1. COP4710 Database Systems
Relational Algebra

2. Why Do We Learn This?
Querying the database: specify what we want from our database
Find all the people who earn more than $1,000,000 and pay taxes in Tallahassee
Could write in C++/Java, but a bad idea
Instead use high-level query languages:
Theoretical: Relational Algebra, Datalog
Practical: SQL
Relational algebra: a basic set of operations on relations that provide the basic principles

3. What is an “Algebra”? Mathematical system consisting of: Examples
Operands --- variables or values from which new values can be constructed
Operators --- symbols denoting procedures that construct new values from given values
Examples
Arithmetic(Elementary) algebra, linear algebra, Boolean algebra ……
What are operands?
What are operators?

4. What is Relational Algebra?
An algebra
Whose operands are relations or variables that represent relations
Whose operators are designed to do common things that we need to do with relations in a database
relations as input, new relation as output
Can be used as a query language for relations!

5. Relational Operators at a Glance
Five basic RA operations:
Basic Set Operations
union, difference (no intersection, no complement)
Selection: s
Projection: p
Cartesian Product: X
When our relations have attribute names:
Renaming: r
Derived operations:
Intersection, complement
Joins (natural join, equi-join, theta join, semi-join, ……)

6. Set Operations Union: all tuples in R1 or R2, denoted as R1 U R2
R1, R2 must have the same schema
R1 U R2 has the same schema as R1, R2
Example:
Active-Employees U Retired-Employees
If any, is duplicate elimination required?
Difference: all tuples in R1 but not in R2, denoted as R1 – R2
R1 - R2 has the same schema as R1, R2
Example
All-Employees - Retired-Employees

7. Selection
Returns all tuples (rows) which satisfy a condition, denoted as sc(R)
c is a condition: =, <, >, AND, OR, NOT
Output schema: same as input schema
Find all employees with salary more than $40,000:
sSalary > (Employee)
SSN
Name
Dept-ID
Salary
Alex 1
30K
Bob
32K
Chris 2
45K
SSN
Name
Dept-ID
Salary
Chris 2
45K

8. Projection
Unary operation: returns certain columns, denoted as P A1,…,An (R)
Eliminates duplicate tuples !
Input schema R(B1, …, Bm)
Condition: {A1, …, An} {B1, …, Bm}
Output schema S(A1, …, An)
Example: project social-security number and names:
P SSN, Name (Employee)
SSN
Name
Dept-ID
Salary
Alex 1
30K
Bob
32K
Chris 2
45K
SSN
Name
Alex
Bob
Chris

9. Selection vs. Projection
Think of relation as a table
How are they similar?
How are they different?
Horizontal vs. vertical?
Duplicate elimination for both? What about in real systems?
Why do you need both?

10. Cartesian Product
Each tuple in R1 with each tuple in R2, denoted as R1 x R2
Input schemas R1(A1,…,An), R2(B1,…,Bm)
Output schema is S(A1, …, An, B1, …, Bm)
Two relations are combined!
Very rare in practice; but joins are very common
Example: Employee x Dependent

11. Example SSN Name 111060000 Alex 754320032 Brandy Employee-SSN
Dependent
SSN
Name
Alex
Brandy
Employee-SSN
Dependent-Name
Chris
David
Employee x Dependent
SSN
Name
Employee-SSN
Dependent-Name
Alex
Chris
David
Brandy

12. Renaming Soc-sec-num, firstname(Employee)
Does not change the relational instance, denoted as Notation: r S(B1,…,Bn) (R)
Changes the relational schema only
Input schema: R(A1, …, An)
Output schema: S(B1, …, Bn)
Example:
Soc-sec-num, firstname(Employee)
SSN
Name
Alex
Bob
Chris
Soc-sec-num
firstname
Alex
Bob
Chris

13. Set Operations: Intersection
Intersection: all tuples both in R1 and in R2, denoted as R1 R2
R1, R2 must have the same schema
R1 R2 has the same schema as R1, R2
Example
UnionizedEmployees RetiredEmployees
Intersection is derived:
R R2 = R1 – (R1 – R2) why ?

