Introduction to Relational Algebra
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This document introduces relational algebra, which is an algebra for modeling and querying data stored in relational databases. Relational algebra uses operators like projection, selection, join, union, difference and rename to filter, combine and transform relations (tables) of data. Several examples are provided to demonstrate how these operators can be combined to solve increasingly complex queries against sample databases. The document concludes by noting that while relational algebra defines the underlying logic, most databases are instead queried using SQL, which is based on this relational algebra.
![lter, slice and combine the data.
I The projection operator, [attr ], returns the relation with only
the chosen attributes, from the relation to which it is applied.
I The selection operator, [cond], returns a relation with only
the tuples matching the conditions provided, from the relation
to which it is applied.
I Note we use ^ for AND, and _ for OR.](https://image.slidesharecdn.com/databases-141204031529-conversion-gate02/85/Introduction-to-Relational-Algebra-9-320.jpg)
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Introduction to Relational Algebra
- 1. An Introduction to Relational Algebra James McMurray PhD Student Department of Empirical Inference 09/12/2014
- 2. Outline of talk Introduction What is Relational Algebra? Relational Algebra operators Core operators Joins Set operators Rename operator Conclusion Conclusion
- 3. What is Relational Algebra? I Algebra for modelling and querying data stored in relational databases. I A relational database consists of a set of named relations, which contain named and typed attributes, and the data stored in tuples. I A relation is a table. I An attribute is a column. I A tuple is a row. I Relational Algebra queries are compositional, so any query returns a relation, which can in turn be queried.
- 4. Database schema I The schema of the database is the names of the relation, names and types of attributes - the structure of the database. I Usually, the schema is
- 5. xed in advance, and the data instances may change with time. I All attributes are typed - i.e. Ints, strings, etc., the implementation depends on the speci
- 6. c database management system (DBMS). I In addition to typed values, there exists the null value which is used for missing values. I A key is an attribute which is unique for every tuple (usually an ID, or row ID), this is important for indexing the relations in practice. I Note there is a debate over where relations (tables) should have singular or plural names. Ruby on Rails uses plurals, Django uses singular names.
- 7. Example database DVD movie rating `Edge Of Tomorrow' 4 `Elysium' 9 `Gattaca' 8 A relation DVD, with the attributes movie (String) and rating (Int). Store name movie price `CheapBay' `Edge Of Tomorrow' 10 `DVD World' `Elysium' 20 `DVD World' `Gattaca' 10 `Prime' `Edge Of Tomorrow' 20 `Prime' `Elysium' 15 A relation Store, with the attributes name (String), movie (String) and price (Int)
- 8. Relational Algebra operators I The simplest query is just the relation name, which returns the relation (table). I Operators act on this to
- 9. lter, slice and combine the data. I The projection operator, [attr ], returns the relation with only the chosen attributes, from the relation to which it is applied. I The selection operator, [cond], returns a relation with only the tuples matching the conditions provided, from the relation to which it is applied. I Note we use ^ for AND, and _ for OR.
- 10. Example of projection and selection DVD movie rating `Edge Of Tomorrow' 4 `Elysium' 9 `Gattaca' 8 ratingDVD = rating 4 9 8 rating=5DVD = movie rating `Elysium' 9 `Gattaca' 8
- 11. Example of combination DVD Movie Rating `Edge Of Tomorrow' 4 `Elysium' 9 `Gattaca' 8 movie (rating=5DVD) = movie `Elysium' `Gattaca'
- 12. Relational Algebra operators (contd.) I The tuple cross-product, , returns a relation with every combination of the tuples. I The natural join, ./, is the cross-product combined with the selection operator to match tuples according to shared attribute names. I Note any relation natural joined with itself, returns the same relation itself. I Note the natural join just syntactic sugar for combining the cross-product and a speci
- 13. c selection operator. I The Theta join, ./, is the cross-product combined with a general selection operator. I This is also just syntactic sugar, and is the standard JOIN ON operator in DBMSs.
