Relational Algebra – Basis for Relational Query Languages Based on presentation by Juliana Freire

Formal relational query languages

Is this the Algebra you know? Algebra -> operators and atomic operands Expressions -> applying operators to atomic operands and/or other expressions Algebra of arithmetic: operands are variables and constants, and operators are the usual arithmetic operators E.g., (x+y)*2 or ((x+7)/(y-3)) + x Relational algebra: operands are variables that stand for relations and relations (sets of tuples), and operations include union, intersection, selection, projection, Cartesian product, etc – E.g., (π c-ownerChecking-account) ∩ (π s-ownerSavings-account)

What is a query? A query is applied to relation instances, and the result of a query is also a relation instance. (view, query) – Schemas of input and output fixed, but instances not. Operators refer to relation attributes by position or name: – E.g., Account(number, owner, balance, type) – Positional notation easier for formal definitions, named-field notation more readable. – Both used in SQL

Relational Algebra Operations The usual set operations: union, intersection, difference Operations that remove parts of relations: selection, projection Operations that combine tuples from two relations: Cartesian product, join Since each operation returns a relation, operations can be composed!

Removing Parts of Relations Selection – rows Projection - columns

Selection: Example

Another selection

Example of Projection

Projection removes duplicates

Set Operations Union Intersection Difference

What happens when sets unite?

Union Operation – Example Relations r, s: r  s: AB  AB  2323 r s AB 

Union Example

Union Compatibility

Intersection

Set Difference Operation – Example Relations r, s: r – s: AB  AB  2323 r s AB  1111

Difference

Another way to show intersection?

Summary so far: E 1 U E 2 : union E 1 - E 2 : difference E 1 x E 2 : cartesian product  c (E 1 ) : select rows, c = condition (book has p for predicate) II s (E 1 ) : project columns : s =selected columns  x(c1,c2) (E 1 ) : rename, x is new name of E 1, c1 is new name of column

Combining Tuples of Two Relations Cross product (Cartesian product) Joins

Cartesian-Product Operation – Example Relations r, s: r x s: AB  1212 AB  CD  E aabbaabbaabbaabb CD  E aabbaabb r s

Cross Product Example

Cross Product Renaming operator:  Rename whole relation: Teacher X  secondteacher (Teacher)  Teacher.t-num, Teacher.t-name, secondteacher.t-num, secondteacher.t-name OR rename attribute before combining: Teacher X  secondteacher(t-num2, t-name2) (Teacher)  t-num, t-name, t-num2, t-name2 OR rename after combining  c(t-num1, t-name1, t-num2, t-name2) (Teacher X Teacher)  t-num1, t-name1, t-num2, t-name2 How to resolve????

Join: Example

Condition Join

Equi and Natural Join

Divide operator

Divide Operation

Divide Definition

When to divide?

Division Example

Dividing without division sign

Working out an example

Assignment operation

Why would we use Relational Algebra?????

Equivalencies help

ER vs RA Both ER and the Relational Model can be used to model the structure of a database. Why is it the case that there are only Relational Databases and no ER databases?

RA vs Full Programming Language Relational Algebra is not Turing complete. There are operations that cannot be expressed in relational algebra. What is the advantage of using this language to query a database?

Summary of Operators updated Summary so far: E 1 U E 2 : union E 1 - E 2 : difference E 1 x E 2 : cartesian product  c (E 1 ) : select rows, c = condition (book has p for predicate) II s (E 1 ) : project columns : s =selected columns  x(c1,c2) (E 1 ) : rename, x is new name of E 1, c1 is new name of column E 1  E 2 : division E 1 E 2 : join, c = match condition

Practice Find names of stars who’ve appeared in a 1994 movie Information about movie year available in Movies; so need an extra join: σ year=1994 (πname(Stars ⋈ AppearIn ⋈ Movies)) An even more efficient solution: π name (Stars ⋈ π name (AppearIn ⋈ (π movieId σ year=1994 (Movies))) A query optimizer can find this, given the first solution! A more efficient solution: π name (Stars ⋈ AppearIn ⋈ (σ year=1994 ( Movies))

Extended Relational Algebra Operations Generalized projection Outer join Aggregate functions

Generalized projection – calculate fields

Aggregate Operation – Example Relatio n r: AB   C g sum(c ) (r) sum(c ) 27

Aggregate Functions on more than one tuple Samples: –Sum –Count-distinct –Max –Min –Count –Avg Use “as” to rename branchname g sum(balance) as totalbalance ( account )

Aggregate Operation – Example Relation account grouped by branch-name: branch_name g sum( balance ) ( account ) branch_nameaccount_numberbalance Perryridge Brighton Redwood A-102 A-201 A-217 A-215 A branch_namesum(balance) Perryridge Brighton Redwood

Outer Join Keep the outer side even if no join Fill in missing fields with nulls

Outer Join – Example Relation loan Relation borrower customer_nameloan_number Jones Smith Hayes L-170 L-230 L loan_numberamount L-170 L-230 L-260 branch_name Downtown Redwood Perryridge

Outer Join – Example Inner Join loan Borrower loan_numberamount L-170 L customer_name Jones Smith branch_name Downtown Redwood Jones Smith null loan_numberamount L-170 L-230 L customer_namebranch_name Downtown Redwood Perryridge Left Outer Join loan Borrower

Outer Join – Example loan_numberamount L-170 L-230 L null customer_name Jones Smith Hayes branch_name Downtown Redwood null loan_numberamount L-170 L-230 L-260 L null customer_name Jones Smith null Hayes branch_name Downtown Redwood Perryridge null Full Outer Join loan borrower Right Outer Join loan borrower

Summary of Operators - Full E 1 U E 2 : union E 1 - E 2 : difference E 1 x E 2 : cartesian product  c (E 1 ) : select rows, c = condition (book has p for predicate) II s (E 1 ) : project columns : s =selected columns separated by commas, can have calculations included  x(c1,c2) (E 1 ) : rename, x is new name of E 1, c1 is new name of column E 1  E 2 : division E 1 E 2 : join, c = match condition E 1 E 2 : outer join, c = match condition, keep the side with the arrows  : assignment – give a new name to an expression to make it easy to read as : rename a calculated column attribute1 g function (attribute2) (E 1 ) : perform function on attribute2 whenever attribute1 changes