1 Relational Algebra Basic Operations Algebra of Bags

2 What is an “Algebra” uMathematical system consisting of: wOperands --- variables or values from which new values can be constructed. wOperators --- symbols denoting procedures that construct new values from given values.

3 What is Relational Algebra? uAn algebra whose operands are relations or variables that represent relations. uOperators are designed to do the most common things that we need to do with relations in a database. wThe result is an algebra that can be used as a query language for relations.

4 Core Relational Algebra uUnion, intersection, and difference. wUsual set operations, but both operands must have the same relation schema. uSelection: picking certain rows. uProjection: picking certain columns. uProducts and joins: compositions of relations. uRenaming of relations and attributes.

5 Selection  R1 := σ C (R2) wC is a condition (as in “if” statements) that refers to attributes of R2. wR1 is all those tuples of R2 that satisfy C. uFor relation R(A 1, …, A n ) the construction of condition C is as follows: wbasic (atomic) conditions: A i θ A j, A i θ α, where α is a constant, θ Є {=, }, wif B 1, B 2 are conditions, then  B 1, B 1  B 2, B 1  B 2 are also conditions.

6 Example: Selection Relation Sells: barbeerprice Joe’sBud2.50 Joe’sMiller2.75 Sue’sBud2.50 Sue’sMiller3.00 JoeMenu := σ bar=“Joe’s” (Sells): barbeerprice Joe’sBud2.50 Joe’sMiller2.75

7 Projection  R1 := π L (R2) wL is a list of attributes from the schema of R2. wR1 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. wEliminate duplicate tuples, if any.

8 Example: Projection Relation Sells: barbeerprice Joe’sBud2.50 Joe’sMiller2.75 Sue’sBud2.50 Sue’sMiller3.00 Prices := π beer,price (Sells): beerprice Bud2.50 Miller2.75 Miller3.00

9 Extended Projection  Using the same π L operator, we allow the list L to contain arbitrary expressions involving attributes: 1.Arithmetic on attributes, e.g., A+B->C. 2.Duplicate occurrences of the same attribute.

10 Example: Extended Projection R = ( AB ) π A+B->C,A,A (R) =CA1A

Exercise uWhat are the names of those drinkers, who like Bud? (Use relation Likes(drinker, beer).) 11

Set Operators uUnion, intersect, difference. uThe definition of these operators is as usual. uWarning. The schema of the operands must be the same. 12

Example: Set Operators 13 ABC abc cde gad ABC abc cde gdf RS ABC gad R - S

Exercise uWhat are the names of those drinkers, who do not like Bud? 14

15 Cross Product  R3 := R1 Χ R2 wPair each tuple t1 of R1 with each tuple t2 of R2. wConcatenation t1t2 is a tuple of R3. wSchema of R3 is the attributes of R1 and then R2, in order. wBut beware attribute A of the same name in R1 and R2: use R1.A and R2.A.

Exercise uWhich beers do those drinkers like, who visit Joe’s bar? (Use relations Likes(drinker, beer) and Frequents(drinker, bar).) 16

17 Example: R3 := R1 Χ R2 R1(A,B ) R2(B,C ) R3(A,R1.B,R2.B,C )

18 Theta-Join  R3 := R1 ⋈ C R2  Take the product R1 Χ R2.  Then apply σ C to the result.  As for σ, C can be any boolean-valued condition. wHistoric versions of this operator allowed only A  B, where  is =, <, etc.; hence the name “theta-join.” uEqui-joins are special theta-joins in which only equality comparisons are used.

19 Example: Theta Join Sells(bar,beer,price )Bars(name,addr ) Joe’sBud2.50Joe’sMaple St. Joe’sMiller2.75Sue’sRiver Rd. Sue’sBud2.50 Sue’sCoors3.00 BarInfo := Sells ⋈ Sells.bar = Bars.name Bars BarInfo(bar,beer,price,name,addr ) Joe’sBud2.50Joe’sMaple St. Joe’sMiller2.75Joe’sMaple St. Sue’sBud2.50Sue’sRiver Rd. Sue’sCoors3.00Sue’sRiver Rd.

Exercise uWhat are the prices of those beers that are sold in a bar on Maple Street? (Use relations Sells(bar, beer, price) and Bars(name, addr).) 20

21 Natural Join uA useful join variant (natural join) connects two relations by: wEquating attributes of the same name, and wProjecting out one copy of each pair of equated attributes.  Denoted R3 := R1 ⋈ R2.

22 Example: Natural Join Sells(bar,beer,price )Bars(bar,addr ) Joe’sBud2.50Joe’sMaple St. Joe’sMiller2.75Sue’sRiver Rd. Sue’sBud2.50 Sue’sCoors3.00 BarInfo := Sells ⋈ Bars Note: Bars.name has become Bars.bar to make the natural join “work.” BarInfo(bar,beer,price,addr ) Joe’sBud2.50Maple St. Joe’sMilller2.75Maple St. Sue’sBud2.50River Rd. Sue’sCoors3.00River Rd.

23 Renaming  The ρ operator gives a new schema to a relation.  R1 := ρ R1(A1,…,An) (R2) makes R1 be a relation with attributes A1,…,An and the same tuples as R2. uSimplified notation: R1(A1,…,An) := R2.

