1. Relational Algebra
Chapter 4 1

2. Relational Query Languages
Query languages: Allow manipulation and retrieval of data from a database.
Relational model supports simple, powerful QLs:
Strong formal foundation based on algebra/logic.
Allows for much optimization.
We’ve seen examples of manipulation commands in Ch. 3.
Expand on logic, deductive databases, prolog and logic programming.
Mention relational completeness rather than Turing completeness.. The joins don’t come out on PC fonts.
Query Languages != programming languages!
QLs not expected to be “Turing complete”.
QLs not intended to be used for calculations.
QLs support easy, efficient access to large data sets. 2

3. Formal Relational Query Languages
Two mathematical Query Languages form the basis for “real” languages (e.g. SQL), and for implementation:
Relational Algebra: More operational, very useful for representing execution plans.
Relational Calculus: Lets users describe what they want, rather than how to compute it. (Non-operational, declarative.) We’ll skip this for now.
Relational Calculus relates to Query By Example

4. Overview Notation Relational Algebra
Relational Algebra basic operators.
Relational Algebra derived operators.

5. Preliminaries
A query is applied to relation instances, and the result of a query is also a relation instance.
Schemas of input relations for a query are fixed
The schema for the result of a given query is also fixed! Determined by definition of query language constructs. 4

6. Preliminaries Positional vs. named-attribute notation:
Positional notation
Ex: Sailor(1,2,3,4)
easier for formal definitions
Named-attribute notation
Ex: Sailor(sid, sname, rating,age)
more readable
Advantages/disadvantages of one over the other?
Cartesian product works well with positional notation. For some joins, need to know names of fields (natural join talks about fields that have the same name).

7. Example Instances R1
“Sailors” and “Reserves” relations for our examples.
We’ll use positional or named field notation.
Assume that names of fields in query results are `inherited’ from names of fields in query input relations. S1 S2
Cut out tables with records to manipulate them 5

8. Relational Algebra

9. Algebra In math, algebraic operations like +, -, x, /.
Operate on numbers: input are numbers, output are numbers.
Can also do Boolean algebra on sets, using union, intersect, difference.
Focus on algebraic identities, e.g.
x (y+z) = xy + xz.
(Relational algebra lies between propositional and 1st-order logic.)
draw more examples on overhead projects.
Matrix multiplication. 3 7 4

10. Relational Algebra Every operator takes one or two relation instances
A relational algebra expression is recursively defined to be a relation
Result is also a relation
Can apply operator to
Relation from database
Relation as a result of another operator

11. Relational Algebra Operations
Basic operations:
Selection ( ) Selects a subset of rows from relation.
Projection ( ) Deletes unwanted columns from relation.
Cross-product ( ) Allows us to combine two relations.
Set-difference ( ) Tuples in reln. 1, but not in reln. 2.
Union ( ) Tuples in reln. 1 and in reln. 2.
Additional derived operations:
Intersection, join, division, renaming: Not essential, but very useful.
Since each operation returns a relation, operations can be composed! (Algebra is “closed”.)
derived operation: like a macro 6

12. Basic Relational Algebra Operations

13. Projection Deletes attributes that are not in projection list.
Schema of result contains exactly the fields in the projection list, with the same names that they had in the (only) input relation.
Projection operator has to eliminate duplicates! (Why??)
Note: real systems typically don’t do duplicate elimination unless the user explicitly asks for it. (Why not?) 7

14. Selection Selects rows that satisfy selection condition.
No duplicates in result! (Why?)
Schema of result identical to schema of (only) input relation.
Selection conditions:
simple conditions comparing attribute values (variables) and / or constants or
complex conditions that combine simple conditions using logical connectives AND and OR.
Show with cut-out records 8

15. Union, Intersection, Set-Difference
All of these operations take two input relations, which must be union-compatible:
Same number of fields.
`Corresponding’ fields have the same type.
What is the schema of result?
Schema of R union S is schema of R. Similar for R intersect S. 9

