COMP 5138COMP 5138

Relational Database Relational Database

Management Systems Management Systems

Semester 2, 2007Semester 2, 2007

Lecture 5ALecture 5A

Relational AlgebraRelational Algebra

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L3L3 Relational Data ModelRelational Data Model Overview

There are many tasks that can be understood as calculating some relation by combining the information in one or more relation instancesRelational algebra defines some operators that can be used to express a calculation like thatA central insight: a query that extracts information can be seen as calculating a relation from the current state of the database

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L3L3 Relational Data ModelRelational Data Model Relational Algebra

Users request information from a database using a query language

Six basic and several additional operators

Basic operations:Selection ( ) Selects a subset of rows from relation.Projection ( ) Extracts only desired columns from relation.Cross-product ( ) Allows us to combine two relations.Set-difference ( ) Tuples in reln. 1, but not in reln. 2.Union ( ) Tuples in reln. 1 or in reln. 2.Rename ( ) Allows us to rename one field to another name.

Additional operations:Intersection, join, division: Not essential, but (very!) useful.

The operators take one or more relations as inputs and give a new relation as result

σπX_

Uρ

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L3L3 Relational Data ModelRelational Data Model Running Example

sid sname rating age

22 dustin 7 45.0

31 lubber 8 55.5 58 rusty 10 35.0

sid sname rating age 28 yuppy 9 35.0 31 lubber 8 55.5 44 guppy 5 35.0 58 rusty 10 35.0

sid bid day

22 101 10/10/96 58 103 11/12/96

Instance R1 of reserves

Instance S1 of sailors

“Sailors” and “Reserves” relations for our examples.

Instance S2 of sailors

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L3L3 Relational Data ModelRelational Data Model Projection

sname rating

yuppy 9 lubber 8 guppy 5 rusty 10

sname, rating(S2)

age

35.0 55.5

age

Deletes attributes that are not in projection list.Schema of result contains exactly the fields in the projection list, with the same names that they had in the (only) input relation.

ππ (S2)

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L3L3 Relational Data ModelRelational Data Model Selection

sid sname rating age 28 yuppy 9 35.0 58 rusty 10 35.0

sname rating yuppy 9 rusty 10

Selects rows that satisfy selection condition.

rating > 8σ (S2)

Result relation can be the input for another relational algebra operation! (Operator composition.)

sname, rating ( )π

rating > 8σ (S2)

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L3L3 Relational Data ModelRelational Data Model Set Operation

All of these operations take two input relationsSame number of fields.`Corresponding’ fields have the same type.Set Union R S

Definition: R U S = { t | t R t S }

Set Intersection R SDefinition: R S = { t | t R t S }

Set Difference R - SDefinition: R - S = { t | t R t S }

sid sname rating age

22 dustin 7 45.0 31 lubber 8 55.5 58 rusty 10 35.0 44 guppy 5 35.0 28 yuppy 9 35.0

sid sname rating age 31 lubber 8 55.5 58 rusty 10 35.0

sid sname rating age

22 dustin 7 45.0

U(S1) (S2) I(S1) (S2)

(S1) (S2)_

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L3L3 Relational Data ModelRelational Data Model Cross-Product

Each row of S1 is paired with each row of R1.Result schema has one field per field of S1 and R1, with field names `inherited’ if possible.

Conflict: Both S1 and R1 have a field called sid.Sometimes called Cartesian product

(sid) sname rating age (sid) bid day

22 dustin 7 45.0 22 101 10/ 10/ 96

22 dustin 7 45.0 58 103 11/ 12/ 96

31 lubber 8 55.5 22 101 10/ 10/ 96

31 lubber 8 55.5 58 103 11/ 12/ 96

58 rusty 10 35.0 22 101 10/ 10/ 96

58 rusty 10 35.0 58 103 11/ 12/ 96

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L3L3 Relational Data ModelRelational Data Model Renaming

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.

Notation 1: ρ x (E)

returns the expression E under the name X

Notation 2: ρx (A1, A2, …, An) (E)

returns the result of expression E under the name X, and with the attributes renamed to A1, A2, …., An.(assumes that the relational-algebra expression E has arity n)

ExampleC( 1 sid1, 5 sid2)( S1XR1)ρ

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L3L3 Relational Data ModelRelational Data Model Joins

Condition Join : R c S c R S>< = ×σ ( )

(sid) sname rating age (sid) bid day

22 dustin 7 45.0 58 103 11/ 12/ 96 31 lubber 8 55.5 58 103 11/ 12/ 96

S RS sid R sid

1 11 1

><. .<

Result schema same as that of cross-product.Fewer tuples than cross-product, might be able to compute more efficientlySometimes called a theta-join.

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L3L3 Relational Data ModelRelational Data Model Joins

Equi-Join: A special case of condition join where the condition c contains only equalities.

sid sname rating age bid day

22 dustin 7 45.0 101 10/ 10/ 96 58 rusty 10 35.0 103 11/ 12/ 96

S Rsid

1 1><

Result schema similar to cross-product, but only one copy of fields for which equality is specified.Natural Join: Equijoin on all common fields.

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aabab

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aaabb

A B C DR

B D ES

=

A B C DR S

E

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aaaab

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L3L3 Relational Data ModelRelational Data Model Division

Not supported as a primitive operator, but useful for expressing queries like:Find sailors who have served on all boats.

Let A have 2 fields, x and y; B have only field y:

A/B =

i.e., A/B contains all x tuples (sailors) such that for every y tuple (boat) in B, there is an xy tuple in A.Or: If the set of y values (boats) associated with an x value (sailor) in A contains all y values in B, the x value is in A/B.

In general, x and y can be any lists of fields; y is the list of fields in B, and x y is the list of fields of A.

{ }x x y A y B| ,∃ ∈ ∀ ∈

∪

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L3L3 Relational Data ModelRelational Data ModelExamples of Division A/B

sno pno s1 p1 s1 p2 s1 p3 s1 p4 s2 p1 s2 p2 s3 p2 s4 p2 s4 p4

pno p2

pno p2 p4

pno p1 p2 p4

sno s1 s2 s3 s4

sno s1 s4

sno s1

A

B1

B2

B3

A/B1 A/B2 A/B3

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L3L3 Relational Data ModelRelational Data Model Equivalence Rules

The following equivalence rules hold:Commutation rules

1. πA ( σp ( R ) ) σp ( πA ( R ) )2. R S S R

1. Association rule1. R (S T) (R S) T

2. Idempotence rules1. πA ( πB ( R ) ) πA ( R ) if A B

2. σp1 (σp2 ( R )) σp1 p2 ( R )

3. Distribution rules1. πA ( R S ) πA ( R ) πA ( S )

2. σP ( R S ) σP ( R ) σP ( S )

3. σP ( R S ) σP (R) S if P only references R

4. πA,B(R S ) πA ( R ) πB ( S ) if join-attr. in (A B)5. R ( S T ) ( R S ) ( R T )

4. These rules are the basis for the automatic optimisation of relational algebra expressions