Relational Algebra SPJRUD

Size: px

Start display at page:

Download "Relational Algebra SPJRUD"

Todd Ford
5 years ago
Views:

1 Relational Algebra SPJRUD Jef Wijsen Université de Mons (UMONS) May 14, 2018 Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

2 Tabular Representation The table A B C is shorthand for the following set of tuples: {A : 1, B : 3, C : 2}, {A : 1, B : 4, C : 1}, {A : 2, B : 4, C : 2}, {A : 2, B : 3, C : 1}. Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

3 Notation Every tuple is a total function from a set of attributes to the set of constants. Therefore, if t = {A : 1, B : 3, C : 2}, then t(b) = 3 and t[{a, C}] = {A : 1, C : 2}. Note that t(b) is a constant, and t[{a, C}] a tuple. In database theory, we often omit curly brackets ({}) and union symbols ( ) in the notation of sets. For example, if A, B, C, D are attributes and X = {A, B}, then XCD denotes the set {A, B, C, D}. Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

4 Algebraic Operators Unary operators: Select, Project, Rename Binary operators: Join, Union, Difference Unary operators take in a single relation; binary operators take in two relations. Every operator returns a single relation. Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

5 Select R A B C σ A= 1 (R) A B C σ A=C (R) A B C This is like SELECT * FROM R WHERE A="1" in SQL. Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

6 Project R A B C π {A,B} (R) A B Since relations are sets, duplicates are removed. Note that SELECT A, B FROM R in SQL does not remove duplicates. Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

7 Join R A B C S B C D R S A B C D R S contains all tuples t such that t[abc] is in R, and t[bcd] in S. This is different from the cross product SELECT * FROM R, S in SQL. Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

8 Rename R A B C ρ C D (R) A B D Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

9 Union R A B C S A B C R S A B C R S is only allowed if R and S have exactly the same attributes. In SQL, (SELECT * FROM R) UNION (SELECT * FROM S) only requires that R and S have the same number of attributes. The result takes the attributes of R. Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

10 Difference R A B C S A B C R S A B C R S is only allowed if R and S have exactly the same attributes. In SQL, (SELECT * FROM R) MINUS (SELECT * FROM S) only requires that R and S have the same number of attributes. The result takes the attributes of R. Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

11 Alphabet Relation names. Each relation name R is associated with a fixed set of attributes, denoted sort(r). Constants. Typically, R, S, T are relation names; A, B, C are attributes; a, b, c are constants. Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

12 Syntax I Relation names Every relation name is an algebra expression. Selection If E is an algebra expression, A sort(e), and c is a constant, then σ A=c (E) is an algebra expression with sort(σ A=c (E)) = sort(e). Selection If E is an algebra expression and A, B sort(e), then σ A=B (E) is an algebra expression with sort(σ A=B (E)) = sort(e). Projection If E is an algebra expression and X sort(e), then π X (E) is an algebra expression with sort(π X (E)) = X. Join If E 1 and E 2 are algebra expressions, then E 1 E 2 is an algebra expression with sort(e 1 E 2 ) = sort(e 1 ) sort(e 2 ). Rename If E is an algebra expression, A sort(e), and B is an attribute not in sort(e), then ρ A B (E) is an algebra expression with sort(ρ A B (E)) = (sort(e) \ {A}) {B}. Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

13 Syntax II Union If E 1 and E 2 are algebra expressions with sort(e 1 ) = sort(e 2 ), then E 1 E 2 is an algebra expression with sort(e 1 E 2 ) = sort(e 1 ). Difference If E 1 and E 2 are algebra expressions with sort(e 1 ) = sort(e 2 ), then E 1 E 2 is an algebra expression with sort(e 1 E 2 ) = sort(e 1 ). Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

14 Semantics I A database I is a total function that maps every relation name R to a relation over sort(r). We denote by R I the relation to which R is mapped by I. For every relation name R, R I = R I. σ A=c (E) I = {t E I t(a) = c}. σ A=B (E) I = {t E I t(a) = t(b)}. π X (E) I = {t[x ] t E I }. Assume sort(e 1 ) = X and sort(e 2 ) = Y. Then, E 1 E 2 I = {t t[x ] E 1 I and t[y ] E 2 I }. ρ A B (E) I contains every tuple that can be obtained by replacing A with B in some tuple of E I. E 1 E 2 I = E 1 I E 2 I, where is the standard union of sets. E 1 E 2 I = E 1 I \ E 2 I. Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

15 Examples VINS Cru Millesime Qualite Chablis 1992 excellent Chablis 1993 bon Rothschild 1993 bon Rothschild 1994 imbuvable ABUS Nom Cru Annee Ed Chablis 1992 Ed Chablis 1993 An Chablis 1993 An Rothschild 1993 Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

16 Qui n a jamais bu un vin excellent? Soit E 1 = ρ Annee Millesime (ABUS) E 2 = σ Qualite= excellent (VINS) E 3 = π Nom (E 1 E 2 ) E 3 rend les personnes ayant déjà bu un vin excellent. La requête demandée est donc : π Nom (ABUS) E 3 Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

17 Discussion Can we express all interesting queries in SPJRUD, or should we add more operators? Why don t we have intersection? Why don t we have σ A B R? S Show the following: if we leave out one of the 6 operators, then we can express strictly less queries. Hint: For union, consider two relations R A 1 and A. Show that if E is an 0 A algebraic expression that does not contain, then E will not return 0 on 1 input R and S (and therefore, E does not express R S). Jef Wijsen (Université de Mons (UMONS)) SPJRUD May 14, / 1

Relational Algebra on Bags. Why Bags? Operations on Bags. Example: Bag Selection. σ A+B < 5 (R) = A B

Relational Algebra on Bags. Why Bags? Operations on Bags. Example: Bag Selection. σ A+B < 5 (R) = A B Relational Algebra on Bags Why Bags? 13 14 A bag (or multiset ) is like a set, but an element may appear more than once. Example: {1,2,1,3} is a bag. Example: {1,2,3} is also a bag that happens to be a