purchase WHERE buyer=name AND store=“The Bon”)

Similarly, you can use INTERSECT and EXCEPT.

You must have the same attribute names (otherwise: rename).

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Conserving Duplicates

(SELECT name FROM Person WHERE City=“Seattle”) UNION ALL(SELECT name FROM Person, Purchase WHERE buyer=name AND store=“The Bon”)

The UNION, INTERSECTION and EXCEPT operators operate as sets, not bags.

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Subqueries

A subquery producing a single tuple:

SELECT Purchase.productFROM PurchaseWHERE buyer = (SELECT name FROM Person WHERE ssn = “123456789”);

In this case, the subquery returns one value.

If it returns more, it’s a run-time error.

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Can say the same thing without a subquery:

SELECT Purchase.productFROM Purchase, PersonWHERE buyer = name AND ssn = “123456789”

Is this query equivalent to the previous one ?

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Subqueries Returning Relations

SELECT Company.name FROM Company, Product WHERE Company.name=maker AND Product.name IN (SELECT product FROM Purchase WHERE buyer = “Joe Blow”);

Here the subquery returns a set of values

Find companies who manufacture products bought by Joe Blow.

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SELECT Company.name FROM Company, Product, Purchase WHERE Company.name=maker AND Product.name = product AND buyer = “Joe Blow”

Equivalent to:

Is this query equivalent to the previous one ?

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SELECT name FROM Product WHERE price > ALL (SELECT price FROM Purchase WHERE maker=“Gizmo-Works”)

Product ( pname, price, category, maker)Find products that are more expensive than all those producedBy “Gizmo-Works”

You can also use: s > ALL R s > ANY R EXISTS R

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Question for Database Fans

• Can we express this query as a single SELECT-FROM-WHERE query, without subqueries ?

• Hint: show that all SFW queries are monotone (figure out what this means). A query with ALL is not monotone

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Conditions on Tuples

SELECT Company.name FROM Company, Product WHERE Company.name=maker AND (Product.name,price) IN (SELECT product, price) FROM Purchase WHERE buyer = “Joe Blow”);

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Correlated Queries

SELECT title FROM Movie AS x WHERE year < ANY (SELECT year FROM Movie WHERE title = x.title);

Movie (title, year, director, length) Find movies whose title appears more than once.

Note (1) scope of variables (2) this can still be expressed as single SFW

correlation

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Complex Correlated Query

Product ( pname, price, category, maker, year)

• Find products (and their manufacturers) that are more expensive than all products made by the same manufacturer before 1972

SELECT pname, makerFROM Product AS xWHERE price > ALL (SELECT price FROM Product AS y WHERE x.maker = y.maker AND y.year < 1972);

Powerful, but much harder to optimize !

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Exercises: write RA and SQL expressions

Product ( pname, price, category, maker)Purchase (buyer, seller, store, product)Company (cname, stock price, country)Person( per-name, phone number, city)

Ex #1: Find people who bought telephony products.Ex #2: Find names of people who bought American productsEx #3: Find names of people who bought American products and did not buy French productsEx #4: Find names of people who bought American products and they live in Seattle.Ex #5: Find people who bought stuff from Joe or bought products from a company whose stock prices is more than $50.

Documents

Lecture 3: Relational Algebra and SQL Tuesday, March 25, 2008