Fundamentals/ICY: Databases2012/13

WEEK 11(relational operators & relational algebra)

John BarndenProfessor of Artificial IntelligenceSchool of Computer ScienceUniversity of Birmingham, UK

Relational Database Operators

Relational algebra Defines theoretical way of manipulating tables using

“relational operators” that mainly manipulate the relations in the tables.

• SELECT

• PROJECT

• JOIN (various sorts)

• INTERSECT Use of relational algebra operators on existing tables

produces new tables

• UNION

• DIFFERENCE

• PRODUCT

• ((DIVIDE))

Select [better name would be Select-Rows]

SQL: SELECT * FROM … WHERE … Note: it’s the WHERE part that is actually doing the selection

according to a criterion.

Relational algebra notation in handout: Result table is C(T) where T is the given table and C is the

selection criterion. More compact than SQL notation. Avoids notation private to

particular versions of particular programming languages.

Project [better name would be Select-Columns]

SQL: SELECT …column specs … FROM …

Relational algebra notation in handout: Result table is X(T) where T is the given table and X

is the list of selected attributes (columns).

But this always removes row duplications from the result, and so does not exactly correspond to the full DB notion of projection.

Union, Intersection and Difference SQL:

UNION, INTERSECT, EXCEPT (or MINUS) UNION ALL, INTERSECT ALL, EXCEPT (or MINUS) ALL

Relational algebra notation in handout: Result tables are T1 T2, T1 T2 and T1 T2 where T1 and

T2 are the given tables.

Maths of relations: Result relations are R1 R2, R1 R2 and R1 R2 in the

non-ALL cases. where R1and R2 are the relations in the given tables.

Problem: relations don’t account for duplicates of rows, so don’t handle the ALL versions.

Some “Relational Operations”:Set Operations Applied to Relations

Union of relations R and S: R S = the set of tuples that are in R or S (or both). NB: no repetitions created!

Intersection of relations R and S: R S = the set of tuples that are in both R and S.

Difference of relations R and S: R S = the set of tuples that are in R but not S.

Relational Operations: contrast to SQL Those operations do NOT themselves require R and S to

have similar tuples in order to be well-defined. E.g., R could be binary and on integer sets, S could be ternary and

on character-string sets.

But the corresponding DB table operations (which are usually called “relational operators”) do require the tables to have the same shape (same number of columns, same domains for corresponding columns).

Relational Operators (continued)

Join (various types)

Allows us to join related rows from two or more tables

It’s an important feature of the relational database idea

Joining has been implicitly important in some of the module handouts, because of the use of WHERE to test for attribute equality between tables.

Relational Operators (continued) Product or Cross Join

Yields a table containing all concatenations of whole rows from first given table with whole rows from second given table.

Product

If second table also had a PRICE attribute, then the product would have a Table1.PRICE attr. and a Table2.PRICE attr.

Product or Cross Join (continued) SQL:

SELECT * FROM …two [or more] tables … NB: it’s the mere listing of the tables that does the Product,

but it’s possible also to write: SELECT * FROM T1 CROSS JOIN T2 CROSS JOIN ...

Relational algebra notation: Result table is T1 T2 where T1 and T2 are the given tables.

Maths of relations: Result relation is R1 R2 where R1and R2 are the relations

in the given tables. Problem: relations don’t account for duplicates of rows.

So, I want … ….. to define the non-standard notion of “flattened

Cartesian product” of two relations R and S. Notated by the symbol (underlined multiplication symbol).

R S = the set of tuples that are the concatenations of members of R and members of S.

E.g., if <a,b,c> is in R and <d,e,f> is in S then <a,b,c,d,e,f> is in R S.

Contd. If A is the People relation and B is the Organizations relation, and

A has members of form E156, ‘Sam’, ‘Finks’, I678> and

B has members of form I459, ‘Dell’, ‘UK’>

THEN

A B has members of form

E156, ‘Sam’, ‘Finks’, I678>, I459, ‘Dell’, ‘UK’> >

BUT

A B has members of form

E156, ‘Sam’, ‘Finks’, I678, I459, ‘Dell’, ‘UK’>

Two Tables That Will Be Used to Illustrate Other Joins

Natural Join (continued)

SQL: SELECT …all the attributes but including only one version of each

shared one … FROM T1, T2 WHERE … explicit condition of equalities for ALL the shared attributes ...

SELECT * FROM T1 NATURAL JOIN T2; Instead of using *, can choose columns, and can add a WHERE

Relational algebra notation: Result table is T1 T2 where T1 and T2 are the given tables.

is the “bow tie” symbol.

Correspondence to your SQL experience:

SELECT sid, office FROM staff, lecturing WHERE staff.sid = lecturing.sid; Does a natural join (because sid is the only shared attribute) followed by

a projection onto sid, office.

