2. 5.1 Relational Operations on Bags
Bags or multisets
(See Fig. 5.1)
5.1.1 Why Bags?
Some relational operations are considerably more efficient if we use the bag model; e.g., union or projection.
(See Fig. 5.1 and Fig. 5.2)
5.1.2 Union, Intersection, and Difference of Bags
R in which tuple t appears n times
S in which tuple t appears m times
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3. 5.1 Relational Operations on Bags
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4. 5.1 Relational Operations on Bags
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5. 5.1 Relational Operations on Bags
R∪S: n+m times
R∩S: min(n, m) times
R - S: max(0, n-m) times
Example
R A B S A B R∪S: A B
3 4
R∩S: A B R - S: A B 3 4
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6. 5.1 Relational Operations on Bags
5.1.3 Projection of Bags
πA,B(R)
(See Fig. 5.1 and Fig. 5.2)
5.1.4 Selection of Bags
Example
R A B C A B C
σC≧6(R) ⇒
1 2 7
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7. 5.1 Relational Operations on Bags
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8. 5.1 Relational Operations on Bags
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9. 5.1 Relational Operations on Bags
5.1.5 Product of Bags
Example
(See Fig. 5.3)
5.1.6 Joins of Bags
R∞S A B C R∞R.B<S.BS A R.B S.B C
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10. 5.1 Relational Operations on Bags
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11. 5.2 Extended Operators of Relational Algebra
5.2.1 Duplicate Elimination
δ(R)
5.2.2 Aggregation Operators
SUM, AVG, MIN, MAX, COUNT
Example
R A B SUM(B)=10
AVG(A)=1.5
MIN(A)=1
MAX(B)=4
COUNT(A)=4
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12. 5.2 Extended Operators of Relational Algebra
5.2.3 Grouping
Example
(See Fig. 5.4)
5.2.4 The Grouping Operator
γL(R)
A list L of elements, each of which is either:
1) A grouping attribute: an attribute of the relation R to
which the γ is applied.
2) An aggregated attribute: an attribute of the relation R to
which an aggregate operator is applied.
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13. 5.2 Extended Operators of Relational Algebra
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14. 5.2 Extended Operators of Relational Algebra
Constructing as follows:
1) Partition the tuples of R into groups.
2) For each group, produce one tuple consisting of:
i) The grouping attributes' values for that group and
ii) The aggregations, over all tuples of that group, for the
aggregated attributes on list L.
Example
γ starName, MIN(year)→minYear, COUNT(title)→ctTitle (StarsIN)
(See Fig. 5.5)
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15. 5.2 Extended Operators of Relational Algebra
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16. 5.2 Extended Operators of Relational Algebra
5.2.5 Extending the Projection Operator
πL(R)
Projection lists can have the following kinds of elements:
1) A single attribute of R
2) An expression x → y
3) An expression E → z
Example
πA, B+C→X(R)
πB-A→X, C-B→Y(R)
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17. 5.2 Extended Operators of Relational Algebra
5.2.6 The Sorting Operator
τL(R)
Example
τC,B(R)
5.2.7 Outerjoins
R⊙S
The dangling tuples are padded with a special null symbol, ⊥, in all the attributes that they do not possess but that appear in the join result.
(See Fig. 5.6)
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18. 5.2 Extended Operators of Relational Algebra
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19. 5.2 Extended Operators of Relational Algebra
Left outerjoin: R⊙LS
Right outerjoin: R⊙RS
Example
R⊙A >V.CS
(See Fig. 5.7)
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20. 5.2 Extended Operators of Relational Algebra
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21. The End.
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