1. More Complex Operators
Relational Algebra #2
More Complex Operators
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2. More Algebra Operators
Set Operators: Union, Intersection, Difference
Cross (Cartesian) Product
Select, Project
Join: Natural Join, Theta Join, (Left/Right) Outer Join
Renaming, Duplicate Elimination
Aggregation: MIN, MAX, COUNT, SUM, AVG
Grouping
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3. Natural Join: R ⋈ S Consider relations Let A = AR ∩ AS, and
R with attributes AR, and
S with attributes AS.
Let A = AR ∩ AS, and
A = {A1, A2, …, An}
R ⋈ S can be defined as :
πAR – A, A, AS - A (σR.A1 = S.A1 AND R.A2 =S.A2 AND … R.An=S.An (R X S))
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4. Natural Join example R1 R2 R1 ⋈ R2 A B 1 2 B C 2 3 4 5 A B C 1 2 3
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5. Theta Join: R ⋈C S Theta Join is Cross product, with condition C.
It is defined as :
R ⋈C S = (σC (R X S))
R1 ⋈ R1.B<R2.BR2 R1 R2 A
R1.B
R2.B C 1 2 4 5 3 A B 1 2 3 B C 2 3 4 5
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6. Outer Join: R ⋈o S
Similar to natural join (plus with extra tuples that preserve all content of R and S)
R1 ⋈o R2 R1 R2 A B C D 1 2 3 10 11 4 5 6
null 7 8 9 12 A B C 1 2 3 4 5 6 7 8 9 B C D 2 3 10 11 6 7 12
But if a tuple in R has no “matching” tuple in S, then tuple appears in output with NULL for attributes in S, and vice versa.
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7. Left Outer Join: R ⋈oLS Similar to natural join (not symmetric !)
But if a tuple in R1 has no “matching” tuple in R2, then tuple appears in output , with NULL for attributes in R2
R1 ⋈oL R2 R1 R2 A B C D 1 2 3 10 11 4 5 6
null 7 8 9 A B C 1 2 3 4 5 6 7 8 9 B C D 2 3 10 11 6 7 12
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8. Right Outer Join: R ⋈oRS
Similar to natural join (not symmetric !)
But if a tuple in R2 has no “matching” tuple in R1, then tuple appears in output , with NULL for attributes in R1
R1 ⋈oR R2 R1 R2 A B C D 1 2 3 10 11
null 6 7 12 A B C 1 2 3 4 5 6 7 8 9 B C D 2 3 10 11 6 7 12
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9. Renaming ρS (R) changes relation name from R to S
ρS(A1, A2, …, An) (R) renames also attributes of R to A1, A2, …, An.
ρS(X, C, D) (R2)
ρS (R2) R2 S S B C D 2 3 10 11 6 7 12 X C D 2 3 10 11 6 7 12 B C D 2 3 10 11 6 7 12
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10. Duplicate Elimination:  (R)
Convert a bag to a set.
 (R) R A B 1 2 3 4 A B 1 2 3 4
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11. Extended Projection: πL (R)
Here L can be :
An attribute (just like simple projection)
An expression x → y, where x and y are names of attributes This renames attribute x to y.
An expression E → z, where E is any expression with attributes, constants, and arithmetic operators This generates an attribute called z whose values are given by E.
π C, B→A, C+D→X (R)
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12. Extended Projection: πL (R)
πB→A, C+D→X, C, D (R) R A X C D 2 13 3 10 14 11 6 19 7 12 B C D 2 3 10 11 6 7 12
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13. Aggregation operators
MIN, MAX, COUNT, SUM, AVG
AGGB (R) considers only non-null values of R.
SUMB (R)
MINB (R)
COUNTB (R) R
MINB (R) 2
SUMB (R) 9
COUNTB (R) 3 A B 1 2 3 4
null
AVGB (R)
COUNT* (R)
MAXB (R)
AVGB (R) 3
COUNT* (R) 4
MAXB (R) 4
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14. Making Aggregation Useful
AVG gpa ( Students) returns one average for all students.
What if I would like the average gpa for freshman, sophomore, junior and senior students, separately?
What if I would like the average gpa for students from different origins (country / city / etc)?
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15. Grouping Operator: GL, AL (R)
GL, AL (R) groups all attributes in GL, and performs aggregation specified in AL.
 starName, MIN (year)→year, COUNT(title) →num (StarsIn)
StarsIn
title
year
starName
SW1 77 HF
Matrix 99 KR
6D&7N 93
SW2 79
Speed 94
starName
year
num HF 77 3 KR 94 2
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16. Summary : Algebra Operators
Set Operators: Union, Intersection, Difference
Cartesian Product
Select, Project
Join: Natural Join, Theta Join, (Left/Right) Outer Join
Renaming, Duplicate Elimination
Aggregation: MIN, MAX, COUNT, SUM, AVG
Grouping
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