Querying Relational Data: Algebraavid.cs.umass.edu/courses/645/s2010/lectures/02-RelationalAlg.pdf · Division • A derived operator useful for queries like: Find students who have

Querying Relational Data: Algebra

Gerome MiklauUMass Amherst

CMPSCI 645 – Database Systems

Jan 21, 2010

Some slide content courtesy of Zack Ives, Ramakrishnan & Gehrke, Dan Suciu, Ullman & Widom

Thursday, January 21, 2010

Next lectures

• Today– Relational model, relational algebra

• Next Tuesday– SQL

• Homework 1 will be on these topics


Relational Database: Definitions

• Relational database: a set of relations• Relation: made up of 2 parts:

– Instance : a table, with rows and columns. – Schema : specifies name of relation, plus

name and type/domain of each column.

Restriction: all attributes are of atomic type, no nested tables

Students(sid: string, name: string, login: string, age: integer, gpa: real).


Relational instances: tablesArity (number of attributes) is 5

Students

column, attribute, field

row, tuple

Attribute value

A relation is a set of tuples: no tuple can occur more than once– Real systems may allow duplicates for efficiency or other

reasons – we’ll come back to this.


Relational Query Languages• Query languages: Allow manipulation and retrieval

of data from a database.• Query Languages != programming languages!

– QLs not expected to be “Turing complete”.– QLs not intended to be used for complex calculations.– QLs support easy, efficient access to large data sets.


Preliminaries

• A query is applied to one or more relation instances

• The result of a query is a relation instance.• Input and output schema:

– Schema of input relations for a query are fixed – The schema for the result of a given query is also fixed:

determined by definition of query language constructs.

Query Q: R1..Rn → R’


What is an “Algebra”

• Mathematical system consisting of:– Operands --- variables or values from

which new values can be constructed.– Operators --- symbols denoting procedures

that construct new values from given values.


What is the Relational Algebra?

• An algebra whose operands are relations or variables that represent relations.

• Operators are designed to do the most common things that we need to do with relations in a database.– The result is an algebra that can be used

as a query language for relations.


Relational Algebra• Operates on relations, i.e. sets

– Later: we discuss how to extend this to bags• Five operators:

– Union: ∪– Difference: -– Selection: σ– Projection: Π – Cartesian Product: ×

• Derived or auxiliary operators:– Intersection, complement– Joins (natural, equi-join, theta join)– Renaming: ρ– Division: /


Example Database

sid name

1 Jill2 Bo3 Maya

fid name

1 Diao2 Saul8 Weems

sid cid

1 6451 6833 635

cid name sem

645 DB F05683 AI S05635 Arch F05

fid cid

1 6452 6838 635

STUDENT Takes COURSE

PROFESSOR Teaches


1. Union and 2. Difference

sid name1 Jill2 Bo3 Maya

R1 sid name1 Jill4 Bob

R2

sid name2 Bo3 Maya

sid name1 Jill2 Bo3 Maya4 Bob

R1 – R2R1 ∪ R2


What about Intersection ?

• It is a derived operator• R1 ∩ R2 = R1 – (R1 – R2)• Also expressed as a join (we’ll see

later)

R1 R2 R1 – R2


3. Selection• Returns all tuples which satisfy a

condition• Notation: σc(R)• Examples

σCID > 600 (Course)σname = “AI” (Course)

• The condition c can be =, <, ≤, >, ≥, <>

cid name sem

645 DB F05683 AI S05635 Arch F05

Course


4. Projection• Eliminates columns, then removes duplicates• Notation: Π A1,…,An (R)• Example: project cid and name

Π cid, name (Course)Output schema: Answer(cid, name)

cid name sem

645 DB F05683 AI S05645 DB S05

Coursecid name

645 DB683 AI

Answer

Π


5. Cartesian Product

• Each tuple in R1 with each tuple in R2

• Notation: R1 × R2

• Very rare in practice; mainly used to express joins

Also called “Cross Product”


Cartesian Product

16

sid cid

1 6451 6833 635

sid name1 Jill2 Bo

Student TakesStudent × Takes

sid name sid cid1 Jill 1 6451 Jill 1 6831 Jill 3 6352 Bo 1 6452 Bo 1 6832 Bo 3 635


Renaming

• Changes the schema, not the instance• Notation: ρ B1,…,Bn (R)• Example:

ρcourseID, cname, term (Course)

cid name sem

645 DB F05683 AI S05645 DB S05

CoursecourseID cname term645 DB F05683 AI S05645 DB S05

ρ


Natural Join• Notation: R1 R2

• Meaning: R1 R2 = ΠA(σC(R1 × R2))

• Where:– The selection σC checks equality of all

common attributes– The projection eliminates the duplicate

common attributes


Natural join example

19


sid cid

1 6451 6833 635

Takes

Student

sid name cid

1 Jill 6451 Jill 6833 Maya 635

Student Takes


Example Database

sid name

1 Jill2 Bo3 Maya

fid name

1 Diao2 Saul8 Weems

sid cid

1 6451 6833 635

cid name sem

645 DB F05683 AI S05635 Arch F05

fid cid

1 6452 6838 635

STUDENT Takes COURSE

PROFESSOR Teaches


Natural join questions

• 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 ?