- 14. Example database DVD movie rating `Edge Of Tomorrow' 4 `Elysium' 9 `Gattaca' 8 A relation DVD, with the attributes movie (String) and rating (Int) Store name movie price `CheapBay' `Edge Of Tomorrow' 10 `DVD World' `Elysium' 20 `DVD World' `Gattaca' 10 `Prime' `Edge Of Tomorrow' 20 `Prime' `Elysium' 15 A relation Store, with the attributes name (String), movie (String) and price (Int)
- 15. Example cross-product I The cross-product returns every possible combination of the tuples. (nameStore) (MovieDVD) = name movie `CheapBay' `Edge Of Tomorrow' `CheapBay' `Elysium' `CheapBay' `Gattaca' `DVD World' `Edge Of Tomorrow' `DVD World' `Elysium' `DVD World' `Gattaca' `Prime' `Edge Of Tomorrow' `Prime' `Elysium' `Prime' `Gattaca'
- 16. Example join I The natural join combines the tuples in the relations, where the joint attribute (movie) is equal. Store ./ DVD = name movie price rating `CheapBay' `Edge Of Tomorrow' 10 4 `DVD World' `Elysium' 20 9 `DVD World' `Gattaca' 10 8 `Prime' `Edge Of Tomorrow' 20 4 `Prime' `Elysium' 15 9
- 17. Relational Algebra operators (contd.) I The union operator, [, concatenates/stacks the tuples. I The dierence operator, , subtracts the second set of tuples from the
- 18. rst . I i.e. A B returns tuples in A but not in B. I The intersection operator, , returns the tuples present in both sets. I Note the intersection operator is just syntactic sugar, we can always write: A B = A (A B)
- 19. Example dierence I We want to
- 20. nd all stores which only sell movies for less than 12 dollars. I We can do this by
- 21. nding all stores which sell any movie for 12 dollars or more, and then subtracting this from the set of all stores. (nameStore) (name(price=12Store)) = name `CheapBay'
- 22. Relational Algebra operators (contd.) I The rename operator, [newnames], returns the relation with the attributes assigned to the new names. I This is not just syntactic sugar. I Technically we need attribute names to be the same to carry out the natural join. I More importantly, it is necessary for disambiguation in self-joins. newname ratingDVD = newname 4 9 8
- 23. Self-join example I We want to
- 24. nd every distinct pair of names of movies which have been rated - i.e. (`Edge Of Tomorrow', `Elysium') I In order to do this we have to use a self-join, and the rename operator is critical (also note the importance of the greater than condition): moviemovie2 ((movieDVD) (movie2 movieDVD)) = movie movie2 `Edge Of Tomorrow' `Elysium' `Edge Of Tomorrow' `Gattaca' `Elysium' `Gattaca'
- 25. More complicated example I We want to
- 26. nd the maximum rating of all
- 27. lms sold by Prime. I The trick we use is to
- 28. nd the set of ratings which are ever less than another rating (via a self-join), and then subtract this from the set of all ratings of
- 29. lms sold by Prime (denoted by PrimeRatings). PrimeRatings := rating ((name=`prime'Store) ./ DVD) Result := PrimeRatings rating ratingrating2( PrimeRatings rating2PrimeRatings ) = rating 9
- 30. Even harder example I We want to
- 31. nd the names of all the stores which sell ALL of the
- 32. lms which have ratings higher than 7. I Trick is to calculate all possible, relevant store-movie tuples, then calculate the could have been tuples which were not observed, then
- 33. nd which stores these were present for, and subtract this from all the stores. I This is Relational Division. GoodMovie := movie rating7DVD Result :=nameStore name(((nameStore) GoodMovie) name,movieStore) = name `DVD World'
- 34. Conclusion I Relational algebra de
- 35. nes an algebra for querying relational databases. I Operators may seem simple but queries can become complicated. I Many other common extensions have been omitted here, such as the semijoin, antijoin and relational division - these are all syntactic sugar. I In practice we do not query relational databases with relational algebra directly, but with a query language based upon it. I The most popular is SQL, the Structured Query Language. I Though there is a Relational Algebra interpeter called RA for SQLite (used for these examples). Questions?