4/10/2018. Relational Algebra (RA) 1. Selection (σ) 2. Projection (Π) Note that RA Operators are Compositional! 3.

Transcription

1 Lecture 33: The Relational Model 2 Professor Xiannong Meng Spring 2018 Lecture and activity contents are based on what Prof Chris Ré of Stanford used in his CS 145 in the fall 2016 term with permission Relational Algebra (RA) Five basic operators: 1. Selection: s 2. Projection: P 3. Cartesian Product: 4. Union: 5. Difference: - We ll look at these first! Derived or auxiliary operators: Intersection, complement Joins (natural,equi-join, theta join, semi-join) Renaming: r Division And also at one example of a derived operator (natural join) and a special operator (renaming) 1. Selection (σ) 2. Projection (Π) Returns all tuples which satisfy a condition Notation: s c (R) Examples s Salary > (Employee) s name = Smith (Employee) The condition c can be =, <,, >,, <> Students WHERE gpa > 3.5; σ gpa >3.5 (Students) Eliminates columns, then removes duplicates Notation: P A1,,An (R) project social-security number and names: P SSN, Name (Employee) Output schema: Answer(SSN, Name) sname, gpa Students; Π sname,gpa (Students) Note that RA Operators are Compositional! 3. Cross-Product ( ) sname, gpa Students WHERE gpa > 3.5; How do we represent this query in RA? Π sname,gpa (σ gpa>3.5 (Students)) σ gpa>3.5 (Π sname,gpa ( Students)) Are these logically equivalent? Each tuple in R1 with each tuple in R2 Notation: R1 R2 Employee Dependents Rare in practice; mainly used to express joins Students, People; Students People 1

2 Another example: People ssn pname address John 216 Rosse Bob 217 Rosse Students People Students sid sname gpa ssn pname address sid sname gpa John 216 Rosse Bob 217 Rosse John 216 Rosse Bob 216 Rosse Renaming (ρ) Changes the schema, not the instance A special operator- neither basic nor derived Notation: r B1,,Bn (R) Note: this is shorthand for the proper form (since names, not order matters!): r A1 B1,,An Bn (R) SELECT sid AS studid, sname AS name, gpa AS gradeptavg Students; ρ studid,name,gradeptavg (Students) We care about this operator because we are working in a named perspective Another example: Students sid sname gpa ρ studid,name,gradeptavg (Students) Students studid name gradeptavg Natural Join ( ) Notation: R 1 R 2 Joins R 1 and R 2 on equality of all shared attributes If R 1 has attribute set A, and R 2 has attribute set B, and they share attributes A B = C, can also be written: R 1 C R 2 Our first example of a derived RA operator: Meaning: R 1 R 2 = P A U B (s C=D (ρ C D(R 1 ) R 2 )) Where: The rename ρ C D renames the shared attributes in one of the relations The selection s C=D checks equality of the shared attributes The projection P A U B eliminates the duplicate common attributes Students(sid,name,gpa) People(ssn,name,address) ssid, S.name, gpa, ssn, address Students S, WHERE S.name = P.name; Students People Another example: Students S Natural Join sid S.name gpa ssn P.name address John 216 Rosse Bob 217 Rosse Given schemas R(A, B, C, D), S(A, C, E), what is the schema of R S? Students People sid S.name gpa ssn address Rosse Rosse Given R(A, B, C), S(D, E), what is R S? Given R(A, B), S(A, B), what is R S? 2

3 Example: Converting SFW Query -> RA People(ssn,sname,address) Logical Equivalence of RA Plans Given relations R(A,B) and S(B,C): gpa, address Students S, WHERE gpa > 3.5 AND sname = pname; How do we represent this query in RA? Π gpa,address (σ gpa>3.5 (S P)) Here, projection & selection commute: σ A=5 (Π A (R)) = Π A (σ A=5 (R)) What about here? σ A=5 (Π B (R))? = Π B (σ A=5 (R)) We ll look at this in more depth later in the lecture RDBMS Architecture How does a SQL engine work? RDBMS Architecture How does a SQL engine work? SQL Query Relational Algebra (RA) Plan Optimized RA Plan Execution SQL Query Relational Algebra (RA) Plan Optimized RA Plan Execution We saw how we can transform declarative SQL queries into precise, compositional RA plans We ll look at how to then optimize these plans later in this lecture RDBMS Architecture How is the RA plan executed? SQL Query Relational Algebra (RA) Plan Optimized RA Plan We already know how to execute all the basic operators! Execution RA Plan Execution Natural Join / Join: We saw how to use memory & IO cost considerations to pick the correct algorithm to execute a join with (BNLJ, SMJ, HJ )! Selection: We saw how to use indexes to aid selection Can always fall back on scan / binary search as well Projection: The main operation here is finding distinct values of the project tuples; we briefly discussed how to do this with e.g. hashing or sorting We already know how to execute all the basic operators! 3