– R(A,B,C,D,E)

– Cartesian Product

– Intersection


Theta Join

• A join that involves a predicate• R1 θ R2 = σ θ (R1 × R2)

• Here θ can be any condition: =, <, ≠, ≤, >, ≥

Example: Student sid<sid Takes


Equi-join

• A theta join where θ is an equality• R1 A=B R2 = σ A=B (R1 × R2)• Very useful join in practice

• Example: Student sid=sid Takes


Semijoin

R S = Π A1,…,An (R S)where A1, …, An are the attributes in R

The semijoin of R and S is the set of tuples of R that agree with at least one tuple of S on all attributes common to the schema of R and S.


sid cid1 6451 6833 635

TakesStudent

sid name

1 Jill3 Maya

Student Takes


Division• A derived operator useful for queries like:

Find students who have enrolled in all systems courses.

• Let R have 2 fields, x and y; S have only field y:• R/S = • i.e., R/S contains all x tuples (students) such that for

every y tuple (course) in S, there is an xy tuple in R.• Or: If the set of y values (courses) associated with an x

value (student) in R contains all y values in S, the x value is in R/S.

• In general: attributes of S must be subset of attributes of R: • R(A1 ... An, B1, ... Bm) and S(B1 ... Bm)

{ (x) | ∀ (y) ∈ S, ∃ (x,y) ∈ R }


Division examplessno pnos1 p1s1 p2s1 p3s1 p4s2 p1s2 p2s3 p2s4 p2s4 p4

pnop2

pnop2p4

pnop1p2p4

snos1s2s3s4

snos1s4

snos1

A

B1B2

B3

A/B1 A/B2 A/B3


Expressing division using basic operators

Idea: For R/S, compute all x values that are not `disqualified’ by some y value in S. an x value is disqualified if, by attaching y value from S,

we obtain an xy tuple that is not in R.

Disqualified x values: Πx((Πx(R) × S)-R)

R / S: Πx(R) - all disqualified tuples


Combining operators: complex expressions

Πname,sid (σname=”DB” (Students (Takes Course)))

Students CourseTakes

σname=”DB”

Πname,sid


Algebraic Equivalences

• Relational algebra has laws of commutativity, associativity, etc. that imply certain expressions are equivalent.

Definition: Query Equivalence

Two queries Q and Q’ are equivalent if:

for all instances D, Q(D) = Q’(D)


Query OptimizationIs Based on Algebraic Equivalences

• Equivalent expressions may be different in cost of evaluation!

σc ∧ d(R) ≡ σc( σd(R) )

σc (R ⋈ S) ≡ σc(R) ⋈ S

• Query optimization finds the most efficient representation to evaluate (or one that’s not bad)

R ⋈ (S ⋈ T) ≡ (R ⋈ S) ⋈ T)

cascading selection

join associativity

pushing selections


Operations on BagsA bag = a set with repeated elementsRelational Engines work on bags, not sets !All operations need to be defined carefully on bags• {a,b,b,c}∪{a,b,b,b,e,f,f}={a,a,b,b,b,b,b,c,e,f,f}• {a,b,b,b,c,c} – {b,c,c,c,d} = {a,b,b}• σC(R): preserves the number of occurrences

• ΠA(R): no duplicate elimination

• Cartesian product, join: no duplicate elimination


Beware: Bag Laws != Set Laws

• Some, but not all algebraic laws that hold for sets also hold for bags.

• Example: the commutative law for union (R ∪ S = S ∪ R ) does hold for bags.– Since addition is commutative, adding the

number of times x appears in R and S doesn’t depend on the order of R and S.


Example of the Difference

• Set union is idempotent, meaning that S ∪ S = S.

• However, for bags, if x appears n times in S, then it appears 2n times in S ∪ S.

• Thus S ∪ S != S in general.


Relational calculus

•What is a “calculus”?– The term "calculus" means a system of

computation – The relational calculus is a system of

computing with relations

34


Relational calculus (in 1 slide)

We will study another logic-based formalism for queries called Datalog later.

Name and sid of students who are taking the course “DB”English:

{xname, xsid | ∃xcid∃xterm Students(xsid,xname) ∧ Takes(xsid,xcid) ∧ Course(xcid,”DB”, xterm) }RC:

RA: Πname,sid (Students Takes σname=”DB” (Course)

Where are the joins?


Algebra v. Calculus

• Relational Algebra: More operational; very useful for representing execution plans.

• Relational Calculus: More declarative, basis of SQL

• The calculus and algebra have equivalent expressive power (Codd)

A language that can express this core class of queries is called Relationally Complete


What can’t you express in RA,RC?

• Can I get from Oakland to Boston in 2 flights?

• Can I get from Oakland to Reno?

37

depart arrive

NYC Reno

NYC Oakland

Boston Tampa

Oakland Boston

Tampa NYC


Querying Relational Data: Algebraavid.cs.umass.edu/courses/645/s2010/lectures/02-RelationalAlg.pdf · Division • A derived operator useful for queries like: Find students who have

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