1

Lecture 12:Further relational algebra,

further SQL

www.cl.cam.ac.uk/Teaching/current/Databases/

2

Today’s lecture

• Where does SQL differ from relational model?

• What are some other features of SQL?

• How can we extend the relational algebra to match more closely SQL?

3

Duplicate rows

• Consider our relation instances from lecture 6, Reserves, Sailors and Boats

• Consider SELECT rating,age

FROM Sailors;

• We get a relation that doesn’t satisfy our definition of a relation!

• RECALL: We have the keyword DISTINCT to remove duplicates

4

Multiset semantics

• A relation in SQL is really a multiset or bag, rather than a set as in the relational model– A multiset has no order (unlike a list), but

allows duplicates– E.g. {1,2,1,3} is a bag– select, project and join work for bags as well

as sets• Just work on a tuple-by-tuple basis

5

Bag operations

• Bag union:– Sum the number of times that an element appears in

the two bags, e.g.• {1,2,1}{1,2,3} = {1,1,1,2,2,3}

• Bag intersection:– Take the minimum of the number of occurrences in

each bag, e.g.• {1,2,1}{1,2,3,3} = {1,2}

• Bag difference:– Proper-subtract the number of occurrences in the two

bags, e.g.• {1,2,1}-{1,2,3,3} = {1}

6

Laws for bags

• Note that whilst some of the familiar (set-theoretic) laws continue to hold, some of them do not

• Example: R(ST) = (RS)(RT) ??

7

Extended relational algebra

Add features needed for SQL

1. Bag semantics

2. Duplicate elimination operator, 3. Sorting operator, 4. Grouping and aggregation operator, 5. Outerjoin operators, oV, Vo, oVo

8

Duplicate-elimination operator

(R) = relation R with any duplicated tuples removed

• R= (R)=

• This is used to model the DISTINCT feature of SQL

A B

1 2

3 4

1 2

A B

1 2

3 4

9

Sorting L1,… Ln

(R) returns a list of tuples of R, ordered according to the

attributes L1, …, Ln

• Note: does not return a relation• R= B(R)= [(5,2),(1,3),(3,4)]

• ORDER BY in SQL, e.g. SELECT * FROM Sailors WHERE rating>7 ORDER BY age, sname;

A B

1 3

3 4

5 2

10

Extended projection

• SQL allows us to use arithmetic operators SELECT age*5

FROM Sailors;

• We extend the projection operator to allow the columns in the projection to be functions of one or more columns in the argument relation, e.g.

• R= A+B,A,A(R)=

A B

1 2

3 4

A+B A.1 A.2

3 1 1

7 3 3

11

Arithmetic

• Arithmetic (and other expressions) can not be used at the top level– i.e. 2+2 is not a valid SQL query

• How would you get SQL to compute 2+2?

12

Aggregation

• SQL provides us with operations to summarise a column in some way, e.g.

SELECT COUNT(rating) FROM Sailors;

SELECT COUNT(DISTINCT rating) FROM Sailors;

SELECT COUNT(*) FROM Sailors WHERE rating>7;

• We also have SUM, AVG, MIN and MAX

13

Grouping

• These aggregation operators have been applied to all qualifying tuples. Sometimes we want to apply them to each of several groups of tuples, e.g.– For each rating, find the average age of the

sailors– For each rating, find the age of the youngest

sailor

14

GROUP BY in SQL

SELECT [DISTINCT] target-list

FROM relation-list

WHERE qualification

GROUP BY grouping-list;

• The target-list contains 1. List of column names

2. Aggregate terms– NOTE: The variables in target-list must be

contained in grouping-list

15

GROUP BY cont.

For each rating, find the average age of the sailors SELECT rating,AVG(age)

FROM Sailors

GROUP BY rating;

For each rating find the age of the youngest sailor SELECT rating,MIN(age)

FROM Sailors

GROUP BY rating;

16

Grouping and aggregation

L(R) where L is a list of elements that are either– Individual column names (“Grouping

attributes”), or– Of the form (A), where is an aggregation

operator (MIN, SUM, …) and A is the column it is applied to

• For example,rating,AVG(age)(Sailors)

17

Semantics

• Group R according to the grouping attributes

• Within each group, compute (A)

• Result is the relation consisting of one tuple for each group. The components of that tuple are the values associated with each element of L for that group

18

Example

• Let R=

• Compute beer,AVG(price)(R)

bar beer price

Anchor 6X 2.50

Anchor Adnam’s 2.40

Mill 6X 2.60

Mill Fosters 2.80

Eagle Fosters 2.90

19

Example cont.

1. Group according to the grouping attribute, beer:

2. Compute average of price within groups:

bar beer price

Anchor 6X 2.50

Mill 6X 2.60

Anchor Adnam’s 2.40

Mill Fosters 2.80

Eagle Fosters 2.90

beer price

6X 2.55

Adnam’s 2.40

Fosters 2.85

20

NULL values

• Sometimes field values are unknown (e.g. rating not known yet), or inapplicable (e.g. no spouse name)

• SQL provides a special value, NULL, for both these situations

• This complicates several issues– Special operators needed to check for NULL– Is NULL>8? Is (NULL OR TRUE)=TRUE?– We need a three-valued logic– Need to carefully re-define semantics

21

NULL values

• Consider INSERT INTO Sailors (sid,sname) VALUES (101,”Julia”);

SELECT * FROM Sailors;

SELECT rating FROM Sailors;

SELECT sname FROM Sailors WHERE rating>0;

22

Entity integrity constraint

• An entity integrity constraint states that no primary key value can be NULL

23

Outer join

• Note that with the usual join, a tuple that doesn’t ‘join’ with any from the other relation is removed from the resulting relation

• Instead, we can ‘pad out’ the columns with NULLs

• This operator is called an full outer join, written oVo

24

Example of full outer join

• Let R= Let S=

• Then RVS =

• But RoVoS =

A B

1 2

3 4

B C

4 5

6 7

A B C

3 4 5

A B C

1 2 NULL

3 4 5NULL 6 7

25

Outer joins in SQL

• SQL/92 has three variants:– LEFT OUTER JOIN (algebra: oV)– RIGHT OUTER JOIN (algebra: Vo)– FULL OUTER JOIN (algebra: oVo)

• For example: SELECT * FROM Reserves r LEFT OUTER JOIN Sailors s ON r.sid=s.sid;

26

Views

• A view is a query with a name that can be used in further SELECT statements, e.g.

CREATE VIEW ExpertSailors(sid,sname,age) AS SELECT sid,sname,age

FROM Sailors

WHERE rating>9;

• Note that ExpertSailors is not a stored relation• (WARNING: mysql does not support views )

27

Querying views

• So an example query SELECT sname

FROM ExpertSailors

WHERE age>27;

• is translated by the system to the following: SELECT sname

FROM Sailors

WHERE rating>9 AND age>27;

28

Summary

You should now understand:• Multi-set semantics• Conditions• Aggregation• GROUP BY• NULLs, entity ICs and outer joins• Views• Extensions to the core relational algebra