1. CS257
Query Optimization

2. 2.4 An Algebraic Query Language

3. 2.4 An Algebraic Query Language
2.4.1 Why Do We Need a Special Query Language?
2.4.2 What is an Algebra?
2.4.3 Overview of Relational Algebra
2.4.4 Set Operations on Relations
2.4.5 Projection
2.4.6 Selection
2.4.7 Cartesian Product
2.4.8 Natural Joins
2.4.9 Theta-Joins
Combining Operations to Form Queries
Naming and Renaming
Relationships Among Operations
A Linear Notation for Algebraic Expressions
Exercises for Section 2.4

4. 2.4.1 Why Do We Need a Special Query Language?

5. 2.4.2 What is an Algebra? + - * / Projection Selection Natural Join
Cartesian Product
Union
Intersect
Minus

6. 2.4.3 Overview of Relational Algebra
Both numerical algebra and relational algebra are defined by (respective sets of)
Algebraic Laws.

7. 2.4.3 Overview of Relational Algebra
Algebraic laws of Numbers
that we have learned in calculus, algebra and etc since high school days

8. 2.4.3 Overview of Relational Algebra
Algebraic laws
of relational algebra are
“New” to most of us

9. 2.4.3 Overview of Relational Algebra
Special Relational Operators
Projection
Selection
Natural Join
Traditional Set Operators
4. Cartesian Product
5. Union
6. Intersect
7. Minus

11. QUERY OPTIMIZATION
In query optimization the query is transformed by compiler into such a form that can solve the problem “fastest ”

12. QUERY OPTIMIZATION Illustration: 2 * 3 + 5 * 3
From the laws of Numerical algebra, it can be computed as follows:.
Method I - It computes in three operations
2 * * 3 = 6 (first) + 15(second)
= 21 (third)

13. QUERY OPTIMIZATION Illustration: 2 * 3 + 5 * 3
From the laws of Numerical algebra, it can Also be computed as follows:.
Method II- transform into equivalent one.
= (2 + 5) * 3 = 7 (first) * 3 = 21 (second)
It computes in two operations
So compiler choose Method II

14. Roles of Relational Algebra
QUERY OPTIMIZATION
Similar Idea is used in Relational Algebra
So we need to know the
Algebraic Laws
of Relational Algebra well
That is the main goal for Ch5 and part of Ch16

15. 2.4.4 Set Operations on Relations
Recall Venn Diagram Y X b c d e a

16. Query Algebra In DBMS a query is turn into
a sequence of operators on U
These operators forms the query algebra (Extended RA)

17. Query Algebra RS  SUM (# of times in R, - - - - - - S)
Bag operations:
RS  SUM (# of times in R, S)
R∩S  Min (# of times in R, S)

18. Examples Two bags: R = {A, B, B} S = {C, A, B, C}
R  S ={A, A, B, B, B, C, C}
R ∩ S = {A, B}
R—S ={B} (Obvious?)

19. Selection: C(R) from R where C C can involve computable formulas
SQL: Select *
from R
where C
C can involve computable formulas

20. Projection: L(R)
Select L
from R

21. Projection: L(R) L can have : 1. A single attribute of R.
2. An expression x  y: It means we take the attribute x of R and rename it as y.
3. An expression E  z where E is an expression involving attributes of R and z is a new name for the attribute that results from the calculation
. Ex: a+b  x

22. Theta Join:  Theta Join: R c S = σc(RS) R c S = πL(σc(RS)) Where
Natural Join (Special Case of Theta Join)
R c S = πL(σc(RS))
Where
L: The list that meets the condition c of equality and redundant attributes are dropped

23. SORTING OPERATOR ()  L (R)
This is the SQL ORDER BY clause and denoted by the operator 
 L (R)
where R is a relation and L a list of some of R’s attributes in the relation R but with the tuples of R sorted in the order indicated by L.
If L is a1, a2…an, then tuples are first sorted by a1, then a2 until an. By default sorting is in ascending order.

24. Grouping and Aggregation: L(R)
Select L(=A,B)
from R
group by A
A: aggregating attribute
B: aggregated attributes

25. Grouping and Aggregation: L(R)
It returns a relation that partitions the tuples
of R in to groups. Each group consists of all
tuples having one particular assignment of
values to the grouping attributes A in L.
L also contains B, Aggregated attributes, in the form: Aggregation operator  Name

26. Grouping and Aggregation
Aggregation operators :
AVG, SUM, COUNT, MIN, MAX
Grouping: GROUP BY clause in SQL
Having clause must follow a GROUP BY clause

27. Grouping and Aggregation
Grouping and aggregation are generally implemented together. So we have a single operator defining it
It is a generalized Projection Operator
Delicate-elimination operator  is a special Aggregation operator.

