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Database Systems &Applications

Lec13Extended Operators of Relational Algebra

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Relational Algebra on Bags

A bag (or multiset) is like a set, but anelement may appear more than once.

Example: {1,2,1,3} is a bag. Example: {1,2,3} is also a bag that

happens to be a set.

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Why Bags?

SQL, the most important query languagefor relational databases, is actually a bag

language. Some operations, like projection, are much

more efficient on bags than sets.

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Operations on Bags

Selection applies to each tuple, so its

effect on bags is like its effect on sets.

Projection also applies to each tuple, but

as a bag operator, we do not eliminate

duplicates.

Products andjoins are done on each pair

of tuples, so duplicates in bags have noeffect on how we operate.

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Example: Bag Selection

R( A, B )1 2

5 6

SELECTA+B

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Example: Bag Projection

R( A, B )1 2

5 61 3

PROJECTA(R) = A

151

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Example: Bag Product

R( A, B ) S( B, C )1 2 3 45 6 7 8

R * S = A R.B S.B C1 2 3 41 2 7 8

5 6 3 45 6 7 8

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Example: Bag Theta-Join

R( A, B ) S( B, C )1 2 3 45 6 7 8

R JOIN R.B

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Bag Union

An element appears in the union of two

bags the sum of the number of times it

appears in each bag.

Example: {1,2,1} UNION {1,1,2,3,1} =

{1,1,1,1,1,2,2,3}

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Bag Intersection

An element appears in the intersection of

two bags the minimum of the number of

times it appears in either. Example: {1,2,1,1} INTER {1,2,1,3} =

{1,1,2}.

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Bag Difference

An element appears in the difference

A B of bags as many times as it appears

inA, minus the number of times it appears

in B.

Example: {1,2,1,1} {1,2,3} = {1,1}.

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Beware: Bag Laws != Set Laws

Some, but notall algebraic laws that holdfor sets also hold for bags.

Example: the commutative law for union(R UNION S = S UNION R ) does hold forbags.

Since addition is commutative, adding

the number of timesxappears in R andS doesnt depend on the order ofR andS.

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Example of the Difference

Set union is idempotent, meaning that

S UNION S=S.

However, for bags, ifxappears n times in

S, then it appears 2n times in S UNION S.

Thus S UNION S!=S in general.

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The Extended Algebra

1 . : eliminate duplicates from bags.1 . : sort tuples.1. Extended projection: arithmetic, duplication of

columns.

1 . : grouping and aggregation.1. OUTERJOIN: avoids dangling tuples = tuples that do

not join with anything.

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Example: Duplicate Elimination

R = A B1 23 41 2

(R) = A B1 23 4

R1 := (R2) R1 consists of one copy of each tuple that appears in

R2 one or more times.

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Sorting

R1 := L (R2). L is a list of some of the attributes of

R2.

R1 is the list of tuples of R2 sorted first onthe value of the first attribute on L, thenon the second attribute ofL, and so on.

is the only operator whose result isneither a set nor a bag.

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Example: Sorting

R = ( A B )1 2

3 45 2

B(R) = [(5,2), (1,2), (3,4)]

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Example: Extended Projection

R = A B1 2

3 4

A+BC,AA1,AA2 (R) = C A1 A2

3 1 1

7 3 3

Using the same L operator, we allow thelist L to contain arbitrary expressionsinvolving attributes, for example:

1. Arithmetic on attributes, e.g., A+B.

2. Duplicate occurrences of the same

attribute.

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Aggregation Operators

Aggregation operators are notoperators of relational algebra.

Rather, they apply to entire columns

of a table and produce a single

result.

The most important examples: SUM,

AVG, COUNT, MIN, and MAX.

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Example: Aggregation

R = ( A, B )1 33 43 2

SUM(A) = 7COUNT(A) = 3MAX(B) = 4

AVG(B) = 3

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Grouping Operator

R1 := L (R2)

L is a list of elements that are either:1. Individual (grouping ) attributes.

2. AGG(A), where AGG is one of theaggregation operators andA is an

attribute.

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Applying L (R) Group R according to all the

grouping attributes on list L. i.e., form one group for each distinct list

of values for those attributes in R.

Within each group, compute AGG(A )for each aggregation on list L.

Result has one tuple for each group:

1. The grouping attributes and

2. Their groups aggregations.

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Example:Grouping/Aggregation

R = A B C1 2 34 5 6

1 2 5

A,B,AVG(C) (R) = ??

First, group R :A B C

1 2 31 2 5

4 5 6

Then, average C within

groups:

A B AVG(C)1 2 44 5 6

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Example: Grouping/Aggregation

SpreePL (Sport, year, Player)

For each player who has participated in at least threesports give the earliest year in which he or sheparticipated.

First we group, using Playeras a groupingattribute.

Then, we compute the MIN(year) for each group.

Also, we need to compute the COUNT(Sport)aggregate for each group, for filtering out thosePlayers with less than three Sports.

Player,minYear(ctSport 3( Player,MIN(year)minYear,COUNT(Sport)ctSport(SpreePL)))

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Outerjoin

Motivation Suppose we join R S.

A tuple of R which doesn't join with any tuple of S is saidto be dangling.

Similarly for a tuple of S. Problem: We loose dangling tuples.

Outerjoin

Preserves dangling tuples by padding them with a specialNULL symbol in the result.

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Example: Outerjoin

R = A B S = B C1 2 2 34 5 6 7

(1,2) joins with (2,3), but the other two tuplesare dangling.

R S = A B C

1 2 34 5 NULLNULL 6 7

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R S --- Full outerjoin pad dangling tuples fromboth the tables R and S

R L S -- This left outerjoin: Only pad dangling

tuples from the left table.

R R S -- This right outerjoin: Only pad dangling

tuples from the right table.

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Outer Join Example

Relation loan

Relation borrower

customer_name loan_number

Abhiram

Kavitha

Madhu

L-170

L-230

L-155

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Outer Join Example

Natural Join

loan Borrower

Left Outer Join

loan Borrower

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Outer Join Example

Full Outer Join

loan borrower

Right Outer Join

loan borrower

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Problem Consider the relational schemas:

Email (IDNO, email_id)Student (IDNO, Name, Hostel, Room)

The EMAIL relation is maintained by ARC, where asthe Student relation is maintained by SWD. It isrequired that the information in the two relations be

combined into a new relation with schema:Student_email (IDNO, email_id, Name, Hostel, Room)

It is observed that there are some IDNOs in theStudent relation that are not in the Email relation.Write a Relational Algebra expression to create thenew relation such that no IDNO from the Student

relation is left out.

(Student_email, (student left outer join email))