1. Relational Algebra Prof. Yin-Fu Huang CSIE, NYUST Chapter 7

2. Advanced Database SystemYin-Fu Huang Eight operators of the relational algebra: 1. The traditional set operators union, intersection, difference, and Cartesian product. 2. The special relational operators restrict, project, join, and divide. (See Fig. 7.1) 7.1Introduction

3. Advanced Database SystemYin-Fu Huang

4. Advanced Database SystemYin-Fu Huang 7.2Closure Revisited The output from any given relational operation is another relation. Nested relational expressions

5. Advanced Database SystemYin-Fu Huang 7.3The Original Algebra: Syntax (See Page178-179) ::= Relation { } | | ( ) ::= | ::= { [All But] } ::= | | | | | | |

6. Advanced Database SystemYin-Fu Huang 7.3The Original Algebra: Syntax (Cont.) ::= Rename ( ) ::= Union ::= Intersect ::= Minus ::= Times ::= Where ::= Join ::= Divideby Per ::= | (, ) ::= With : ::= As

7. Advanced Database SystemYin-Fu Huang Tuple-homogeneous Union, Intersect, and Difference (See Fig. 7.2) Product (See Fig. 7.3) If we need to construct the Cartesian product of two relations that do have any such common attribute names, we must use the Rename operator first to rename attributes appropriately. Restrict (See Fig. 7.4) Project (See Fig. 7.5) 7.4 The Original Algebra: Semantics

8. Advanced Database SystemYin-Fu Huang Fig. 7.2Union, intersection, and difference

9. Advanced Database SystemYin-Fu Huang Fig. 7.3Cartesian product example

10. Advanced Database SystemYin-Fu Huang Fig. 7.4Restriction examples

11. Advanced Database SystemYin-Fu Huang Fig. 7.5Projection examples

12. Advanced Database SystemYin-Fu Huang Join natural join (See Fig. 7.6) θ-join (See Fig. 7.7) ((S Rename City As Scity) Times (P Rename City As Pcity)) Where Scity > Pcity Divide (See Fig. 7.8) 7.4 The Original Algebra: Semantics (Cont.)

13. Advanced Database SystemYin-Fu Huang Fig. 7.6&Fig. 7.7

14. Advanced Database SystemYin-Fu Huang Fig. 7.8Division examples

15. Advanced Database SystemYin-Fu Huang 7.5Examples Exam 1: ((Sp Join S) Where P#=P#(‘P2’)) {Sname} Exam 2: (((P Where Color=Color(‘Red’)) Join Sp ) {S#} Join S) {Sname} Exam 3: ((S {S#} Divideby P {P#} Per Sp {S#, P#}) Join S) {Sname} Exam 4: S {S#} Divideby (Sp Where S#=S#(‘S2’)) {P#} Per Sp {S#, P#} Exam 5: (((S Rename S# As Sa) {Sa, City} Join (S Rename S# As Sb) {Sb, City}) Where Sa < Sb) {Sa, Sb} Exam 6: ((S {S#} Minus (Sp Where P#=P#(‘P2’)) {S#}) Join S) {Sname}

16. Advanced Database SystemYin-Fu Huang 7.6What Is the Algebra For? The operators join, intersect, and divide can be defined in terms of the other five. Of the remaining five, however, none can be defined in terms of the other four, so we can regard those five as constituting a primitive or minimum set. Some possible applications: 1. Defining a scope for retrieval 2. Defining a scope for update 3. Defining integrity constraints 4. Defining derived relvars 5. Defining stability requirements 6. Defining security constraints

17. Advanced Database SystemYin-Fu Huang 7.6What Is the Algebra For? (Cont.) A high-level, symbolic representation of the user‘s intent  Transformation rules ((Sp Join S) Where P#=P#(‘P2’)) {Sname} ((Sp Where P#=P#(‘P2’)) Join S) {Sname} The algebra thus serves as a convenient basis for optimization. A language is said to be relationally complete if it is at least as powerful as the algebra.

18. Advanced Database SystemYin-Fu Huang 7.7Further Points Associativity and Commutativity Associative: Union, Intersect, Times, Join e.g. (A Union B) Union C = A Union (B Union C) = A Union B Union C Commutative: Union, Intersect, Times, Join e.g. A Union B = B Union A Some Equivalences e.g. r { } = Table_Dum if r=empty, Table_Dee otherwise (a nullary projection) r Join Table_Dee = Table_Dee Join r = r r Times Table_Dee = Table_Dee Times r = r

19. Advanced Database SystemYin-Fu Huang 7.7Further Points (Cont.) Some Generalizations If s contains no relations at all, then: The join of all relations in s is defined to be Table_Dee. The union of all relations in s is defined to be the empty relation. The intersection of all relations in s is defined to be the “universal” relation.

20. Advanced Database SystemYin-Fu Huang (See Page 196) ::= Semijoin ::= Semiminus ::= Extend Add ( ) ::= As ::= Summarize Per Add ( ) ::= [ ( ) ] As ::= Count | Sum | Avg | Max | Min | All | Any | Countd | Sumd | Avgd| … ::= Tclose 7.8 7.8Additional Operators

21. Advanced Database SystemYin-Fu Huang 7.8 7.8Additional Operators (Cont.) Semijoin  (a Join b) {X, Y} e.g. S Semijoin (Sp Where P#=P#(‘P2’)) Semidifference  a Minus (a Semijoin b) e.g. S Semiminus (Sp Where P#=P#(‘P2’)) Extend e.g. Extend P Add (Weight ＊ 454) As Gmwt (See Fig. 7.9)

22. Advanced Database SystemYin-Fu Huang 7.8 7.8Additional Operators (Cont.)  Exam 1: Extend S Add ‘Supplier’ As Tag  Exam 2: Extend (P Join Sp) Add (Weight ＊ Qty) As Shipwt  Exam 3: (Extend S Add City As Scity) {All But City} Rename  Exam 4: Extend P Add (Weight ＊ 454 As Gmwt, Weight ＊ 16 As Ozwt)  Exam 5: Extend S Add Count((Sp Rename S# As X) Where X=S#) As Np(See Fig. 7.10)

23. Advanced Database SystemYin-Fu Huang 7.8 7.8Additional Operators (Cont.)  Exam 1: Summarize Sp Per P {P#} Add (Sum(Qty) As Totqty, Avg(Qty) As Avgqty)  Exam 2: Summarize Sp Per S {S#} Add Count As Np Summarize is not a primitive operator. Extend  Exam 3: Summarize S Per S {City}Add Avg(Status) As Avg_Status  Exam 4: Summarize Sp Per Sp { }Add Sum(Qty) As Grandtotal Summarize e.g. Summarize Sp Per P {P#} Add Sum(Qty) As Totqty (See Fig. 7.11) Tclose  the transitive closure of a

24. Advanced Database SystemYin-Fu Huang Group e.g. SP Group (P#, Qty) As PQ(See Fig. 7.12) Ungroup e.g. SPQ Ungroup PQ The reversibility of the Group and Ungroup operations(See Fig. 7.13)  Functionally dependency 7.9 7.9Grouping and Ungrouping

25. Advanced Database SystemYin-Fu Huang The End.