Transcript Algebra

Database Management Systems
Chapter 4
Relational Algebra
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
1
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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Preliminaries

A query is applied to relation instances, and 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 the result of a given query is also
fixed! Determined by definition of query language
constructs.
Positional vs. named-field notation:


Positional notation easier for formal definitions,
named-field notation more readable.
Both used in SQL
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Example Schema
Sailors (sid: integer, sname: string, rating:
integer, age: real);
 Boats (bid: integer, bname: string, color:
string);
 Reserves (sid: integer, bid: integer, day: date);

Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Example Instances

R1 sid
22
58
We’ll use positional or
sid
S1
named field notation,
22
assume that names of fields
in query results are
31
`inherited’ from names of
58
fields in query input
relations.
S2 sid
28
31
44
58
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
bid
day
101 10/10/96
103 11/12/96
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
5
Relational Algebra

Basic operations:








Additional operations:


Selection ( ) Selects a subset of rows from relation.
Projection ( ) Deletes unwanted 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 and in reln. 2.
Intersection, join, division, renaming: Not essential, but
(very!) useful.
Since each operation returns a relation, operations
can be composed!
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.
No duplicates in result!
(Why?)
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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Set Operation - Union


R U S returns a relation instance
containing all tuples that occur
in either relation instance R or
relation instance S (or both), and
the schema of the result is
defined to be identical to the
schema of R.
R and S are union-compatible
S1
S2
 They have the same number of
the fields,
 Corresponding fields, taken in
order from left to right, have
the same domains.
sid
22
31
58
sid
28
31
44
58
sname
dustin
lubber
rusty
rating
7
8
10
age
45.0
55.5
35.0
sname rating age
yuppy
9
35.0
lubber
8
55.5
guppy
5
35.0
rusty
10 35.0
sid sname
rating age
22
31
58
44
28
7
8
10
5
9
dustin
lubber
rusty
guppy
yuppy
45.0
55.5
35.0
35.0
35.0
S1 S2
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Set Operation - Intersection
returns a relation
instance containing all
tuples that occur in
both R and S, and the
schema of the result is
defined to be identical
to the schema of R.
R and S must be unioncompatible.
S1
 R S

S2
sid
22
31
58
sid
28
31
44
58
sname
dustin
lubber
rusty
rating
7
8
10
age
45.0
55.5
35.0
sname rating age
yuppy
9
35.0
lubber
8
55.5
guppy
5
35.0
rusty
10 35.0
sid sname rating age
31 lubber 8
55.5
58 rusty
10
35.0
S1S 2
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Set Operation - Set-Difference


sid
22
31
58
sname
dustin
lubber
rusty
rating
7
8
10
age
45.0
55.5
35.0
R-S returns a relation
S1
instance containing all
tuples that occur in R but
not in S, and the schema
sid sname rating age
28 yuppy
9
35.0
of the result is defined to
S2 31 lubber 8 55.5
be identical to the schema
44 guppy
5
35.0
58 rusty
10 35.0
of R.
The relations R and S
must be unionsid sname rating
compatible.
22 dustin 7
age
45.0
S1 S2
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Set Operation – Cross-product
R1 sid bid

R X S returns a relation
instance whose schema
contains all the fields of
R followed by all the
fields of S, this is one
tuple <r,s> for each pair
of tuples r∈R, s∈S.
day
101 10/10/96
103 11/12/96
22
58
sid
S1
22
31
58
sname rating age
dustin
7
45.0
lubber
8
55.5
rusty
10 35.0
(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
S1 X R1
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Renaming

Renaming operator ρ takes an relational
algebra expression E and returns an instance
of a relation R.
( R(1 sid1, 5 sid 2), S1 R1)
position
New name
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Joins
One of most useful operations and most
commonly used to combine information from
two or more relations.
 A join operation is defined as a cross-product,
then a selection, and projection.

R S
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Condition Join
R  c S   c ( R  S)
(sid) sname rating age
22
dustin 7
45.0
31
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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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
S1.sid  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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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 1 field y:




A/B contains all x tuples such that for every y tuple in B,
there is an x y tuple in A.
Or: If the set of y values associated with an x value 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
pno
p1
p2
p3
p4
p1
p2
p2
p2
p4
pno
p2
sno
s1
s2
s3
s4
sno
s1
s4
sno
s1
A
A/B1
A/B2
A/B3
pno
p2
p4
B1
B2
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
pno
p1
p2
p4
B3
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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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Q1:
Find names of sailors who’ve reserved boat #103


Solution 1:
Solution 2:
 sname((
bid 103
 (Temp1, 
Reserves)  Sailors)
C
bid  103
Re serves)
 ( Temp2, Temp1  Sailors)
C
 sname (Temp2)

Solution 3:
 sname (
bid 103
(Re serves  Sailors))
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
C
20
Q2:
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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Q5:
Find sailors who’ve reserved a red or a green boat

Can identify all red or green boats, then find
sailors who’ve reserved one of these boats:
 (Tempboats, (
color ' red '  color ' green '
Boats))
 sname(Tempboats  Re serves  Sailors)

Can also define Tempboats using union! (How?)

What happens if  is replaced by  in this query?
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Q6:
Find sailors who’ve reserved a red and a green boat

Previous approach won’t work! Must identify
sailors who’ve reserved red boats, sailors
who’ve reserved green boats, then find the
intersection (note that sid is a key for Sailors):
 (Tempred, 
sid
 (Tempgreen, 
((
sid
color ' red '
((
Boats)  Re serves))
color ' green'
Boats)  Re serves))
 sname((Tempred  Tempgreen)  Sailors)
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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Q9:
Find the names of sailors who’ve reserved all boats

Uses division; schemas of the input relations
to / must be carefully chosen:
 (Tempsids, (
sid, bid
Re serves) / (
bid
Boats))
 sname (Tempsids  Sailors)

To find sailors who’ve reserved all ‘Interlake’ boats:
.....
/
bid
(
bname ' Interlake'
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
Boats)
24
Summary
The relational model has rigorously defined
query languages that are simple and
powerful.
 Relational algebra is more operational; useful
as internal representation for query
evaluation plans.
 Several ways of expressing a given query; a
query optimizer should choose the most
efficient version.

Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke
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