v Conceptual Design: ER model v Logical Design: ER to relational model v Querying and manipulating data

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1 Outline Conceptual Design: ER model Relational Algebra Calculus Yanlei Diao UMass Amherst Logical Design: ER to relational model Querying and manipulating data Practical language: SQL Declarative: say what you want, not how to get it Formal languages: Relational Algebra and Calculus Mathematical foundation for query processing Slides Courtesy of R. Ramakrishnan and J. Gehrke 1 2 What is an Algebra? What is Relational Algebra A mathematical system consisting of: Operands: variables or values from which new values can be constructed. Operators: symbols denoting procedures that construct new values from given values. Relational Algebra: Operands are relations. Each operator takes 1 or 2 relations and produces a relation. Closure property: relational algebra is closed under the relational model. Relational operators can be arbitrarily composed! 3 4 Relational Algebra Basic operations: Selection ( σ Selects a subset of rows from a relation. Projection ( π Deletes unwanted columns from a relation. Cross-product ( Allows us to combine two relations. Set-difference ( - Tuples in reln. 1, but not in reln. 2. Union ( Tuples in reln. 1 or in reln. 2. Additional operations: Join (, Intersection (, Division ( /, Renaming ( ρ Can be derived from basic operators. Not essential, but useful for writing simpler queries. Example Instances Sailors R1 Reserves sid bid day /10/ /12/96 5 6

2 Projection Retain only attributes in the projection list; delete others. Schema of result contains exactly the fields in projection list. sname rating yuppy 9 lubber 8 guppy 5 rusty 10 π ( sname, rating Projection (contd. age π age ( Projection operator has to eliminate duplicates! Real systems typically don t do duplicate elimination unless the user explicitly asks for it. (Why not? 7 8 Selection Select rows that satisfy the selection condition; discard others. Schema of result identical to schema of input. σ S rating >8 ( 2 9 Selection (contd. σ (rating > 5 rating < 5 age < 50 ( 10 A Simple Example of Composition Union, Intersection, Set-Difference Composition: result relation of an operator can be the input to another operator. σ S rating >8 ( 2 π sname rating yuppy 9 rusty 10 σ ( ( S sname, rating rating> Set operations: Union ( Intersection ( Set difference ( - Take two input relations, which must be union-compatible: Same number of fields. Corresponding fields have the the same type. What is the schema of result? 12

3 Example Set Operations Duplicate elimination: remove tuples that have same values in all attributes. Example Set Operations Duplicates? Cross (Cartesian Product Cross-Product (contd. R1: Each row of is paired with each row of R1. R1 sid R1 bid day /10/ /12/96 (sid sname rating age (sid bid day /10/ /12/ /10/ /12/ /10/ /12/96 15 (sid sname rating age (sid bid day /10/ /12/ /10/ /12/ /10/ /12/96 Result schema inherits all fields of and R1. Conflict: Both and R1 have a field called sid. Renaming operator: ρ ( C( 1 sid1, 5 sid2, R1 What if we want to find pairs s.t..sid =.sid? 16 Join Join Condition (theta Join: R θ S = σ θ (R S (sid sname rating age (sid bid day /12/ /12/96 S 1 R sid R1 sid 1. <. Equi-Join: A special case of condition join where the condition θ contains only equalities. bid day /10/ /12/96 R1 sid Result schema same as that of cross-product. But often fewer tuples, more efficient for computation. Result schema contains only one copy of fields for which equality is specified. Natural Join ( R w S : equi-join on all common fields

4 Self-Join Example Schema ρ(,s ρ(,s.rating>.rating.age<.age age: integer Boats(bid: integer, color: string Reserves(sid: integer, bid: integer, day: date Find names of sailors who ve reserved boat 103 Find names of sailors who ve reserved boat 103 age: integer Boats(bid: integer, color: string Reserves(sid: integer, bid: integer, day: date How many relations do we need to access? If more than 1, use cross-products or joins. Filtering condition? Use selection. Attributes to return? Use projection. 21 Solution 1: π sname (( σ Re serves Sailors bid =103 Solution 2: ρ ( Temp1, σ Re serves bid =103 ρ ( Temp2, Temp1 Sailors π sname ( Temp2 Solution 3: π sname( σ (Re serves Sailors bid =103 Algebraic equivalence! 22 Find names of sailors who ve reserved a red boat age: integer Boats(bid: integer, color: string Reserves(sid: integer, bid: integer, day: date Find names of sailors who ve reserved a red or a green boat age: integer Boats(bid: integer, color: string Reserves(sid: integer, bid: integer, day: date Boat color is only available in Boats; so need an extra join: π sname (( σ Boats serves Sailors color ' red ' Re = 23 24

5 Find names of sailors who ve reserved a red or a green boat Can identify all red or green boats, then find sailors who ve reserved one of these boats: ρ ( Tempboats, ( σ color = ' red ' color = ' green' Boats π sname ( Tempboats Re serves Sailors Can also define Tempboats using union. How? Find names of sailors who ve reserved a red boat and a green boat A single selection won t work. Why? Instead, intersect sailors who ve reserved red boats and sailors who ve reserved green boats. sid is a key for Sailors ρ ( Tempred, π (( σ Boats Re serves sid color = ' red ' ρ ( Tempgreen, π (( σ Boats Re serves sid color = ' green' π sname (( Tempred Tempgreen Sailors 25 Can we project onto name and then do the intersection? 26 Find the names of sailors who ve reserved all boats Relational Calculus Query has the form: ' (! x1, x2,..., xn p x1, x2,..., xn " $ % +, - domain variables, or constants formula Answer includes all tuples x1, x2,..., xn that make the formula! p x1, x2,..., xn " $ % be true Formulas Formula is recursively defined: Atomic formulas: retrieving tuples from relations, or making comparisons of values Logical connectives:,, Quantifiers:, (optional o x F(x is equivalent to x F(x Find names of sailors rated > 7 who have reserved boat 103 Relational Algebra: π sname (( σ Reserves ( σ Sailors bid = 103 rating> 7 Relational Calculus: {X sname X sid, X rating, X age Sailors(X sid, X sname, X rating, X age X rating > 7 X bid, X day Reserves(X sid, X bid, X day X bid =103 } Where is the join? Use to find a tuple in Reserves that `joins with the Sailors tuple under consideration

6 Find names of sailors who ve reserved all boats Sailors(sid: integer, sname: string, rating: integer, age: integer Boats(bid: integer, color: string Reserves(sid: integer, bid: integer, day: date X bid, X color Boats ( X day X sid, X bid, X day Reserves } To find sailors who ve reserved all red boats: X bid, X color Boats (X color ʹ redʹ X day X sid, X bid, X day Reserves } Unsafe Queries, Expressive Power Some syntactically correct calculus queries can have an infinite number of answers! Such queries are called unsafe. ( " %, e.g., S $ S Sailors' - + $ Equivalence result: every query that can be expressed in relational algebra can be expressed as a safe query in relational calculus; the converse is also true. '. Query language (e.g., SQL can express every query that is expressible in relational algebra/calculus Find names of sailors who ve reserved all boats X bid, X color Boats ( X day X sid, X bid, X day Reserves } X bid, X color Boats ( X day X sid, X bid, X day Reserves } Find the names of sailors who ve reserved all boats Step 1: for each sailor, check if there exists a boat that he has not reserved (called formula F. ρ(s _ neg, π sid ( (π sid Re serves (π bid Boats (π sid,bid Re serves Step 2: find sailors for which F is not true and retrieve their names π sname ( (π sid Re serves S _ neg Sailors - : the only way to express negation in relational algebra! 33 34