1. The Relational Algebra
Textbook Ch
© 1999 UW CSE
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2. Overview A non-procedural language Operations on whole relations
Inputs are sets, output is a set
Can nest arbitrarily complex expressions
Basis for SQL
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3. The RA Operators SELECT , PROJECT 
on a single relation
Set union , intersection , difference 
on pair of compatible relations
Cartesian product  and various flavors of JOIN
on any pair of relations
Division  sort of the inverse of product
Aggregate functions (unofficial)
group tuples of one relation, then operate
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4. Set Operations Set union , intersection , difference 
on compatible relations: corresponding columns have same types
Note there is no "set complement"!
What would the complement of a table be??
Databases tend to operate on a "closed world" assumption: "If it's not in the DB, it doesn't exist."
In Prolog: "If you can't prove it, it isn't true."
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5. condition(relationname)
Select
Unary operation
Select a subset of tuples from a relation, based upon a condition
use AND, OR, NOT for compound conditions
Result: a table with same attributes as original: a proper subset
may be given a name (temporary)
Notation:
condition(relationname)
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6. col-list(relationname)
Project
Unary operation
Select a subset of columns
Result: a table with same number of rows as original
not actually a subset of the original (unlike )
may be given a name (temporary)
Notation:
col-list(relationname)
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7. Join A binary operation on relations Notation (these slides):
R1JN join-condition R2
Result is a whole relation
All the columns of R1 followed by all the columns of R2
General description: a  followed by a .
The  condition equates or otherwise relates common attributes between the two relations
Often a superfluous common attribute is removed
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8. "Equi" and "Natural" join
Equi-join: Common attributes are compared for equality
no need to specify a join condition
could join on more than one attribute
need to list attributes if names are not the same
superfluous column of 2nd relation removed
"Natural join": Equi-join on all common fields
Notation (some texts): R1 * attr-lists R2
When people say “join” natural join is usually what they mean.
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9. Semi Joins (not in book)
Left semi-join of R1 and R2:
Each row of R1 is including in the output (with nulls in R2 columns if necessary), even if there are no matches in R2 for that row.
Right semi-join:
Each row of R2 is included, etc.
Quiz: suppose R1 and R2 have no matches in their common columns. What is the cardinality of: Cartesian product; natural join; left, right semi-joins?
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10. Division: R1  R2 Sort of the reverse of Cartesian product
Like integer division in that any "remainder" is discarded
Main idea: find all the tuples in R1 which are joined to all the values in R2
the R2 attributes are discarded from the output
Same thing can be accomplished with combination of , , 
Can you figure out how?
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11. Division Details of R = R1R2
R1 (dividend): attribute set X  Y, |R1| rows
R2 (divisor): attribute set Y, |R2| rows
R (quotient):
attribute set X, i.e., the attributes of R1 not in R2
at most |R1|/|R2| rows
A row is in the answer (R) if that row (X attributes) occurs in R1 with each combination of the rows (Y attributes) of R2.
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12. Division Examples
Who would have lots to talk about with Bessie? "Find (all) customers who have rented (all) the same movies as Bessie has."
What airlines compete with Horizon Air? "Find the airlines which serve a city also served by Horizon" (not a division query).
Which airline is best positioned to put Horizon Air out of business? "Find the airlines which fly to (all) the cities served by Horizon" (a division query).
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13. Aggregate Functions (not in book)
Technically, not part of R.A.
Actual query languages will implement many of these
(Usually) unary operators, take a whole relation and compute a value
COUNT, AVERAGE, MAX, MIN
Result is returned as a relation with one row and one column
i.e., not as a scalar number
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14. Grouping and Aggregates
Rows may be grouped based on attribute values
Think of it as a sort on those attributes
Aggregate functions can be applied to the grouped relation
Computes a value for each group
Result returned as a relation with:
one row for each group (unique value),
one column for each aggregate function
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15. Grouping Notation and Example
<grouping attributes>  <agg. function list> (relation)
"List number of employees and average salary for each department"
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16. Relational Completeness
A query language is "relationally complete" if any R.A. query can be expressed in it.
Does not mean all R.A. operations are available
The concept is analogous, but not identical, to Turing-completeness
Commercial query languages are mostly much more powerful than the R.A.
Aggregates, grouping, etc.
We'll look later at relational calculus and QBE, which are also relationally complete
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17. Notice Something Missing?
Yes: iteration/recursion
This is typical of relational languages
Some fairly natural queries are impossible
“Find all the children of X” -- no problem
“Find all the grandchilden of X” -- still OK
“Find all the descendents of X” -- no way!
This is an example of a "transitive closure"
Solution: embed the query in a procedural language (COBOL, C, etc).
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18. Looking Ahead: Query Processing
Order of operations affects efficiency
Example: (R1) *(R2) probably much faster than (R1 * R2)
result set is the same
Large joins can be particularly taxing
Ideally, we do not let this affect how we write queries!
Smart DBMSs do "query optimization"
automatically reorder operations for efficiency
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