1. An Algebraic Query Language
Relational Algebra
An Algebraic Query Language
Lu Chaojun, SJTU

2. Operations on Data Recall: A data model provides
structures
operations
constraints
Relational model provides operations on relations, expressed in
Relational algebra
Relational calculus
SQL
 "syntactically sugared" RA (+RC)
Lu Chaojun, SJTU

3. Relational Algebra Algebra operands+operatorsexpressions
operands: relations
relation = a set of tuples
operators = set op + special relational op
Lu Chaojun, SJTU

4. Relational Algebra Expressions
Relations as operands, represented by
relation names, or
constants (a list of tuples)
Apply operators to relations/subexpressions
Define the schema and set of tuples of the resulting relation!
Query = relational algebra expression
Lu Chaojun, SJTU

5. Simplest Query Our running example: Relation name itself is a query.
Student(sno,name,age,dept)
Course(cno,name,credit)
SC(sno,cno,grade)
Relation name itself is a query.
E.g. Student
returns the whole relation as the result
Lu Chaojun, SJTU

6. Set Operations Union: RS Intersection: RS Difference: RS
Insertion
Duplicates elimination
Intersection: RS
Difference: RS
Deletion
Compatibility condition:
R and S must have schemas with identical set of attribute:type pairs.
Before computation, make the order of attributes the same for R and S. 6
Lu Chaojun, SJTU

7. Example: Union R: S: RS: Note that one copy of (,2) is eliminated. AB A B   1 2   2 3 A B   1 2 3
Note that one copy of (,2) is eliminated. 7
Lu Chaojun, SJTU

8. Example: IntersectionB A B   1 2   2 3 A B  2
Lu Chaojun, SJTU

9. Example: Difference R: S: RS: A B A B   1 2   2 3 A B   1 9
Lu Chaojun, SJTU

10. Projection Projected on some columns: L(R)
L is a list of attributes of R
E.g. name,age(Student)
Duplicates elimination
E.g. age(Student)
Lu Chaojun, SJTU

11. Example: Projection R: A,C (R): A B C   10 20 30 40 1 2 A C A C  = 11
Lu Chaojun, SJTU

12. Selection Selection of some rows: C(R) Compute C(R):
C is a conditional expression involving the attributes of R
attributes, constants,=,<,>,AND,OR,NOT
E.g. age>20 AND dept=‘CS’(Student)
Compute C(R):
Apply C to each tuple t of R by substituting t[A] for each A in C.
If t makes C true, then t is in C(R).
Lu Chaojun, SJTU

13. Example: Selection R: A=B ^ D>5 (R) A B C D     1 5 12 23 7 310 A B C D     1 23 7 10 13
Lu Chaojun, SJTU

14. Cartesian Product Combining tables: RS RR is possible.
Union of attributes
Use qualified name to avoid ambiguities
e.g. Student.name and Course.name
Pairing tuples in all possible combinations
e.g. Student  Course
RR is possible.
It looks odd to glue unrelated tuples together, so  is not useful.
Lu Chaojun, SJTU

15. Example: Cartesian Product
R: S:
RS: A B C D E   1 2    10 20 a b A B C D E   1 2    10 20 a b 15
Lu Chaojun, SJTU

16. Theta-Join Join: pairing tuples that match.
joined tuple
dangling tuple
Special kind of product: R  S
(RS)
Use  to restrict possible pairing
Example :-)
name(Student Student.sno=SC.sno AND age>grade SC)
Most often:  is an equation involving shared attributes of R and S.
Lu Chaojun, SJTU

17. Example: Theta-Join R: S: R A=C S: A B C D E   1 2    10 20 a b A17
Lu Chaojun, SJTU

18. Natural Join Special kind of Theta-Join: R S Example
Implicitly,  is: equating shared attributes of R and S
Only one copy of shared attributes is left in the resulting relation
Example
Student.name,Course.name,grade(Student SC Course)
Lu Chaojun, SJTU

19. Example: Natural Join R: S: R S: A B C D B D E     1 2 4    a b3 2 a c b      A B C D E   1 2    a c    19
Lu Chaojun, SJTU

20. Naming and Renaming Control the names of relations/attributes S(R)
S(A1,…,An)(R)
S(expression)
S(A1,…,An)(expression)
Lu Chaojun, SJTU

21. Combining Operations Apply op. to subexpressions.
More than one way to write a query
query optimizer
Operator precedence
Unary ops have highest precedence: , , 
“Multiplicative” ops are next : ,  ,
“Additive” ops are lowest: , , 
Use parentheses !
Expression tree
Lu Chaojun, SJTU

22. Quotient T = RS Defined for R(X,Y) and S(Y) T(X)
T = {t | t  X(R) AND sS(ts  R)}
Lu Chaojun, SJTU

23. Example: Quotient(1) R: S: RS: A B B A 1 2        1 2 3 4 6 23
Lu Chaojun, SJTU

24. Example: Quotient(2) R: S: RS: A B C D E D E    a    a b 1 3 a24
Lu Chaojun, SJTU

25. Example: Quotient(3)
Find those students who have selected all the courses.
sno,cno(SC)  cno(Course)
Find those students who have selected at least the courses that S007 has taken.
sno,cno(SC)  cno(sno=‘S007’(SC))
Exercise
RS = X(R)  X((X(R)  S)  R)
Lu Chaojun, SJTU

26. Semijoin
R S
R’s tuples that agree with at least one tuple in S on all shared attributes
Exercise 2.4.8
Equivalent expressions:
(1) R(R S)
(2) ?
(3) ?
Lu Chaojun, SJTU

27. Example: Semijoin R: S: R S: A B C D B D E     1 2 4    a b c 13 2 a c b      A B C D   1 2    a c 27
Lu Chaojun, SJTU

28. Independent Set of Operators
Some op. can be expressed in terms of others.
Independent set of operators:
{, , , , , }
R  S = R  (R  S)
R C S = C (R  S)
R S = L (C (R  S))
C: equating shared attributes
L: eliminating duplicate attributes
Lu Chaojun, SJTU

29. Assignment A linear notation for complex RA expressions
R(a,b,c) := exp.
R is a temporary relation
Answer(...) for the resulting relation
Example
R(no,n,a,d) := dept=‘CS’ (Student)
Answer(sno,name) := no,n(R)
Lu Chaojun, SJTU

30. RA as a Constraint Language
Two ways to express constraints
equal-to-the-emptyset
R = 
set-containment
R  S
The two ways are equivalent:
R =  iff R  
R  S iff RS = 
Lu Chaojun, SJTU

31. Referential Integrity Constraints
The object/entity referenced by another object/entity must really exist.
Example
In SC(sno,cno,grade), if there’s a tuple ('001','CS123',90), then 'CS123' must exist in Course.
We can express this constraint by
cno(SC)  cno(Course)
Lu Chaojun, SJTU

32. Key Constraint Example: Student(sno,name,age,dept)
No two tuples agree on sno component.
S1.sno=S2.sno AND S1.deptS2.dept(S1 S2) = 
where S1 and S2 are the results of renaming Student:
S1(Student)
S2(Student)
Similar for other attributes.
Lu Chaojun, SJTU

33. Some Other Constraints
Domain constraint: permitted values for an attribute.
Example
age<0 OR age>200(Student) = 
More complex constraints
age<18 AND cno=‘C1’ (Student SC) = 
Lu Chaojun, SJTU

34. End