Relational Algebra: Relational Algebra CIS 4301 Lecture Notes Lecture 13 - 2/23/2006


Database Query Languages: Database Query Languages Use 'Campus' schema Given a database, ask questions, get data as answers Ex: Get all students with GPA andgt; 3.7 who applied to Berkeley and Stanford and nowhere else Ex: Get all humanities departments at campuses in Florida with andlt; 1000 applicants Ex: Get the campus with highest average accept rate over the last five years Some questions are easy to pose, some are not Some questions are easy for DBMS to answer, some are not. 'Query language' also used to update the database


Relational Query Languages: Relational Query Languages Formal: relational algebra, relational calculus, Datalog Actual: SQL, Quel, Query-by-Example (QBE) In ALL languages, a query is executed over a set of relations, get single relation as the result


Relational Algebra: Relational Algebra Notation for describing queries in the relational model Relational model has concrete set of 'standard' operations Operations are not 'Turing Complete' Not a defect, helps with query processing and optimization FYI, a language is Turing Complete if it is powerful enough to implement any Turing machine. It's widely believed that Turing machines can do any calculation that can be performed by a modern computer program Start by introducing operations of relational algebra, SQL next Algebra applies to sets of tuples, i.e., relations Commercial DBMS use different notation of relations which are multisets


Relational Algebra: Relational Algebra Construct new relations from old ones Set of operators Relations are operands Build progressively more complex expressions by applying operators to relations or to rela. algebra expressions (which are relations as well) Query is an expression of relational algebra First concrete example of a query language Four broad classes of operations Set operations, selection operations, operations that combine data from two relations, rename operation


Sample Relational Schema: Sample Relational Schema Movie (Title,Year,length,filmType, studioName,producerC#) StarsIn (MovieTitle,MovieYear,StarName) MovieStar(Name,address,gender,birthdate) MovieExec(name,address,Cert#,netWorth) Studio (Name,address,presC#)


Basics: Basics Operations of traditional relational algebra fall into four broad classes: Set operations Operations that remove parts of a relation Operations that combine tuples of two relations Renaming


Set Operations: Set Operations Union (binary, commutative, associative) R  S Intersection (binary, commutative, associative) R  S Set Difference (binary) R - S Set of elements in R but not in S R-S  S-R !! R(A1,A2,…,An), S(B1,B2,…,Bn) must be union compatible R and S are of the same degree for each i, dom(Ai) = dom(Bi) Columns of R and S must be ordered so that order of attributes is same for both relations


Example: Example name address gender birthdate Carrie Fisher Mark Hamil 123 Maple St., Hollywood 456 Oak Rd., Brentwood F M 9/9/99 8/8/88 R name address gender birthdate Carrie Fisher Harrison Ford 123 Maple St., Hollywood 789 Palm Dr., Beverly Hills F M 9/9/99 7/7/77 S


Sample Operations: Sample Operations name address gender birthdate Carrie Fisher 123 Maple St., Hollywood F 9/9/99 R  S name address gender birthdate Carrie Fisher Harrison Ford 123 Maple St., Hollywood 789 Palm Dr., Beverly Hills F M 9/9/99 7/7/77 R  S Mark Hamil 456 Oak Rd., Brentwood M 8/8/88 name address gender birthdate R - S Mark Hamil 456 Oak Rd., Brentwood M 8/8/88


Relational Operator: Project: Relational Operator: Project Project (unary) andlt;attr listandgt; (R) andlt;attr listandgt; is a list of attributes (columns) from R only Ex: title, year, length (Movie) 'horizontal restriction' A1 A2 A3 … An ... i A1 A2… Ak ... j, i  j  n k


Project: Project PROJECT can produce many tuples with same value Relational algebra semantics says remove duplicates SQL does not -- one difference between formal and actual query languages


Relational Operator: Select: Relational Operator: Select Select or Restrict (unary, commutative) andlt;predicateandgt; (R) andlt;predicateandgt; is a conditional expression of the type that we are familiar with from conventional programming languages andlt;attributeandgt; andlt;opandgt; andlt;attributeandgt; andlt;attributeandgt; andlt;opandgt; andlt;constantandgt; attribute in R op  {=,,andlt;,andgt;,, …, AND, OR} Ex: length100 (Movie) vertical restriction'


Pictorially: Pictorially A1 A2 A3 … An ... i A1 A2 A3 … An ... j, i  j  title year length filmType Star Wars Mighty Ducks Wayne’s World 1977 1991 1992 124 104 95 color color color Movie result set # of selected tuples is referred to as the selectivity of the condition


Cartesian Product: Cartesian Product Cartesian Product (binary, commutative, associative) R x S Sets of all pairs that can be formed by choosing the first element of the pair to be any element of R, the second any element of S Relation schema is union of schemas for R and S Resulting schema may be ambiguous Use R.A or S.A to disambiguate an attribute that occurs in both schemas


Example: Example A B 1 2 3 4 B C 2 5 4 7 D 6 8 9 10 11 x A R.B S.B C D R S 1 2 2 5 6 1 2 4 7 8 1 2 9 10 11 3 4 3 4 3 4 2 5 6 4 7 8 9 10 11


Join Operations: Join Operations Natural Join (binary) R join S Match only those tuples from R and S that agree in whatever attributes are common to the schemas of R and S If r and s from r(R) and s(S) are successfully paired, result is called a joined tuple This join operation is the same we used in earlier section to recombine relations that had been projected onto two subsets of their attributes (e.g., as a result of a BCNF decomposition)


Example: Example A B 1 2 3 4 B C 2 5 4 7 D 6 8 9 10 11 join Resulting schema has attributes from R, either R or S (i.e., joining attribute(s)), and S Tuples that fail to pair with any tuple of the other relation are called dangling tuples A B C D R S 1 2 5 6 3 4 7 8


Join Operations: Join Operations Theta Join (binary) R joinC S, where C is an arbitrary join condition Step 1: take the product of R and S Step 2: Select from the product only those tuples that satisfy condition C As with the product operation, the schema for the result is the union of the schemas of R and S


Example: Example B C 2 3 2 3 D 4 5 7 8 10 joinAandlt;D AND U.BV.B A B 1 2 6 7 C 3 8 9 7 8 U V V.B V.C D A U.B U.C 1 2 3 7 8 10


Final Word on Join: Final Word on Join DBMS often implements theta-join as basic operation Use of term 'join' in implementation circles usually refers to theta-join or sometimes to cross-product