Relational Algebra. B term 2004: lecture 10, 11

Transcription

1 Relational lgebra term 00: lecture 0, Nov, 00 asics Relational lgebra is defined on bags, rather than relations. ag or multiset allows duplicate values; but order is not significant. We can write an expression using relational algebra operators with parentheses: we need closure an operator on bag returns a bag. Relational algebra includes set operators, and other operators specific to relational model. Nov, 00

2 Set Operators Union, Intersection, ifference, cross product Union, Intersection and ifference are defined only for union compatible relations. Two relations are union compatible if they have the same set of attributes and the types (domains) of the attributes are the same. Eg of two relations that are not union compatible: Student (snumber, sname) ourse (cnumber, cname) Nov, 00 Union: onsider two bags R and R that are unioncompatible. Suppose a tuple t appears in R m times, and in R n times. Then in the union, t appears m + n times. R R Nov, 00 R R

3 Intersection: onsider two bags R and R that are unioncompatible. Suppose a tuple t appears in R m times, and in R n times. Then in the intersection, t appears min (m, n) times. R R R R Nov, 00 ifference: - onsider two bags R and R that are unioncompatible. Suppose a tuple t appears in R m times, and in R n times. Then in R R, t appears max (0, m - n) times. R R R R Nov, 00

4 ag semantics vs Set semantics Union is idempotent for sets: (R R) R = R R Union is not idempotent for bags. Intersection and difference are idempotent for sets and bags. For sets and bags, R R = R (R R ). Nov, 00 ross Product (artesian Product): onsider two bags R and R. Suppose a tuple t appears in R m times, and a tuple t appears in R n times. Then in R X R, t t appears mn times. R R R X R R. R. Nov, 00

5 asic Relational Operations Select, Project, Join Select: denoted (R): selects the subset of tuples of R that satisfies selection condition. can be any boolean expression, its clauses can be combined with N, OR, NOT. R ( ) (R) Nov, 00 Select Select is commutative: ( (R)) = ( (R)) Select is idempotent: ( (R)) = (R) We can combine multiple select conditions into one condition. ( ( n (R) )) = N N n (R) Nov, 00

6 Project:,,, n (R) onsider relation (bag) R with set of attributes R.,,, n (R), where,,, n R returns the tuples in R, but only with columns,,, n. R, (R) Nov, 00 Project: ag Semantics vs Set Semantics For bags, the cardinality of R = cardinality of,,, n (R). For sets, cardinality of R,,, n (R). For sets and bags project is not commutative project is idempotent cardinality of Nov, 00

7 Natural Join: R onsider relations (bags) R with attributes R, and S with attributes S. Let = R S. R S can be defined as R,, S - ( R. = S. N R. =S. N R.n=S.n (R X S)) where = {,,, n} The above expression says: select those tuples in R X S that agree in values for each of the attributes, and project the resulting tuples such that we have only one value for each attribute. S Nov, 00 Natural Join example R R R R Nov, 00

8 Theta Join: R S Theta Join is similar to natural join, except that we can specify any condition. It is defined as R S = ( (R X S)) R R R R.<R. R R. R. Nov, 00 Outer Join: R o S Similar to natural join, however, if there is a tuple in R, that has no matching tuple in S, or a tuple in S that has no matching tuple in R, then that tuple also appears, with null values for attributes in S (or R). R o R R 9 R 0 null 9 0 null null Nov, 00

9 Left Outer Join: R o L S Similar to natural join, however, if there is a tuple in R, that has no matching tuple in S, then that tuple also appears, with null values for attributes in S (note: a tuple in S that has no matching tuple in R does not appear). R 9 R 0 R o L R 9 0 null null Nov, 00 Right Outer Join: R o R S Similar to natural join, however, if there is a tuple in S, that has no matching tuple in R, then that tuple also appears, with null values for attributes in R (note: a tuple in R that has no matching tuple in S does not appear). R 9 R 0 null R o R R 0 Nov, 00 9

10 Renaming: S(,,, n) (R) Rename relation R to S, attributes of R are renamed to,,, n S (R) renames relation R to S, keeping the attributes same. R 0 X S(X,, ) (R ) S 0 S (R ) S 0 Nov, 00 Example: Introducing new relations Find the semijoin of relations R, S. Semijoin denoted R S is defined as the tuples in R, such that for a tuple t in R, if there exists a tuple t in S that t and t agree in all attributes common to R and S, then t appears in the result. R = R S R = R (R) R S = R R Nov, 00 0

11 uplicate Elimination: δ (R) onvert a bag to a set. R δ (R) Nov, 00 Extended Projection: Here L can be L (R) n attribute (just like simple projection) n expression x y, where x and y are names of attributes, this renames attribute x to y. n expression E z, where E is any expression involving attributes, constants, and arithmetic and string operators. This has an attribute called z whose values are given by E. R 0, + X,, (R) X 9 0 Nov, 00

12 ggregation operators MIN, MX, OUNT, SUM, VG The aggregate operators aggregate the values in one column of a relation. R MIN () = MX () = OUNT () = SUM () = 0 VG () =. Nov, 00 Grouping Operator: γ GL, L (R) Usually we want to perform aggregation not on all the values of a column, but on a subgroup of the values of a column. γ GL, L (R) groups all attributes in GL, and performs the aggregation specified in L. title SW Matrix &N SW Speed StarsIn γ starname, MIN (year) year, OUNT(title) num (StarsIn) year starname HF KR HF HF KR starname HF KR year 9 num Nov, 00

13 Sorting Operator: τ L (R) It sorts the tuples in R. If L is list,,, n, it first sorts by, then by, and so on. Sort is used as a last operator in an expression. R τ, (R) Nov, 00