1. Database Management Systems 1 Raghu Ramakrishnan Relational Algebra Chpt 4 Xin Zhang

2. Database Management Systems 2 Raghu Ramakrishnan Relational Query Languages v Query languages: Allow manipulation and retrieval of data from a database. v Relational model supports simple, powerful QLs: –Strong formal foundation based on logic. –Allows for much optimization. v Query Languages != programming languages! –QLs not expected to be “Turing complete”. –QLs not intended to be used for complex calculations. –QLs support easy, efficient access to large data sets.

3. Database Management Systems 3 Raghu Ramakrishnan 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. * Understanding Algebra is key to understanding * SQL, query processing!

4. Database Management Systems 4 Raghu Ramakrishnan Preliminaries v 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 (but query will run regardless of instance!) –The schema for the result of a given query is also fixed! Determined by definition of query language constructs. v Positional vs. named-field notation: –Positional notation easier for formal definitions, named-field notation more readable. –Both used in SQL

5. Database Management Systems 5 Raghu Ramakrishnan Example Instances R1 S1 S2 v “Sailors” and “Reserves” relations for our examples. v 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.

6. Database Management Systems 6 Raghu Ramakrishnan Relational Algebra v 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. v Additional operations: –Intersection, join, division, renaming: Not essential, but (very!) useful.

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

8. Database Management Systems 8 Raghu Ramakrishnan Selection v Selects rows that satisfy selection condition. v No duplicates in result! v Schema of result identical to schema of (only) input relation. v Result relation can be the input for another relational algebra operation! ( Operator composition. )

9. Database Management Systems 9 Raghu Ramakrishnan Union, Intersection, Set-Difference B AUBAUB AnBAnBA-B A B-A

10. Database Management Systems 10 Raghu Ramakrishnan Union, Intersection, Set-Difference v All of these operations take two input relations, which must be union-compatible : –Same number of fields. –`Corresponding’ fields have the same type. v What is the schema of result? Occur in either S1 or S2 or both Occur in both S1 and S2 Occur in S1 but not in S2

11. Database Management Systems 11 Raghu Ramakrishnan More Example Instances sidsnameratingage 23sandy721.0 32windy819.5 59rope1018.0 sidsnameratingage 23sandy721.0 32windy819.5 45jib525.0 68topsail1025.0 S3 S4 v Work out on board SS34  SS34  SS34 

12. Database Management Systems 12 Raghu Ramakrishnan Union, Intersection, Set-Difference (continued) v Answers sidsnameratingage 23sandy721.0 32windy819.5 59rope1018.0 45jib525.0 68topsail1025.0 sidsnameratingage 23sandy721.5 32windy819.5 SS34  SS34  sidsnameratingage 59rope1018.0 SS34 

13. Database Management Systems 13 Raghu Ramakrishnan Cross-Product v Each row of S1 is paired with each row of R1. v 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. * Renaming operator :

14. Database Management Systems 14 Raghu Ramakrishnan Joins v Condition Join : v Result schema same as that of cross-product. v Fewer tuples than cross-product, might be able to compute more efficiently v Sometimes called a theta-join.

15. Database Management Systems 15 Raghu Ramakrishnan Joins v Equi-Join : A special case of condition join where the condition c contains only equalities. v Result schema similar to cross-product, but only one copy of fields for which equality is specified. v Natural Join : Equijoin on all common fields.

16. Database Management Systems 16 Raghu Ramakrishnan Division v Not supported as a primitive operator, but useful for expressing queries like: Find sailors who have reserved all boats. v Let A have 2 fields, x and y ; B have only field y : – A/B contains all x tuples (sailors) such that for every y tuple (boat) in B, there is an xy tuple in A. – Or : If the set of y values (boats) associated with an x value (sailor) in A contains all y values in B, the x value is in A/B. v 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.

17. Database Management Systems 17 Raghu Ramakrishnan Examples of Division A/B A B1 B2 B3 A/B1 A/B2A/B3

18. Database Management Systems 18 Raghu Ramakrishnan Example Instances R1 S1 S2 bidbnamebcolor 101Gippy red 103Fullsail green B1

19. Database Management Systems 19 Raghu Ramakrishnan Find names of sailors who’ve reserved boat #103 v Solution 1: v Solution 2 : v Solution 3 :

20. Database Management Systems 20 Raghu Ramakrishnan Find names of sailors who’ve reserved a red boat v Information about boat color only available in Boats; so need an extra join: v A more efficient solution: * A query optimizer can find this given the first solution!

21. Database Management Systems 21 Raghu Ramakrishnan Find sailors who’ve reserved a red or a green boat v Can identify all red or green boats, then find sailors who’ve reserved one of these boats: v Can also define Tempboats using union! v Select boats color=red and union with color=green v What happens if is replaced by in this query? v Nothing selected because color can’t be red & green

22. Database Management Systems 22 Raghu Ramakrishnan Find sailors names who’ve reserved a red and a green boat v 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):

23. Database Management Systems 23 Raghu Ramakrishnan v Uses division; schemas of the input relations to / must be carefully chosen: Find the names of sailors who’ve reserved all boats v To find sailors who’ve reserved all ‘Interlake’ boats:.....

24. Database Management Systems 24 Raghu Ramakrishnan Summary v The relational model has rigorously defined query languages that are simple and powerful. v Relational algebra is more operational; useful as internal representation for query evaluation plans. v Several ways of expressing a given query; a query optimizer should choose the most efficient version.