16.2.Algebraic Laws for
Improving Query Plans
16.2 Algebraic Laws for Improving
Query Plans
•
•
•
•
•
•
•
•
16.2.1 Commutative and Associative Laws
16.2.2 Laws Involving Selection
16.2.3 Pushing Selections
16.2.4 Laws Involving Projection
16.2.5 Laws About Joins and Products
16.2.6 Laws Involving Duplicate Elimination
16.2.7 Laws Involving Grouping and Aggregation
16.2.8 Exercises for Section 16.2
16.2.4 Laws Involving Projection
• Projection, like selection can be pushed
down through many other operators
• Pushing Projection usually involves
introducing a new projection somewhere
below an existing projection.
• Projection differs from selection in the
aspect that projection reduces the length
of the tuples whereas selection reduces
the number of the tuples
16.2.4. Laws involving Projection
• Consider term π E  x
– E : attribute, or expression involving attributes and constants.
– All attributes in E are input attributes of projection and x is output
attribute
• Simple projection: if a projection consists of only attributes.
– Example: π a,b,c (R) is simple. a,b,c are input and output
attributes.
• Projection can be introduced anywhere in expression
tree as long as it only eliminates attributes that are never
used.
16.2.4. Laws involving Projection (cont….)
• πL(R
S) = πL(πM(R) πN(S)) ; M and N are all
attributes of R and S that are either join (in schema of
both R and S) or input attributes of L
• πL(R c S) = πL(πM(R) c πN(S)) ; M and N are all
attributes of R and S that are either join(mentioned in
condition of C ) or input attributes of L
• πL(R x S) = πL(πM(R) x πN(S)) ; M and N are all
attributes of R and S that are input attributes of L
Projections cannot be pushed below set unions or either of
set or bag versions of intersection or difference at all.
16.2.4 Laws Involving Projection
SELECT starName FROM StarsIn WHERE year = 1996
π
starName
σ
movieYear = 1996
StarsIn
Fig : Logical query plan for the above query
We can introduce a projection in the above Figure
16.2.4 Laws Involving Projection
π
starName
σ
movieYear = 1996
π
starName, movieYear
StarsIn
Convert the tree into relational algebra, then simplify as much as you
can
16.2.5 Laws About Joins and
Products
• RCS= C(R S)
• RS= L (C(R S))
Where C is the condition that equates each
pair of atrribute from R and S with the
same name, and L is the list that includes
one attribute from each equted attributed
and all other attributes of R and S.
16.2.6 Laws Involving Duplicate
Elimination
• The operator δ , which eliminates duplicates
from a bag can be pushed through only some of
the operators
• Moving δ down the tree reduces the size of
intermediate relation and may therefore be
beneficial
• In some cases, we can move δ to a position
where it can be eliminated because it is applied
to a relation that does not have any duplicates
16.2.6 Laws Involving Duplicate
Elimination
• δ( R ) = R if R has no duplicates
Important cases of such a relation R include
1. A stored relation with a declared primary
key
2. A relation that is the result of a γ
operation ,since grouping creates a
relation with no duplicates
• δ cannot be moved across the operators
like U , - , π.
16.2.6 Laws involving duplicate elimination
Laws that “push” δ (delta) through other operator
• δ(R x S) = δ(R) x δ(S)
• δ(R S) = δ(R) δ(S)
• δ(R c S) = δ(R) c δ(S)
• δ( c(R)) = c(δ(R))
δ eliminates duplicates from a bag, but cannot be pushed through
all the operators
16.2.7 Laws Involving Grouping
and Aggregation
• While using grouping and aggregation ,the
applicability of many transformation
depends on the details of the aggregation
used.
• Due to the above ,we cannot state laws in
generality.
• One exception is the law below that γ
absorbs δ
δ(γL(R)) = γL ( R )
16.2.7 Laws Involving Grouping
and Aggregation
• We may project useless attributes prior to
applying γ operation
•
γL ( R ) = γL(πM (R )
where M is the list containing at least all
those attributes of R that are mentioned in
L.
Laws involving grouping and aggregation (cont…)
• Some aggregations like MIN and MAX are not affected
by presence or absence of duplicates
• Others like SUM,COUNT,AVG produce different values if
duplicates are eliminated prior to aggregation.