28.  pnum, sum(qty)sum(SP)
Select pnum, sum(qty) as sum
from SP
group by pnum;

29. Some rules on selection:
1.σc1 and c2(R) = σc1(σc2 (R) )
2.σc1 or c2(R) = (σc1R) s (σc2 R)
whenever c apply to R or S

30. Some Rules about Selection:
3.σc(R  S) = (σcR )  (σc S)
4.σc(R  S) = (σcR)  (σc S)
5.σc(R  S) = (σcR)  (σcS)
whenever c apply to R or S

31. Rules about (generalized) Projection:
6. We may introduce a (generalized) projection anywhere in an expression tree, as long as it eliminates only attributes that are never used by any operator above

32. Some Rules about Projection:
7. L(R c S ) =L(M(R) c N(R))
8. L(R  S ) =L(R) L(S)

33. Query Optimization Example
Select p.pname, p.pnum, sum(sp.qty) as sum
from Parts p, Shipments sp
where p.pnum = sp.pnum
and p.weight > 10
group by p.pname, p.pnum
having sum(sp.qty) >= 200;

34. Translating to Query Algebra
Step 1 (P  SP)
Step 2σP.PNUM = SP.PNUM and P.WEIGHT >10(PSP) = 
Step 3σSUM >= 200 (γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM ())=, 
Step 4 πP.PNAME, P. PNUM, SUM () = 

35. P SP πP.PNAME, P. PNUM, SUM σSUM >= 200
σSUM >= 200
γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM
σP.PNUM = SP.PNUM and P.WEIGHT >10 P SP

36. Computing in QA
 = πP.PNAME, P. PNUM, SUM(   γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
σP.WEIGHT >10 and P.PNUM = SP.PNUM (PSP)   )
=(1)πP.PNAME, P. PNUM, SUM(   γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
σP.WEIGHT >10 (σP.PNUM = SP.PNUM (PSP)   )

37. πP.PNAME, P. PNUM, SUM(   γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
Computing in QA
πP.PNAME, P. PNUM, SUM(   γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
σP.WEIGHT >10 (σP.PNUM = SP.PNUM (PSP)   )=(6)
σP.WEIGHT >10 (πLσP.PNUM = SP.PNUM (PSP)   )

38. πP.PNAME, P. PNUM, SUM(   γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
Computing in QA
πP.PNAME, P. PNUM, SUM(   γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
σP.WEIGHT >10 (πLσP.PNUM = SP.PNUM (PSP)   )=D
πP.PNAME, P. PNUM, SUM(   γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM ( σP.WEIGHT >10 (P P.PNUM = SP.PNUM SP   )

39. πP.PNAME, P. PNUM, SUM(   γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
Computing in QA
πP.PNAME, P. PNUM, SUM(   γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM ( σP.WEIGHT >10 (P P.PNUM = SP.PNUM SP   )=(5)
πP.PNAME, P. PNUM, SUM(   γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
σP.WEIGHT >10 (P) P.PNUM = SP.PNUM σP.WEIGHT >10 (SP)   )

40. πP.PNAME, P. PNUM, SUM (   γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
Computing in QA
πP.PNAME, P. PNUM, SUM (   γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
(σP.WEIGHT >10 (P) P.PNUM = SP.PNUM σP.WEIGHT >10 (SP)   ) =D
πP.PNAME, P. PNUM, SUM(    γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
σP.WEIGHT >10 (P) P.PNUM = SP.PNUM (σP.WEIGHT >10 =1)(SP)  )=

41. σSUM >= 200 (γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
Final Expression
= πP.PNAME, P. PNUM, SUM(
σSUM >= 200 (γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM (
σP.WEIGHT >10 (P) P.PNUM = SP.PNUM (SP)  )= 

42. γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM
σSUM >= 200
γP.PNAME, P. PNUM, SUM(SP.QTY) → SUM
P.PNUM = SP.PNUM
σp.weight SP P

43.   Final Expression is computationally cheaper than
σP.WEIGHT >10 (P) is smaller than P

44. σp.weight SP P πP.PNAME, P. PNUM, SUM σSUM >= 200 P.PNUM = SP.PNUM
σSUM >= 200
P.PNUM = SP.PNUM
σp.weight
γP. PNUM, SUM(SP.QTY) → SUM SP P

45. P.PNUM = SP.PNUM σp.weight P SP πP.PNAME, P. PNUM, SUM σSUM >= 200
P.PNUM = SP.PNUM
σp.weight
σSUM >= 200 P
γP. PNUM, SUM(SP.QTY) → SUM SP

47. EXPRESSION TREES
Generated by combining several Qerry Algebra operators into one expression by applying one operator to the result(s) of one or more operators.
The leaves of this tree are names of relations.
Interior nodes are operators, which are applied to the relations represented by its child or children