Databases 2012 The Relational Algebra - Computer …csj/files/dDB/slides/algebra-2012-3.pdf · Databases 2012 The Relational Algebra Christian S. Jensen Computer Science, Aarhus University

Databases 2012

The Relational Algebra

Christian S. Jensen

Computer Science, Aarhus University

2 The Relational Algebra

What is an Algebra?

An algebra consists of

• values

• operators

• rules

Closure: operations yield values

Examples

• integers with +, ,

• sets with , , \,

• matrices with +, ,

• functions with , O , -1

• relations with query operators


Mathematical Relations

An n-ary relation on a set S is a subset of Sn

Examples

• is a binary relation on R, a subset of RR

{ (1.2, 3.4), (34, 117.363), (53, 0.1234), ... }

• divides is a binary relation on N, a subset of NN

{ (2, 4), (3, 9), (3, 12), (17, 34), (1237, 21029), ... }

• negative is a binary relation on N, a subset of NN

{ (3,-3), (-17,17), (0,0), (2, -2), (-2,2), (87, -87), ...}

• sum is a ternary relation on N, a subset of NNN

{ (3,5,8), (23,14,37), (0,123,123), (42,87,129), ... }

• married to is a binary relation on people

{ (Hillary, Bill), (Bill, Hillary), (Angelina, Brad), ... }


Tables as Relations

A database relation on a data set D consists of

• a schema of attribute names (a1, a2, ..., an)

• a finite n-ary relation on D, a subset of Dn

A relation is like a table where

• all columns have the same generic type

• no duplicates are allowed

• no other constraints are imposed

We implicitly allow permutations of the attributes

Database relations form an algebra with the operators

• union:

• intersection:

• difference: \

• projection:

• renaming:

• selection:

• Cartesian product:

• natural join: ⋈

These provide an abstract model of database queries


Relational Operators


Union, Intersection, Difference

The arguments must have the same schema

The result has again that schema

R S

R S

R \ S

They compute the set operations on the relations


(R)

Assume the schema of R is (a1,...,an,b1,...,bm)

The schema of the result is (a1,...,an)

The result relation is

{ (d1, ..., dn) | (d1, ..., dn+m) R }

Projection

a1,...,an


Renaming

ab(R)

The name a must occur as ai in the schema of R

The name b must not occur in the schema of R

Schema of the result: (a1, ..., ai-1, b, ai+1, ..., an)

The result relation is unchanged

ab,cd,ef(R) = ab(cd(ef(R)))


Selection

C(R)

C is a condition of the attributes of R

The resulting schema is unchanged

The relation part is: { r | r R C(r) }


Cartesian Product

R S

Assume

• R has schema (a1, ..., am)

• S has schema (b1, ..., bn)

The new schema is (a1, ..., am, b1, ..., bn)

The relation part is

{ (c1, ..., cm+n) | (c1, ..., cm) R (cm+1, ..., cm+n) S }


Natural Join

R ⋈ S

Assume

• R has schema (a1, ..., ak, c1, ..., cn)

• S has schema (c1, ..., cn, b1, ..., bm)

• {ai} {bi} =

The new schema is (a1, ..., ak, c1, ..., cn, b1, ..., bm)

The relation part is

{ (d1, ..., dk, e1, ..., en, f1, ..., fm) |

(d1, ..., dk, e1, ..., en) R (e1, ..., en, f1, ..., fm) S }

R ⋈ S = R ∩ S = R ∖ (R ∖ S)

• when the schemas are identical

R ⋈ S = R ⨉ S

• when the schemas are disjoint

R ⋈𝚹 S = 𝚹(R ⨉ S)

• the theta join

SELECT DISTINCT X1, …, Xk

FROM R1, …, Rn

WHERE C

= x1, …, xk(C (R1 ⨉… ⨉ Rn)


Derived Operators

In which meetings do the owners participate?

what,meetid(status=’a’(

owneruserid(Meetings) ⋈ piduserid(Participants)))


Query Trees

owneruserid piduserid

Meetings Participants

what,meetid

status=’a’


Limitations

The relational algebra cannot answer all queries

Flights

Which cities can be reached from Copenhagen

in one or more flights?

from to

Copenhagen Madrid

Rome London

Madrid Athens

Athens Rome

... ...