16.2.7 Laws Involving Grouping
and Aggregation
• Suppose we have the relation
•
MovieStar(name ,addr ,gender ,birthdate)
•
StarsIn(movieTitle ,movieYear ,starName)
• Consider the query below
•
Select movieYear ,MAX(birthDate)
•
FROM MovieStar ,StarsIn
•
WHERE name = starName
•
GROUP BY movieYear
16.2.7 Laws Involving Grouping
and Aggregation
• The FROM list is expressed by a product and
the WHERE clause by a selection above it.
• The grouping and aggregation are expressed by
the γ.
• Combine the selection and product into an
equijoin
• Generate a δ below the γ ,since the γ is
duplicate-impervious
• Generate a π between the γ and the introduced
δ to project onto movieYear and birthDate ,the
only attributes relevant to the γ
16.2.7 Laws Involving Grouping
and Aggregation
γmovieYear ,MAX(birthDate)
σname = starName

MovieStar
StarsIn
1. Use 16.2.5. (and following 2 reasons) we can rewrite the tree
2. There is no duplication in output (because γ), we can add 
3. By projection law We can add .
16.2.7 Laws Involving Grouping
and Aggregation
γmovieYear ,MAX(birthDate)
movieYear ,birthDate
δ
name = starName
MovieStar
Figure : Second query plan
StarsIn
16.2.7 Laws Involving Grouping
and Aggregation
γ movieYear ,MAX(birthDate)
π movieYear ,birthDate
name = starName
δ
πbirthDate,name
MovieStar
δ
πbirthDate,name
StarsIn
Figure : Third query plan  can be push down
16.2.7a Additional Example
• From DB1
16.2.7a Laws Involving Grouping
and Aggregation
• SELECT
• FROM
• GROUP
PNUM, SUM(QTY)
SHIPMENTs, Parts
BY PNAME;
16.2.7b Laws Involving
Grouping and Aggregation
γpname ,SUM(qty) sum
σpnum = pnum

Shipments (Sh)
Parts(P)
Figure : Initial Logical query plan for the query
γpname ,SUM(qty)  sum(σsh.pnum=p.pnum (ShipmentsParts))
16.2.7c Laws Involving
Grouping and Aggregation
γpname ,SUM(qty)sum
pname ,QTY
δ
Shipments
Parts
γpname ,SUM(qty)sum ( pname.qty (Shipments Parts))
16.2.7d Laws Involving
Grouping and Aggregation
γpname ,SUM(qty)sum
π pname ,qty
δ
πqty,pnum
Shipments
δ
pnum, pname
Parts
γpname ,Sum(qty) sum ( pname.qty ((  pname.qty (Shipments)) (  pname.qty (Parts))))
16.2.8 Exercises for Section 16.2
16.1. SQL(not RAE) Figure 16.2
• select movietitle
• from starsIn
• where starname in
•
(select name
•
from moviestar
•
where birthdate
•
like '%1960');
16.3. SQL/RAE of Figure 16.19
•
•
•
•
•
•
select movietitle
from starsIn a, (select name
from moviestar
where birthdate like '%1960') temp
where a.starname = temp.name;
RAE=Relational Algebra Expressible SQL
16.3 Figure 16.19
γpname ,SUM(qty)sum
π pname ,qty
δ
πqty,pnum
Shipments
δ
pnum, pname
Parts
γpname ,Sum(qty) sum ( pname.qty ((  pname.qty (Shipments)) (  pname.qty (Parts))))
SQL in Figure 16.20
Select distinct m1.movieTitle, m1.movieYear
From StarsIn m1
Where m1.movieYear – 40 <=
(
Select AVG(birthdate)
From StarsIn m2, Moviestar s
Where m2.starName=s.name AND
m1.movieTitle = m2.movieTitle AND
m1.movieyear = m2.movieyear
);
SQL in Figure 16.22
Select distinct m.movieTitle, m.movieYear
From StarsIn m1, (Select m2.movieTitle,
m2.movieyear, AVG(birthdate) as ave
From StarsIn m2, Moviestar s
Where m2.starName=s.name
Group by m2.movieTitle,
m2.movieyear ) m
Where m1.movieTitle = m.movieTitle and
m1.movieYear – 40 <=ave;