SELECT sid, office FROM staff, lecturing WHERE staff.sid = lecturing.sid AND year > 2001; In effect, does a natural join followed by a further (row) selection

followed by a projection.

SELECT sid, office FROM staff NATURAL JOIN lecturing WHERE year > 2001; Does same thing.

Natural Join

The common attributes or columns are called the join attributes or columns): just the AGENT_CODE attribute in above example

Can be thought of as the result of a three-stage process:

the PRODUCT of the tables is created a SELECT is performed on the resulting table to yield only the

rows for which the join-attribute values (e.g. AGENT_CODE values) are equal

a PROJECT is now performed to yield a single copy of each join attribute, thereby eliminating duplicate columns

Natural Join, Step 1: PRODUCT

Note the two AGENT_CODE columns

Natural Join, Step 2: SELECT

Natural Join, Step 3: PROJECT

Natural Join (continued)

A row in one of the given tables that does not match any row in the other given table on the join attributes does not lead to a row in the result table.

Note that if the two tables have no attributes in common, then every row of each table trivially matches every row of the other table!

So in this case the result is the PRODUCT (CROSS JOIN) of the two tables!!

Following stuff on Joins is optional (but important in practice)

Other Forms of Join Equijoin

Links tables on the basis of an equality condition that compares SPECIFIED attributes of each table, rather than automatically taking the common attributes.

Result does not eliminate duplicate columns that are not involved in the join condition.

Theta join Like equijoin but using a non-equality join condition.

Outer joins (left, right, and full) Equijoin or theta join plus unmatched rows from left table,

right table or both, padding them out with NULLs to fit the result table.

Equijoin and Theta Join (continued)

SQL: SELECT * FROM T1, T2 WHERE … explicit join condition,

stating (non)equality of the CHOSEN attributes ...

SELECT * FROM T1 JOIN T2 ON … such a condition …

SELECT * FROM T1 JOIN T2 USING (… some common attribs …)

[for equijoin only]

Relational algebra notation: T1 C T2 where C is the join condition.

Outer Join Of CUSTOMER and AGENT, using equal AGENT_CODE

Left outer Uses all the rows in the CUSTOMER table, by doing equijoin on

AGENT_CODE but also including non-matching CUSTOMER rows.

Right outer Uses all the rows in the AGENT table, doing equijoin on

AGENT_CODE but also including non-matching AGENT rows.

Full outer Using all the rows in the AGENT and CUSTOMER tables, doing

equijoin on AGENT_CODE but also including non-matching rows from each table.

Union of Left Outer Join result and Right Outer Join result.

Left Outer Join

Same as an equijoin with the addition of the “extra”, last, row shown above

Right Outer Join:

Full Outer Join:Would have the “extra” row of this table as well as the extra row of the Left Outer Join table

Outer Joins (continued) SQL:

SELECT * FROM T1, T2 WHERE … explicit join condition … UNION … a SELECT expression that gets the extra LEFT rows UNION … a SELECT expression that gets the extra RIGHT rows

SELECT * FROM T1 LEFT/RIGHT/FULL JOIN T2 USING (… some shared attribs …) // ON … explicit join cond …

Relational algebra notation: Variants of bow tie symbol. See R,C&C sec. 4.2.3 (though

their symbols need a subscript stating the join condition unless natural).

Note on SQL Join Queries Can of course do your own extra projection (= attrib

selection) in the SELECT, and can add a WHERE.

E.g.:

SELECT …attribs … FROM T1 LEFT JOIN T2 USING (… some shared attribs …) WHERE … ;

Following stuff on DIVIDE is optional

Towards the DIVIDE operation

It’s analogous to the “integer division” of an integer T by an integer S, included in many programming languages.

T div S = the largest integer Q such that

S Q T

So 7 div 3 = 22.

DIVIDE operation on DB tables

The only value of LOC that is associated in T with both values ‘A’ and ‘B’ of CODE is 5.

Simplest case: 2-col table by 1-col table

T S QQ

Divide DIVIDE T by S: the attributes X1 … XM of table S must be

some but not all of those of T’s.

Gives a table QQ having the remaining attributes Y1 … YN of T.

QQ holds the values of Y1 … YN that T associates with every row (X1 … XM) in S.

So the rows of the Product of S with QQ form a subset of the rows (suitably re-ordered) of T, and QQ is maximal in this respect (i.e., adding further rows to QQ would stop the Product’s rows all being in T)

So QQ is the largest table such that

S Q Q T (with rows suitably re-ordered)

using to mean: has some or all rows of.

Divide (continued) SQL:

Not standardly included. Effect can be simulated.

Relational algebra notation: T2 T1

Maths of relations: Result relation R could be described as the maximal set R of tuples such that R1 R R2

where R1 and R2 are the relations in the given tables.