14. Theta Join A join that involves a predicate q, denoted as R1 q R2
Input schemas: R1(A1,…,An), R2(B1,…,Bm)
Output schema: S(A1,…,An,B1,…,Bm)
Derived operator:
R1 q R2 = s q (R1 x R2)
Take the (Cartisian) product R1 x R2
Then apply SELECTC to the result
As for SELECT, C can be any Boolean-valued condition

15. Theta Join: Example Name Address Bar Beer Price AJ’s Bud 2.5 Miller
Sells
Name
Address
AJ's
1800 Tennessee
Michael's Pub
513 Gaines
Bar
Beer
Price
AJ’s
Bud
2.5
Miller
2.75
Michael’s Pub
Corona
3.0
BarInfo := Sells Sells.Bar=Bar.Name Bar
Bar
Beer
Price
Name
Address
AJ’s
Bud
2.5
AJ's
1800 Tennessee
Miller
2.75
Michael’s Pub
Michael's Pub
513 Gaines
Corona
3.0

16. Natural Join Notation: R1 R2
Input Schema: R1(A1, …, An), R2(B1, …, Bm)
Output Schema: S(C1,…,Cp)
Where {C1, …, Cp} = {A1, …, An} U{B1, …, Bm}
Meaning: combine all pairs of tuples in R1 and R2 that agree on the join attributes:
{A1,…,An} {B1,…, Bm} (called the join attributes)

17. Natural Join: Examples
Employee
Dependent
SSN
Name
Alex
Brandy
SSN
Dependent-Name
Chris
David
Employee Dependent =
P SSN, Name, Dependent-Name(sEmployee.SSN=Dependent.SSN(Employee x Dependent)
SSN
Name
Dependent-Name
Alex
Chris
Brandy
David

18. Natural Join: ExamplesB X Y Z V B C Z U V W
R S A B C X Z U V Y W

19. Equi-join
Special case of theta join: condition c contains only conjunctions of equalities
Result schema is the same as that of Cartesian product
May have fewer tuples than Cartesian product
Most frequently used in practice:
R1 A=B R2
Natural join is a particular case of equi-join
A lot of research on how to do it efficiently

20. A Joke About Join
A join query walks up to two tables in a restaurant and asks : “Mind if I join you?”

21. Building Complex Expressions
Algebras allow us to express sequences of operations in a natural way
Example
In arithmetic algebra: (x + 4)*(y - 3)
Relational algebra allows the same
Three notations, just as in arithmetic:
Sequences of assignment statements
Expressions with several operators
Expression trees

22. Sequences of Assignments
Create temporary relation names
Renaming can be implied by giving relations a list of attributes
Example: R3 := R C R2 can be written:
R4 := R1 x R2 (R4: temporary relation)
R3 := sC (R4)

23. Expressions with Several Operators
Example: the theta-join R3 := R1 JOINC R2 can be written: R3 := sC (R1 x R2)
Precedence of relational operators:
Unary operators --- select, project, rename --- have highest precedence, bind first
Then come products and joins
Then intersection
Finally, union and set difference bind last
But you can always insert parentheses to force the order you desire

24. Expression Trees Leaves are operands
either variables standing for relations or particular constant relations
Interior nodes are operators, applied to their child or children

25. Expression Tree: Examples
Given Bars(name, addr), Sells(bar, beer, price), find the names of all the bars that are either on Tennessee St. or sell Bud for less than $3
UNION
RENAMER(name)
PROJECTname
PROJECTbar
SELECTaddr = “Tennessee St.”
SELECT price<3 AND beer=“Bud”
Bars
Sells

26. Summary of Relational Algebra
Why bother ? Can write any RA expression directly in C++/Java, seems easy
Two reasons:
Each operator admits sophisticated implementations (think of and s C)
Expressions in relational algebra can be rewritten: optimized
s(age >= 30 AND age <= 35)(Employees)
Method 1: scan the file, test each employee
Method 2: use an index on age
Employees Relatives
Iterate over Employees, then over Relatives? Or iterate over Relatives, then over Employees?
Sort Employees, Relatives, do “merge-join”
“hash-join”
etc.