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Relational Algebra

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Relational Algebra Chapter 4 - part I – PowerPoint PPT presentation

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Title: Relational Algebra


1
Relational Algebra
  • Chapter 4 - part I

2
Relational Query Languages
  • Query languages Allow manipulation and
    retrieval of data from a database.
  • Relational model supports simple powerful QLs
  • Strong formal foundation based on logic.
  • Allows for much optimization.
  • 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.

3
Formal Relational Query Languages
  • Two mathematical Query Languages form the basis
    for real languages (e.g. SQL) and for
    implementation
  • Relational Algebra
  • More operational, very useful for representing
    execution plans.
  • Relational Calculus
  • Lets users describe what they want, rather than
    how to compute it. (Non-operational, rather
    declarative.)

4
Preliminaries
  • A query is applied to relation instances.
  • The result of a query is also a relation
    instance.
  • Schemas of input relations for a query are fixed
    (but query will run regardless of instance!)
  • The schema for result of given query is also
    fixed! Determined by definition of query language
    .

5
Preliminaries
  • Positional vs. named-field notation
  • Positional field notation e.g., S.1
  • Named field notation e.g., S.sid
  • Pros/Cons
  • Positional notation easier for formal
    definitions, named-field notation more readable.
  • Both used in SQL
  • Assume that names of fields in query results are
    inherited from names of fields in query input
    relations.

6
Example Instances
Sailors
Reserves
R1
S1
Sailors
S2
7
Relational Algebra
  • Basic operations
  • Selection ( ) Selects a subset of rows
    from relation.
  • Projection ( ) Deletes unwanted columns
    from relation.
  • Cartesian-product ( ) Allows us to combine
    two relations.
  • Set-difference ( ) Tuples in reln. 1, but
    not in reln. 2.
  • Union ( ) Tuples in reln. 1 and in reln. 2.
  • Additional operations
  • Intersection, join, division, renaming
    Not essential, but
    (very!) useful.
  • Since each operation returns a relation,
    operations can be composed! (Algebra is
    closed.)

8
Selection
Sailors
S2
9
Selection
  • ?condition (R)
  • Selects rows that satisfy selection condition.
  • attribute op constant
  • attribute op attribute
  • Op is lt,gt,lt,gt, ,
  • No duplicates in result!
  • Schema of result identical to schema of (only)
    input relation.

10
Selection
  • Result relation can be input for another
    relational algebra operation! (Operator
    composition.)

11
Projection
Sailors
S2
12
Projection
  • ? projectlist (R)
  • Deletes attributes that are not in projection
    list.
  • Schema of result contains fields in projection
    list
  • Projection operator has to eliminate duplicates!
    (Why??)
  • Note real systems typically dont do duplicate
    elimination unless the user explicitly asks for
    it. (Why not?)

13
Union, Intersection, Set-Difference
  • All of these operations take two input relations,
    which must be union-compatible
  • Same number of fields.
  • Corresponding fields have same type.
  • What is the schema of result?

14
Example Instances Union
S1
S2
15
Union, Intersection, Set-Difference
  • All of these operations take two input relations,
    which must be union-compatible
  • Same number of fields.
  • Corresponding fields have the same type.
  • What is the schema of result?

16
Difference Operation
S1
S2
17
Union, Intersection, Set-Difference
  • All of these operations take two input relations,
    which must be union-compatible
  • Same number of fields.
  • Corresponding fields have the same type.
  • What is the schema of result?

18
Intersection Operation
S1
S2
19
Union, Intersection, Set-Difference
  • All of these operations take two input relations,
    which must be union-compatible
  • Same number of fields.
  • Corresponding fields have the same type.
  • What is the schema of result?

20
Cross-Product (Cartesian Product)
  • S1 R1 Each row of S1 is paired with each row
    of R1.

Reserves
R1
Sailors
S1
21
Cross-Product (Cartesian Product)
  • S1 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.
  • Renaming operator

22
Why we need a Join Operator ?
  • In many cases,
  • Join Cross-Product Select Project
  • However
  • Cross-product is too large to materialize
  • Apply Select and Project "On-the-fly"

23
Condition Join / Theta Join
  • Condition Join
  • Result schema same as that of cross-product.
  • Fewer tuples than cross-product, more efficient.