4 What you will learn about in this section 1. Set Operations in RA 2. Adv. Relational Algebra 2. Fancier RA 3. Extensions & Limitations Relational Algebra (RA) 1. Union ( ) and 2. Difference ( ) Five basic operators: 1. Selection: s 2. Projection: P 3. Cartesian Product: 4. Union: 5. Difference: - We ll look at these Derived or auxiliary operators: Intersection, complement Joins (natural,equi-join, theta join, semi-join) Renaming: r Division And also at some of these derived operators R1 R2 ActiveEmployees RetiredEmployees R1 R2 AllEmployees - RetiredEmployees R 1 R 2 R 1 R 2 What about Intersection ( )? It is a derived operator R1 R2 = R1 (R1 R2) Also expressed as a join! Example UnionizedEmployees RetiredEmployees R 1 R 2 Fancier RA 4

5 Theta Join ( q ) Equi-join ( A=B ) A join that involves a predicate R1 q R2 = s q (R1 R2) Here q can be any condition Students,People WHERE q; A theta join where q is an equality R1 A=B R2 = s A=B (R1 R2) Employee SSN=SSN Dependents Students S, WHERE sname = pname; Note that natural join is a theta join + a projection. Students θ People Most common join in practice! S sname=pname P Semijoin ( ) R S = P A1,,An (R S) Where A 1,, A n are the attributes in R Employee Dependents sid,sname,gpa Students,People WHERE sname = pname; Semijoins in Distributed Databases Semijoins are often used to compute natural joins in distributed databases Dependents Employee SSN Dname Age SSN Name network Students People Send less data to reduce network bandwidth! Employee ssn=ssn (s age>71 (Dependents)) T = P SSN s age>71 (Dependents) R = Employee T Answer = R Dependents RA Expressions Can Get Complex! P name buyer-ssn=ssn seller-ssn=ssn pid=pid Multisets P ssn P pid s name=fred s name=gizmo Person Purchase Person Product 5

6 Recall that SQL uses Multisets (1, a) (1, a) (1, b) (2, c) (2, c) (2, c) (1, d) (1, d) Equivalent Representations of a Multiset λ X = Count of tuple in X (Items not listed have implicit count 0) λ(x) (2, c) 3 Note: In a set all counts are {0,1}. 31 Generalizing Set Operations to Multiset Operations λ(x) (1, b) 0 (2, c) 3 (1, d) 0 Multiset Y λ(y) (1, a) 5 = (2, c) 2 λ Z = min(λ X, λ Y ) Multiset Z λ(z) (1, b) 0 (2, c) 2 (1, d) 0 For sets, this is intersection 32 Generalizing Set Operations to Multiset Operations Operations on Multisets Multiset Y Multiset Z All RA operations need to be defined carefully on bags λ(x) (1, b) 0 (2, c) 3 (1, d) 0 λ(y) (1, a) 5 = (2, c) 2 λ(z) (1, a) 7 (2, c) 5 s C (R): preserve the number of occurrences P A (R): no duplicate elimination λ Z = λ X + λ Y For sets, this is union Cross-product, join: no duplicate elimination This is important- relational engines work on multisets, not sets! 33 RA has Limitations! Cannot compute transitive closure Name1 Name2 Relationship Fred Mary Father Mary Joe Cousin Mary Bill Spouse Nancy Lou Sister Find all direct and indirect relatives of Fred Cannot express in RA!!! Need to write C program, use a graph engine, or modern SQL 6