24 Example: Renaming Bars(name, addr ) Joe’sMaple St. Sue’sRiver Rd. R(bar, addr ) Joe’sMaple St. Sue’sRiver Rd. R(bar, addr) := Bars

25 Building Complex Expressions uCombine operators with parentheses and precedence rules. uThree notations, just as in arithmetic: 1.Sequences of assignment statements. 2.Expressions with several operators. 3.Expression trees.

26 Sequences of Assignments uCreate temporary relation names. uRenaming can be implied by giving relations a list of attributes.  Example: R3 := R1 ⋈ C R2 can be written: R4 := R1 Χ R2 R3 := σ C (R4)

27 Expressions in a Single Assignment  Example: the theta-join R3 := R1 ⋈ C R2 can be written: R3 := σ C (R1 Χ R2) uPrecedence of relational operators: 1.[ σ, π, ρ ] (highest). 2.[ Χ, ⋈ ]. 3.∩. 4.[ ∪, — ]

28 Expression Trees uLeaves are operands --- either variables standing for relations or particular, constant relations. uInterior nodes are operators, applied to their child or children.

29 Example: Tree for a Query uUsing 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.

30 As a Tree: BarsSells σ addr = “Maple St.” σ price<3 AND beer=“Bud” π name ρ R(name) π bar ∪

31 Example: Self-Join uUsing Sells(bar, beer, price), find the bars that sell two different beers at the same price. uStrategy: by renaming, define a copy of Sells, called S(bar, beer1, price). The natural join of Sells and S consists of quadruples (bar, beer, beer1, price) such that the bar sells both beers at this price.

32 The Tree Sells ρ S(bar, beer1, price) ⋈ π bar σ beer != beer1

33 Schemas for Results uUnion, intersection, and difference: the schemas of the two operands must be the same, so use that schema for the result. uSelection: schema of the result is the same as the schema of the operand. uProjection: list of attributes tells us the schema.

34 Schemas for Results --- (2) uProduct: schema is the attributes of both relations. wUse R.A, etc., to distinguish two attributes named A. uTheta-join: same as product. uNatural join: union of the attributes of the two relations. uRenaming: the operator tells the schema.

35 Relational Algebra on Bags uA bag (or multiset ) is like a set, but an element may appear more than once. uExample: {1,2,1,3} is a bag. uExample: {1,2,3} is also a bag that happens to be a set.

36 Why Bags? uSQL, the most important query language for relational databases, is actually a bag language. uSome operations, like projection, are more efficient on bags than sets.

37 Operations on Bags uSelection applies to each tuple, so its effect on bags is like its effect on sets. uProjection also applies to each tuple, but as a bag operator, we do not eliminate duplicates. uProducts and joins are done on each pair of tuples, so duplicates in bags have no effect on how we operate.

38 Example: Bag Selection R(A,B ) σ B < 3 (R) =AB 12

39 Example: Bag Projection R(A,B ) π A (R) =A 1 5 1

40 Example: Bag Product R(A,B )S(B,C ) R Χ S =AR.BS.BC

41 Example: Bag Theta-Join R(A,B )S(B,C ) R ⋈ R.B<S.B S =AR.BS.BC

42 Bag Union uAn element appears in the union of two bags the sum of the number of times it appears in each bag.  Example: {1,2,1} ∪ {1,1,2,3,1} = {1,1,1,1,1,2,2,3}

43 Bag Intersection uAn element appears in the intersection of two bags the minimum of the number of times it appears in either.  Example: {1,2,1,1} ∩ {1,2,1,3} = {1,1,2}.

44 Bag Difference uAn element appears in the difference A – B of bags as many times as it appears in A, minus the number of times it appears in B. wBut never less than 0 times. uExample: {1,2,1,1} – {1,2,3} = {1,1}.

45 Beware: Bag Laws != Set Laws uSome, but not all algebraic laws that hold for sets also hold for bags.  Example: the commutative law for union (R ∪ S = S ∪ R ) does hold for bags. wSince addition is commutative, adding the number of times x appears in R and S doesn’t depend on the order of R and S.

46 Example: A Law That Fails  Set union is idempotent, meaning that S ∪ S = S.  However, for bags, if x appears n times in S, then it appears 2n times in S ∪ S.  Thus S ∪ S != S in general.  e.g., {1} ∪ {1} = {1,1} != {1}.

Exercises --1 uOur running example: Beers(name, manf) Bars(name, addr, license) Drinkers(name, addr, phone) Likes(drinker, beer) Sells(bar, beer, price) Frequents(drinker, bar) 47

Exercises --2 uWhich are those bars, where Miller is sold for less than 2 dollars? uWhich are those bars, where Miller cannot be bought at all? uWho visits such a bar, where Miller is not sold? uWho are those drinkers, who visit a bar, where they can buy a beer, which they like? 48

Exercises --3 uWhich are those beers that are liked by at least two different drinkers? uWhich are those beers that are liked by exactly one drinker? uWhich is the most expensive beer? uWho likes the most expensive beer? uWho are those, who like every beer appearing in the Likes relation? 49