24
EquiJoin
  • Equi-Join A special case of condition join
    where the condition c contains only equalities.
  • Result schema similar to cross-product, but only
    one copy of fields for which equality is
    specified.
  • An extra project PROJECT ( THETA-JOIN)

25
Natural Join
  • Natural Join Equijoin on all common fields.

26
Division
  • Not supported as a primitive operator, but
    useful for expressing queries like


  • Find sailors who have reserved all boats.

27
Division
  • Let A have 2 fields x and y B have only field
    y
  • A/B
  • A/B contains all x tuples (sailors) such that for
    every y tuple (boat) in B, there is an xy tuple
    in A.
  • If set of y values (boats) associated with an x
    value (sailor) in A contains all y values in B,
    then x value is in A/B.
  • A/B is the largest relation instance Q such that
    Q?B ?A.

28
Division Example
  • e.g., A all parts supplied by
    suppliers,
  • B relation parts
  • A/B suppliers who supply all
    parts listed in B

29
Examples of Division A/B
B1
B2
B3
A/B1
A/B2
A/B3
A
30
Expressing A/B Using Basic Operators
  • Idea For A/B, compute all x values that are not
    disqualified by some y value in B.
  • x value is disqualified if by attaching y value
    from B,
  • we obtain an xy tuple that is not in A.

Disqualified x values
A/B
31
A few example queries
32
Find names of sailors whove reserved boat 103
33
Find names of sailors whove reserved a red boat
Reserves
R1
Sailors
S1
Boats
bid bname color
101 Interlake blue
103 Clipper red
B1
34
Find names of sailors whove reserved a red boat
  • Information about boat color only available in
    Boats so need an extra join

35
Find sailors whove reserved a red or a green boat
  • Can identify all red or green boats, then find
    sailors whove reserved one of these boats

Can also define Tempboats using union! (How?)
What happens if is replaced by in this
query?
36
Find sailors whove reserved a red and a green
boat
  • Previous approach wont work!
  • Must identify sailors who reserved red boats,
    sailors whove reserved green boats,
    then find their intersection


37
Find the names of sailors whove reserved all
boats
  • Uses division schemas of the input relations to
    / must be carefully chosen
  • To find sailors whove reserved all Interlake
    boats

38
Relational Algebra Some More Operators
  • Beyond Chapter 4

39
Generalized Projection
  • ? F1, F2, (R)

? sname, (rating2 as myrating) (Sailors)
40
Aggregation operators
  • MIN, MAX, COUNT, SUM, AVG
  • AGGB (R) considers only non-null values of R.

SUMB (R)
COUNTB (R)
MINB (R)
R
SUMB (R)
9
MINB (R)
2
COUNTB (R)
3
A B
1 2
3 4
1 null
1 3
AVGB (R)
COUNT (R)
MAXB (R)
AVGB (R)
3
COUNT (R)
4
MAXB (R)
4
41
Aggregation Operators
  • MIN, MAX, SUM, AVG must be on any 1 attribute.
    COUNT can be on any 1 attribute or COUNT (R)
  • An aggregation operator returns a bag, not a
    single value ! But SQL allows treatment as a
    single value.

sBMAXB (R) (R)
A B
3 4
42
Grouping Operator ?GL, AL (R)
  • ?GL, AL (R) groups all attributes in GL, and
    performs the aggregation specified in AL.

? starName, MIN (year)?year, COUNT(title) ?num
(StarsIn)
starName year num
HF 77 3
KR 94 2
StarsIn
title year starName
SW1 77 HF
Matrix 99 KR
6D7N 93 HF
SW2 79 HF
Speed 94 KR
43
Summary
  • The relational model has rigorously defined query
    languages that are simple and powerful.
  • Relational algebra is operational useful as
    internal representation for query evaluation
    plans.
  • Several ways of expressing a given query a query
    optimizer should choose most efficient version.
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