1. Algebraic Laws

2. consider physical plans
{P1,P2,…..}
{P1,C1>...}
parse
convert
apply laws
estimate result sizes
consider physical plans
estimate costs
pick best
execute Pi
answer
SQL query
parse tree
logical query plan
“improved” l.q.p
l.q.p. +sizes
statistics

3. Algebraic laws for improving query plans
Commutative and associative laws:
Proofs of the above identities?
Above laws are applicable for sets and bags

4. Theta-join Commutative: Not always associative:
On schema R(a,b), S(b,c), T(c,d)
can not be transformed into:

5. Laws Involving Selection ()

6. If all attributes in the condition C are in R (for binary operators)
Note that in union R and S have to have the same attributes.

7. Example:
Consider relation schemas R(A,B) and S(B,C) and the RA expression below:
(A=1 OR A=3) AND B < C(R ⊳⊲ S)
1. Splitting AND  A=1 OR A=3 (B < C(R ⊳⊲ S))
2. Push  to S  A=1 OR A=3 (R ⊳⊲ B < C(S))
3. Push Cond to R  A=1 OR A=3 (R) ⊳⊲ B < C(S)
4. Splitting OR  (A=1 (R) A=3(R)) ⊳⊲ B < C(S)
Step 4 above may or may not be useful.

8. Some Trivial Laws Watch for some extreme cases: -- an empty relation:
e.g., R⋃S = S, if R = ∅
-- a selection or theta-join whose
condition is always satisfied
e.g., C(R) = R, if C = true
-- a projection on all attributes is
“better” not to be done at all!!

9. Pushing selections
Usually selections are pushed down the expression tree.
The following example shows that it is sometimes useful to pull selection up in the tree.
StarsIn(title,year,starName)
Movie(title,year,length,studioName)
CREATE VIEW MoviesOf1996 AS
SELECT * FROM MOVIE WHERE year=1996;
Query: Which stars worked for which studios in 1996?
SELECT starName,studioName
FROM MoviesOf1996 NATURAL JOIN StarsIN;

10. pull selection up
then push down

11. Laws for (bag) Projection
A simple law: Project out attributes that are not needed later.
i.e. keep only the output attr. and any join attribute.

12. Laws for (bag) Projection …
Bag projections cannot be pushed below ⋃S, and no projection can be pushed below either set/bag versions of ⋂ and –
Example: Consider relation schemas R(A,B) and S(A,B). Supp. R has only (1,2) and S has only (1,3).
 A(R ⋂ S) = A(∅) but A(R) ⋂ A(S) = {(1)}

13. Examples for pushing projection
Schema R(a,b,c), S(c,d,e)

14. Example: Pushing Projection
Schema: StarsIn(title,year,starName)
Query: SELECT starName FROM StarsIn
WHERE year = 1996;
starName
year=1996
StarsIn
starName,year
starName
year=1996
StarsIn
Should we transform to  ?
Depends!

15. Reasons for not pushing the projection
If there is an index on StarsIn.year, such index is useless in the projected relation starName,year(StarsIn)
While such an index is very useful for the selection on “year=1996”

16. Laws for duplicate elimination, grouping, and aggregations
Two general rules: ( absorbs )
, L ⊆ M
Note: (R)=R, if R has no duplicates
(e.g., base relation or result of a grouping)
Defn: L is duplicate-impervious if it is not affected by duplicates, i.e., MIN or MAX. In this case,

17. Example
Schema: StarsIn(title,year,starName) MovieStar(name,address,gender,birthdate)
Query: Find for each year, the birth date of the youngest star to appear in a movie in that year.

18. Improving logical query plans
Push  as far down as possible (sometimes pull them up first).
Do splitting of complex conditions in  in order to push  even further.
Push  as far down as possible, introduce new early  (but take care for exceptions)
Remove or move 
Combine  with  to produce -joins or equi-joins
Choose an order for joins

19. Example of improvement
SELECT title
FROM StarsIn, MovieStar
WHERE starName = name AND birthdate LIKE '%1960';
title
StarsIn
MovieStar
starName=name
birthdate LIKE ‘%1960’
title
starname=name AND birthdate LIKE ‘%1960’
StarsIn 
MovieStar

20. And a better plan introducing a projection
to filter out useless attributes:
title
StarsIn
MovieStar
starName=name
birthdate LIKE ‘%1960’
name