Transitive Closure

The transitive closure of a binary relation R

R = { (x1,xk) | x1,...,xk-1 ((xi,xi+1) R) }

No relational algebra expression computes R

No SQL query can handle it either

• unless SQL is extended with recursion

• or a special closure operator is added

• (some DBMSs do support this)

x x = x idempotence

x y = y x commutativity

x ⋈ x = x idempotence

x ⋈ y = y ⋈ x commutativity

x (y z) = (x y) z associativity

x ⋈ (y ⋈ z) = (x ⋈ y) ⋈ z associativity

x ⋈ (y z) = (x ⋈ y) (x ⋈ z) distributivity


Algebraic Laws (1/3)



C(x y) = C(x) C(y) distributivity

C(x \ y) = C(x) \ C(y) = C(x) \ y distributivity

C(x ⋈ y) = C(x) ⋈ C(y) distributivity

C(x y) = C(x) C(y) distributivity

C(x) = C(C(x)) idempotence

C(D(x)) = D(C(x)) commutativity

CD(x) = C(D(x)) = C(x) ⋈ D(x) splitting

CD(x) = C(x) D(x) splitting

C(x) = x \ C(x) splitting

a(x y) = a(x) a(y) distributivity

(does not hold for ∖ and ∩)

ab(x y) = ab(x) ab(y) distributivity

ab(x \ y) = ab(x) \ ab(y) distributivity

bc(ab(x)) = ac(x) cancellation

ab(cd(x)) = cd(ab(x)) commutativity




Zero and Unit

Define 0 = the empty relation (for each schema)

Define 1 as follows

• the schema is empty

• the relation contains the single empty row

0 x = x 0 = x

0 ⋈ x = x ⋈ 0 = 0

1 ⋈ x = x ⋈ 1 = x


Division


Division Example

Completed dDB student task

Fred Database1

Fred Database2

Fred Compiler1

Eugene Database1

Eugene Compiler1

Eugene Compiler2

Sara Database1

Sara Database2

John Usability1

task

Database1

Database2

CompleteddDB student

Fred

Sara

Those students that have

completed all the dDB tasks


Algebraic Query Optimization

Rewritings may improve efficiency

(A ⋈ B) ⋈ C A ⋈ (B ⋈ C)

C(A B) C(A) C(B)

Depends on the predicates (selectivities) and the

specific instances


Algebraic Query Optimization

10 rows 106 rows 106 rows

1012 rows 10 rows

Rewritings may improve efficiency:

(A ⋈ B) ⋈ C A ⋈ (B ⋈ C)

C(A B) C(A) C(B)

Depends on the predicates (selectivities) and the

specific instances


Rules of Thumb

Push selections down the expressions tree

Push projections down the expression tree

Order joins based on size estimates

In general, search for a good expression tree

• use heuristics

• use statistics: table sizes, distinct values for attributes,

histograms, etc.


Bag Algebra

Allows relations to contain duplicate entries

Sets are replaced by bags

The bag versions of , , and \ count copies

The bag versions of , , and ⋈ keep duplicates

A better match with real-life SQL than sets

Does still not account for the ordering of the tuples

• SQL offers some support for ordering

• Tuples in a relation are stored on disk in some order


Algebraic Laws for Bags

Fewer algebraic laws are valid for the bag algebra

Counter examples

x (y z) = (x y) (x z)

CD(x) = C(x) D(x)

Beware when optimizing bag queries!


Algebraic Laws for Bags

Fewer algebraic laws are valid for the bag algebra

Counter examples

x (y z) = (x y) (x z)

CD(x) = C(x) D(x)

Beware when optimizing bag queries!

a

42 x,y,z =

C,D = true

Documents

Databases 2012 The Relational Algebra - Computer …csj/files/dDB/slides/algebra-2012-3.pdf · Databases 2012 The Relational Algebra Christian S. Jensen Computer Science, Aarhus University