Download Relational Algebra

Relational Algebra
Chapter 4, Part A
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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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. (Nonoperational, declarative.)
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Example Instances
Sailors
Boats
B1
bid
101
103
color
Blue
Red
Reservations
R1
sid bid
day
22 101 10/10/96
58 103 11/12/96
S1 sid
22
31
58
S2 sid
28
31
44
58
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
sname rating age
dustin
7
45.0
lubber
8
55.5
rusty
10 35.0
sname rating age
yuppy
9
35.0
lubber
8
55.5
guppy
5
35.0
rusty
10 35.0
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Set Operations





Entities and Relations are sets (no duplicates).
Usual set operations are Union, Intersection and SetDifference, Cartesian Product.
All of these operations take two input relations, which must
be union-compatible:
 Same number of attributes (degree).
 `Corresponding’ attributes have the same type.
Note that attributes can have different names.
The output is a set.
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Union, Intersection, Set-Difference
sid sname rating age
22
31
58
44
28
dustin
lubber
rusty
guppy
yuppy
7
8
10
5
9
45.0
55.5
35.0
35.0
35.0
sid sname rating age
31 lubber 8
55.5
58 rusty
10
35.0
S1 S2
sid sname
22 dustin
S1 S2
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
rating age
7
45.0
S1 S2
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Cartesian-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.

(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
 Renaming operator:
Introduced later
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Projection



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.
Projection operator has to
eliminate duplicates! (Why??)
 Note: real systems typically
don’t do duplicate elimination
unless the user explicitly asks
for it. (Why not?)
sname
rating
yuppy
lubber
guppy
rusty
9
8
5
10
 sname,rating(S2)
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
age
35.0
55.5
 age(S2)
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Selection



Selects rows that satisfy
selection condition.
Schema of result
identical to schema of
(only) input relation.
Result relation can be
the input for another
relational algebra
operation! (Operator
composition.)
sid sname rating age
28 yuppy 9
35.0
58 rusty
10
35.0
 rating 8(S2)
sname rating
yuppy 9
rusty
10
 sname,rating( rating 8(S2))
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Renaming

Renaming can be represented as a
projection
 sname  first (A)
A
sname
rating
first
rating
yuppy
lubber
guppy
rusty
9
8
5
10
yuppy
lubber
guppy
rusty
9
8
5
10
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Joins -- theta-join -
Condition Join:
(sid)
22
31
R  c S   c ( R  S)
sname rating age
dustin 7
45.0
lubber 8
55.5
S1 
(sid) bid
58
103
58
103
S1. sid  R1. sid
day
11/12/96
11/12/96
R1
Result schema same as that of cross-product.
 Fewer tuples than cross-product, might be
able to compute more efficiently
 Sometimes called a theta-join.

Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Joins – Equi-Join -
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
S1 


R1
sid
Result schema similar to cross-product, but only
one copy of fields for which equality is specified.
Natural Join: Equijoin on all common fields.
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Joins -- Left Outer-Join -
Left Outer-Join: S1
sid
R1
Contains all records of the left table even if the joincondition does not find any matching records in the
right table.
sid
22
58
31
sname
dustin
rusty
lubber
rating
7
10
8
age
45.0
35.0
35.5
bid
101
103
Null
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
day
10/10/96
11/12/96
Null
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Division
Not supported as a primitive operator, but useful for
expressing queries like:
Find sailors who have reserved all boats.
 Let A have 2 fields, x and y; B have only field y:
 A/B =  x |  x , y  A  y  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.
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Examples of Division A/B
sno
s1
s1
s1
s1
s2
s2
s3
s4
s4
A
bno
b1
b2
b3
b4
b1
b2
b2
b2
b4
bno
b2
b4
bno
b2
B2
bno
b1
b2
b4
sno
s1
s4
sno
s1
A/B2
A/B3
B1
sno
s1
s2
s3
s4
A/B1
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
B3
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Division Analogy

r/s = c. cs  r

R/S = C . CS  R
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Find names of sailors who’ve reserved a red boat

Information about boat color only available in
Boats; so need an extra join:
 sname ((
Boats)  Re serves  Sailors)
color ' red '

A more efficient solution:
 sname ( (( 
Boats)  Re s)  Sailors)
sid bid color ' red '
A query optimizer can find this, given the first solution!
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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END
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Expressing A/B Using Basic Operators

Division is not essential op; just a useful shorthand.


(Also true of joins, but joins are so common that systems
implement joins specially.)
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:
 x ( A) 
 x (( x ( A) B)  A)
all disqualified tuples
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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