Relational AlgebraAnnouncementWhat is an “Algebra”What is Relational Algebra?RoadmapCore Relational AlgebraSelectionExampleProjectionSlide 10ProductExample: R3 := R1 * R2Theta-JoinSlide 14Natural JoinSlide 16RenamingSlide 18Building Complex ExpressionsSequences of AssignmentsExpressions in a Single AssignmentExpression TreesSlide 23As a Tree:Other Forms of ExpressionsSlide 26The TreeSchemas for ResultsSchemas for Results --- (2)Relational Algebra on BagsWhy Bags?Operations on BagsExample: Bag SelectionExample: Bag ProjectionExample: Bag ProductExample: Bag Theta-JoinBag UnionBag IntersectionBag DifferenceBeware: Bag Laws != Set LawsExample of the DifferenceThe Extended AlgebraDuplicate EliminationExample: Duplicate EliminationSortingExample: SortingExtended ProjectionExample: Extended ProjectionAggregation OperatorsExample: AggregationGrouping OperatorApplying GAMMAL(R)Example: Grouping/AggregationOuterjoinExample: Outerjoin1Relational AlgebraOperatorsExpression TreesBag Model of Data2AnnouncementProject 1 due todayHomework 3 out, due 5/17Recitation given by Yi Qiao, 5-6pm for the remaining of the quarter3What is an “Algebra”Mathematical system consisting of:Operands --- variables or values from which new values can be constructed.Operators --- symbols denoting procedures that construct new values from given values.4What is Relational Algebra?An algebra whose operands are relations or variables that represent relations.Operators are designed to do the most common things that we need to do with relations in a database.The result is an algebra that can be used as a query language for relations.5RoadmapThere is a core relational algebra that has traditionally been thought of as the relational algebra.But there are several other operators we shall add to the core in order to model better the language SQL --- the principal language used in relational database systems.6Core Relational AlgebraUnion, intersection, and difference.Usual set operations, but require both operands have the same relation schema.Selection: picking certain rows.Projection: picking certain columns.Products and joins: compositions of relations.Renaming of relations and attributes.7SelectionR1 := SELECTC (R2)Denoted as σC(R2)C is a condition (as in “if” statements) that refers to attributes of R2.R1 is all those tuples of R2 that satisfy C.8ExampleRelation Sells:bar beer priceJoe’s Bud 2.50Joe’s Miller 2.75Sue’s Bud 2.50Sue’s Miller 3.00JoeMenu := SELECTbar=“Joe’s”(Sells):bar beer priceJoe’s Bud 2.50Joe’s Miller 2.759ProjectionR1 := PROJL (R2)Denoted as piL(R2)L is a list of attributes from the schema of R2.R1 is constructed by looking at each tuple of R2, extracting the attributes on list L, in the order specified, and creating from those components a tuple for R1.Eliminate duplicate tuples, if any.10ExampleRelation Sells:bar beer priceJoe’s Bud 2.50Joe’s Miller 2.75Sue’s Bud 2.50Sue’s Miller 3.00Prices := PROJbeer,price(Sells):beer priceBud 2.50Miller 2.75Miller 3.0011ProductR3 := R1 * R2Pair each tuple t1 of R1 with each tuple t2 of R2.Concatenation t1t2 is a tuple of R3.Schema of R3 is the attributes of R1 and then R2, in order.But beware attribute A of the same name in R1 and R2: use R1.A and R2.A.12Example: R3 := R1 * R2R1( A, B )1 23 4R2( B, C )5 67 89 10R3( A, R1.B, R2.B, C )1 2 5 61 2 7 81 2 9 103 4 5 63 4 7 83 4 9 1013Theta-JoinR3 := R1 JOINC R2Take the product R1 * R2.Then apply SELECTC to the result.As for SELECT, C can be any boolean-valued condition.Historic versions of this operator allowed only A  B, where  is =, <, etc.; hence the name “theta-join.”14ExampleSells( bar, beer, price ) Bars( name, addr )Joe’s Bud 2.50 Joe’s Maple St.Joe’s Miller 2.75 Sue’s River Rd.Sue’s Bud 2.50Sue’s Coors 3.00 BarInfo := Sells JOIN Sells.bar = Bars.name Bars BarInfo( bar, beer, price, name, addr )Joe’s Bud 2.50 Joe’s Maple St.Joe’s Miller 2.75 Joe’s Maple St.Sue’s Bud 2.50 Sue’s River Rd.Sue’s Coors 3.00 Sue’s River Rd.15Natural JoinA frequent type of join connects two relations by:Equating all attributes of the same name, andProjecting out one copy of each pair of equated attributes.Called natural join.Denoted R3 := R1 JOIN R2.16ExampleSells( bar, beer, price ) Bars( bar, addr )Joe’s Bud 2.50 Joe’s Maple St.Joe’s Miller 2.75 Sue’s River Rd.Sue’s Bud 2.50Sue’s Coors 3.00 BarInfo := Sells JOIN BarsNote Bars.name has become Bars.bar to make the naturaljoin “work.” BarInfo( bar, beer, price, addr )Joe’s Bud 2.50 Maple St.Joe’s Milller 2.75 Maple St.Sue’s Bud 2.50 River Rd.Sue’s Coors 3.00 River Rd.17RenamingThe RENAME operator gives a new schema to a relation.R1 := RENAMER1(A1,…,An)(R2) makes R1 be a relation with attributes A1,…,An and the same tuples as R2.Simplified notation: R1(A1,…,An) := R2.18ExampleBars( name, addr )Joe’s Maple St.Sue’s River Rd. R( bar, addr )Joe’s Maple St.Sue’s River Rd.R(bar, addr) := Bars19Building Complex ExpressionsCombine operators with parentheses and precedence rules.Three notations, just as in arithmetic:1. Sequences of assignment statements.2. Expressions with several operators.3. Expression trees.20Sequences of AssignmentsCreate temporary relation names.Renaming can be implied by giving relations a list of attributes.Example: R3 := R1 JOINC R2 can be written:R4 := R1 * R2R3 := SELECTC (R4)21Expressions in a Single AssignmentExample: the theta-join R3 := R1 JOINC R2 can be written: R3 := SELECTC (R1 * R2)Precedence of relational operators:1. [SELECT, PROJECT, RENAME] (highest).2. [PRODUCT, JOIN].3. INTERSECTION.4. [UNION, DIFFERENCE]But you can always insert parentheses to force the order you desire.22Expression TreesLeaves are operands --- either variables standing for relations or particular, constant relations.Interior nodes are operators, applied to their child or children.23ExampleUsing the relations Bars(name, addr) and Sells(bar, beer, price), find the names of all the bars that are either on Maple St. or sell Bud for less than $3.24As a Tree:Bars SellsSELECTaddr = “Maple St.”SELECTprice<3 AND beer=“Bud”PROJECTnameRENAMER(name)PROJECTbarUNION25Other Forms of Expressions Relational Algebra Expression in single assignmentA sequence of