Rela+onal Algebra. Rela+onal Query Languages. CISC437/637, Lecture #6 Ben Cartere?e

Transcription

1 Rela+onal Algebra CISC437/637, Lecture #6 Ben Cartere?e Copyright Ben Cartere?e 1 Rela+onal Query Languages A query language allows manipula+on and retrieval of data from a database The rela+onal model supports simple but powerful query languages Formal founda+on based on logic Allows for a great deal of op+miza+on A query language is not a programming language Not Turing- complete, not intended for complex calcula+ons Supports easy, efficient access to data Copyright Ben Cartere?e 2 1

2 Formal Query Languages Two mathema+cal formalisms: Rela%onal algebra, based on operators over rela+ons Rela%onal calculus, based on declara+ve statements about data The algebra more directly supports computa+on A rela+onal algebra query implies a sequence of steps that can be taken to execute it SQL is an implementa+on of rela+onal algebra Copyright Ben Cartere?e 3 Preliminaries A query is applied to rela%on instances, and the result of a query is a rela+on instance Schemas of input rela+ons are fixed Schemas of output rela+ons are also fixed, though may be different from input rela+on schemas Posi+onal versus named- field nota+on: Posi+onal nota+on is easier for formal defini+ons; named- field easier to read SQL supports both Copyright Ben Cartere?e 4 2

3 Rela+onal Algebra Basic operators: Selec%on: σ(r) select a subset of records from rela+on R Projec%on: π(r) drop unwanted fields from rela+on R Cross- product: R1 R2 concatenate each record in R1 with each record in R2 Set- difference: R1 R2 return records in R1 that are not in R2 Union: R1 R2 return records in either R1 or R2 Algebra is all about composing operators Every operator takes rela+ons as input and returns rela+ons Algebra is closed under these operators Copyright Ben Cartere?e 5 Rela+onal Algebra More operators: Intersec%on: R1 R2 return records in both R1 and R2 Join: R1 R2 combine informa+on from rela+ons R1 and R2 Division: R1/R2 return records in R1 that match every record in R2 in a subset of fields Renaming: ρ(r(f), E) the rela+on returned by expression E is named R and its fields are renamed according to mapping F Aggregate func%ons: G f (R) calculate aggrega+ng func+on f on rela+on R Copyright Ben Cartere?e 6 3

4 Selec+on & Projec+on Selec+on returns all records matching a logical selec%on condi%on p σ p (R) is a rela+on consis+ng only of records in R for which p is true Schema of σ p (R) is the same as schema of R Projec+on returns a rela+on with only the fields indicated π x (R) is a rela+on with only the fields of R specified in the list x Schema of π x (R) is a subset of the schema of R Records in π x (R) same as records in R, but duplicates are dropped Copyright Ben Cartere?e 7 Projec+on & Selec+on Generalized projec%on allows selec+ng arithme+c func+ons of fields π F (R), where F is an arithme+c (uses +,,, ) func+on of one or more fields and constant values Selec+on and projec+on operators can be composed π x (σ p (R)) returns a rela+on with records in R for which p is true, and with only fields specified in x Copyright Ben Cartere?e 8 4

5 Set Operators Union, intersec+on, and set- difference These take two rela+ons R1 and R2 and return a new rela+on R1 and R2 must be union- compa%ble Same number of fields Corresponding fields must have same domain The schema of the new rela+on is defined to be the schema of R1 Copyright Ben Cartere?e 9 Cross- Product Cross- product R1 R2 pairs each record in R1 with each record in R2 to create a new rela+on of concatenated records Schema is defined to be the concatena+on of R1 and R2 s schemas If both have a field with the same name, refer to that field by number Or rename it using the ρ operator Copyright Ben Cartere?e 10 5

6 Joins Combine two rela+ons R1 and R2 in such a way that all records match some logical condi+on p R1 p R2 σ p (R1 R2) Schema is the same as the cross- product schema (except in special cases) Equijoin: p is an equality condi+on Schema drops duplicate fields that are part of equality Natural join: p is a conjunc+on of equality condi+ons on all fields with the same names Copyright Ben Cartere?e 11 Outer Joins Outer joins preserve records even if they do not match the condi+on p Missing values filled in with nulls LeC outer join R1 p R2 preserves every record in R1 and uses nulls for R2 fields when there is no record in R2 that sa+sfies p Right outer join R1 p R2 preserves every record in R2 and uses nulls for R1 fields when there is no record in R1 that sa+sfies p Full outer join R1 p R2 combines ler and right outer joins, preserving all records in both Copyright Ben Cartere?e 12 6