16. Exercise on Union Number shape holes 1 round 2 square 4 3 rectangle 85
square 6
rectangle 8
Blue blocks (BB)
Yellow blocks(YB)
Which tables are union-compatible?
What is the result of the possible unions?
Bring in lego blocks. Cut out tables and show with record manipulation the result
bottom
top 4 2 6
Stacked(S)

17. Cross-Product Each row of S1 is paired with each row of R1.
Result schema has one field per field of S1 and R1, with field names `inherited’ if possible.
Conflict: Both S1 and R1 have a field called sid.
Show with Records. No two fields must have the same name.
Renaming operator: 10

18. Exercise on Cross-Product
Number
shape
holes 1
round 2
square 4 3
rectangle 8
Number
shape
holes 4
round 2 5
square 6
rectangle 8
Blue blocks (BB)
Write down 2 tuples in BB x S.
What is the cardinality of BB x S?
Show with records
bottom
top 4 2 6
Stacked(S)

19. Derived Operators Join and Division

20. Joins Condition Join: Result schema same as that of cross-product.
Fewer tuples than cross-product, might be able to compute more efficiently. How?
Sometimes called a theta-join.
Π-σ-x = SQL in a nutshell.
Show with cut-outs 11

21. Exercise on Join Number shape holes 1 round 2 square 4 3 rectangle 85
square 6
rectangle 8
Blue blocks (BB)
Yellow blocks(YB)
Show with records
Write down 2 tuples in this join.

22. Joins
Equi-Join: A special case of condition join where the condition c contains only equalities.
Result schema similar to cross-product, but only one copy of fields for which equality is specified.
Natural Join: Equijoin on all common fields. Without specified, condition means the natural join of A and B. 12

23. Example for Natural Join
Number
shape
holes 1
round 2
square 4 3
rectangle 8
shape
holes
round 2
square 4
rectangle 8
Blue blocks (BB)
Yellow blocks(YB)
Show with cut-outs
What is the natural join of BB and YB?

24. Join Examples

25. Find names of sailors who’ve reserved boat #103
Solution 1:
Solution 2:
Solution 3: 16

26. Exercise: Find names of sailors who’ve reserved a red boat
Information about boat color only available in Boats; so need an extra join:
A more efficient solution:
Make this exercise
A query optimizer can find this, given the first solution! 17

27. Find sailors who’ve reserved a red or a green boat
Can identify all red or green boats, then find sailors who’ve reserved one of these boats:
Can also define Tempboats using union! (How?)
What happens if is replaced by in this query? 18

28. Exercise: Find sailors who’ve reserved a red and a green boat
Previous approach won’t work! Must identify sailors who’ve reserved red boats, sailors who’ve reserved green boats, then find the intersection (note that sid is a key for Sailors):
Perhaps make this exercise. Or else add some other exercises. 19

29. Division
Not supported as a primitive operator, but useful for expressing queries like: Find sailors who have reserved all boats.
Typical set-up: A has 2 fields (x,y) that are foreign key pointers, B has 1 matching field (y).
Then A/B returns the set of x’s that match all y values in B.
Example: A = Friend(x,y). B = set of 354 students. Then A/B returns the set of all x’s that are friends with all 354 students.
See also Richard Frank’s notes. 13

30. Examples of Division A/B14

31. Find the names of sailors who’ve reserved all boats
Uses division; schemas of the input relations to / must be carefully chosen:
To find sailors who’ve reserved all ‘red boats:
..... 20

32. Division in General
In general, x and y can be any lists of fields; y is the list of fields in B, and (x,y) is the list of fields of A.
Then A/B returns the set of all x-tuples such that for every y-tuple in B, the tuple (x,y) is in A.
See also Richard Frank’s notes. 13

33. Summary
The relational model has rigorously defined query languages that are simple and powerful.
Relational algebra is more operational; useful as internal representation for query evaluation plans.
Several ways of expressing a given query; a query optimizer should choose the most efficient version.
Book has lots of query examples.