5.1relational operations on bags

Algebraic and Logical Query LanguagesAlgebraic and Logical Query Languages

Prof. Yin-Fu HuangProf. Yin-Fu Huang

CSIE, NYUST CSIE, NYUST

Chapter 5Chapter 5

Database Systems Yin-Fu Huang

5.15.1 Relational Operations on BagsRelational 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


R S: n+m times∪R∩S: min(n, m) timesR - S: max(0, n-m) times

Example

R A B S A B R S: A B ∪ 1 2 1 2 1 2

3 4 3 4 1 2

1 2 3 4 1 2

1 2 5 6 1 2

3 4

R∩S: A B R - S: A B 3 4

1 2 1 2 3 4

3 4 1 2 5 6



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

1 2 5 3 4 6

3 4 6 σC 6≧ (R) 1 2 7⇒

1 2 7 1 2 7

1 2 7



5.1.5 Product of Bags Example

(See Fig. 5.3)

5.1.6 Joins of Bags Example

R∞S A B C R∞R.B<S.BS A R.B S.B C

1 2 3 1 2 4 5

1 2 3 1 2 4 5

1 2 4 5

1 2 4 5



5.2.1 Duplicate Elimination δ(R)

5.2.2 Aggregation Operators SUM, AVG, MIN, MAX, COUNT Example

R A B SUM(B)=10

1 2 AVG(A)=1.5

3 4 MIN(A)=1

1 2 MAX(B)=4

1 2 COUNT(A)=4

5.25.2 Extended Operators of Relational AlgebraExtended 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.



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)



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)



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.

Example

(See Fig. 5.6)



Left outerjoin: R⊙LS

Right outerjoin: R⊙RS

Example

R⊙A >V.CS

(See Fig. 5.7)



The End.

5.1relational operations on bags

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