7 Set Division Suppose A has two fields x and y, B has only field y A/B = { x y B, x, y A } i.e. A/B is the set of tuples such that for every tuple y in B, there is a matching tuple x, y in A In general, y is the list of fields in B, and A has all those fields plus addi+onal ones Schema of B must be a subset of schema of A Schema of A/B consists of fields in A but not in B Copyright Ben Cartere?e 13 Aggrega+on Func+ons Calculate a func+on of a set of values, returning a single value G f (R), where f is a func+on of a field in R, returns a rela+on consis+ng of the value returned by f E.g. G average(gpa) (Students) calculates average GPA over all students Aggregate values by group: g G f (R) calculates f for dis+nct groups defined by values of field g E.g. major G average(gpa) (Students) calculates average GPA for students in each unique major Copyright Ben Cartere?e 14 7

8 Handling Nulls Null values require special handling They should be thought of as unknowns the record has a value, we just don t know what it is Null values in arithme+c: 2 + null = null; 4*null = null Null values in comparisons: Truth value of 2 < null is unknown Null values in Boolean expressions: Follows seman+cs of Boolean and, or, not Copyright Ben Cartere?e 15 Handling Nulls σ p (R): take every row for which p is true, do not include rows for which p is unknown R1 p R2: same as selec+on π x (R): treat nulls as equivalent for elimina+ng duplicates In generalized projec+on, arithme+c expressions involving nulls result in null values in result,, : same as projec+on G f (R): ignore nulls when calcula+ng f If grouping field contains nulls, treat null as a separate group Copyright Ben Cartere?e 16 8

9 sid name gpa age 20 Alice Joe Bob Liz Nelly Alice Jim Jackie Roland Andy Students Courses sid course semester 20 CISC437 S10 20 CISC320 S10 20 ENGL312 F09 20 BISC207 S09 23 CISC437 S10 23 CISC320 F08 23 ENGL312 S08 29 BISC207 F09 29 CISC320 S10 49 BISC207 S10 course %tle level CISC320 algorithms 300 CISC437 databases 400 ENGL312 business wri+ng 300 BISC207 intro biology 200 Enrolled Copyright Ben Cartere?e 17 Rela+onal Calculus Two different types: Tuple rela+onal calculus (TRC) Domain rela+onal calculus (DRC) A calculus consists of variables, constants, comparison operators, logical connec%ves, and logical quan%fiers In TRC, variables range over tuples (records) In DRC, variables range over field domains Copyright Ben Cartere?e 18 9

10 Rela+onal Calculus An expression in the calculus is called a formula A query is expressed as a set of tuples that sa+sfy a formula { x 1, x 2,, x n p( x 1, x 2,, x n ) } in DRC { R p(r) } in TRC The answer is the set of tuples that make the formula true Copyright Ben Cartere?e 19 Tuple Rela+onal Calculus Formulas A formula is built recursively from smaller formulas Atomic formula: R Rela+on; R1.a op R2.b; R1.a op constant Operators are <,, =,, >, Formula: An atomic formula, or p, p q, p q, where p and q are formulas X p(x), where p is a formula with free var X X p(x), where p is a formula with free var X and bind X; an unbound variable is free Copyright Ben Cartere?e 20 10

11 Examples of TRC Queries Find all students with GPA greater than 3.5 { S S Students S.gpa > 3.5 } Find names and ages of students with GPA greater than 3.5 { P S Students, S.gpa > 3.5 P.name = S.name P.age = S.age } Equivalent to algebraic π name,age (σ gpa>3.5 (Students)) Find the names of students taking ENGL312 { P S Students, E Enrollments, E.sid=S.sid E.course= ENGL312 P.name=S.name } Copyright Ben Cartere?e 22 Domain Rela+onal Calculus Formulas A formula is built recursively from smaller formulas Atomic formula: x 1,, x n R; R1.a op R2.b; R1.a op constant Operators are <,, =,, >, Formula: An atomic formula, or p, p q, p q, where p and q are formulas X p(x), where p is a formula with free var X X p(x), where p is a formula with free var X and bind X; an unbound variable is free Copyright Ben Cartere?e 23 11

12 Examples of DRC Queries Find all students with GPA > 3.5 { I, N, G, A I, N, G, A Students G > 3.5 } Find the names of students taking ENGL312 { N I,G,A I, N, G, A Students Ie,Be,D Ie, Be, D Enrolled Ie=I Be= ENGL312 } Copyright Ben Cartere?e 24 12