Relational Algebra. Today s Lecture. 1. The Relational Model & Relational Algebra. 2. Relational Algebra Pt. II

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1 Reltionl Algebr BBM471 Dtbse Mngement Systems Dr. Fut Akl Tody s Lecture 1. The Reltionl Model & Reltionl Algebr 2. Reltionl Algebr Pt. II 2

2 1. The Reltionl Model & Reltionl Algebr Wht you will lern bout in this section 1. The Reltionl Model 2. Reltionl Algebr: Bsic Opertors 3. Execution 4

3 Motivtion The Reltionl model is precise, implementble, nd we cn operte on it (query/updte, etc.) Dtbse mps internlly into this procedurl lnguge. 5 A Little History Reltionl model due to Edgr Ted Codd, mthemticin t IBM in 1970 A Reltionl Model of Dt for Lrge Shred Dt Bnks". Communictions of the ACM 13 (6): Won Turing wrd 1981 IBM didn t wnt to use reltionl model Apprently used in the moon lnding Google for IMS nd the Apollo progrm 6

4 The Reltionl Model: Schemt Reltionl Schem: Students(sid: string, nme: string, gp: flot) Reltion nme String, flot, int, etc. re the domins of the ttributes Attributes 7 The Reltionl Model: Dt Student An ttribute (or column) is typed dt entry present in ech tuple in the reltion sid nme gp 001 Bob Joe Mry Alice 3.5 The number of ttributes is the rity of the reltion 8

5 The Reltionl Model: Dt Student sid nme gp 001 Bob Joe Mry Alice 3.5 The number of tuples is the crdinlity of the reltion A tuple or row (or record) is single entry in the tble hving the ttributes specified by the schem 9 The Reltionl Model: Dt Student sid nme gp 001 Bob Joe Mry 3.8 Recll: In prctice DBMSs relx the set requirement, nd use multisets. 004 Alice 3.5 A reltionl instnce is set of tuples ll conforming to the sme schem 10

6 To Reiterte A reltionl schem describes the dt tht is contined in reltionl instnce Let R(f 1 :Dom 1,,f m :Dom m ) be reltionl schem then, n instnce of R is subset of Dom 1 x Dom 2 x x Dom n In this wy, reltionl schem R is totl function from ttribute nmes to types 11 One More Time A reltionl schem describes the dt tht is contined in reltionl instnce A reltion R of rity t is function: R : Dom 1 x x Dom t à {0,1} i.e. returns whether or not tuple of mtching types is member of it Then, the schem is simply the signture of the function Note here tht order mtters, ttribute nme doesn t We ll (mostly) work with the other model (lst slide) in which ttribute nme mtters, order doesn t! 12

7 A reltionl dtbse A reltionl dtbse schem is set of reltionl schemt, one for ech reltion A reltionl dtbse instnce is set of reltionl instnces, one for ech reltion Two conventions: 1. We cll reltionl dtbse instnces s simply dtbses 2. We ssume ll instnces re vlid, i.e., stisfy the domin constrints 13 Remember the CMS Reltion DB Schem Students(sid: string, nme: string, gp: flot) Courses(cid: string, cnme: string, credits: int) Enrolled(sid: string, cid: string, grde: string) Note tht the schems impose effective domin / type constrints, i.e. Gp cn t be Apple Sid Nme Gp 101 Bob Mry 3.8 Students Reltion Instnces sid cid Grde A Enrolled cid cnme credits Courses 14

8 2 nd Prt of the Model: Querying SELECT S.nme FROM Students S WHERE S.gp > 3.5; Find nmes of ll students with GPA > 3.5 We don t tell the system how or where to get the dt- just wht we wnt, i.e., Querying is declrtive To mke this hppen, we need to trnslte the declrtive query into series of opertors we ll see this next! Actully, I showed how to do this trnsltion for much richer lnguge! 15 Virtues of the model Physicl independence (logicl too), Declrtive Simple, elegnt, clen: Everything is reltion Why did it tke multiple yers? Doubted it could be done efficiently. 16

9 RDBMS Architecture How does n SQL engine work? SQL Query Reltionl Algebr (RA) Pln Optimized RA Pln Execution Declrtive query (from user) Trnslte to reltionl lgebr expresson Find logiclly equivlent- but more efficient- RA expression Execute ech opertor of the optimized pln! 17 RDBMS Architecture How does SQL engine work? SQL Query Reltionl Algebr (RA) Pln Optimized RA Pln Execution Reltionl Algebr llows us to trnslte declrtive (SQL) queries into precise nd optimizble expressions! 18

10 Reltionl Algebr (RA) Five bsic opertors: Selection: s Projection: P Crtesin Product: Union: È Difference: - Derived or uxiliry opertors: Intersection, complement Joins (nturl,equi-join, thet join, semi-join) Renming: r Division 19 Keep in mind: RA opertes on sets! RDBMSs use multisets, however in reltionl lgebr formlism we will consider sets! Also: we will consider the nmed perspective, where every ttribute must hve unique nme àttribute order does not mtter Now on to the bsic RA opertors 20

11 Selection (!) Returns ll tuples which stisfy condition Nottion: s c (R) Exmples s Slry > (Employee) s nme = Smith (Employee) The condition c cn be =, <,, >, ³, <> Students(sid,snme,gp) SQL: SELECT * FROM Students WHERE gp > 3.5; RA:! "#$ &'.) (+,-./0,1) 21 Another exmple: SSN Nme Slry John Smith Fred s Slry > (Employee) SSN Nme Slry Smith Fred

12 Projection (Π) Elimintes columns, then removes duplictes Nottion: P A1,,An (R) Exmple: project socil-security number nd nmes: P SSN, Nme (Employee) Output schem: Answer (SSN, Nme) Students(sid,snme,gp) SQL: SELECT DISTINCT snme, gp FROM Students; RA: Π 45$67,"#$ (+,-./0,1) 23 Another exmple: SSN Nme Slry John John John P Nme,Slry (Employee) Nme Slry John John

13 Note tht RA Opertors re Compositionl! Students(sid,snme,gp) SELECT DISTINCT snme, gp FROM Students WHERE gp > 3.5; How do we represent this query in RA? Π 45$67,"#$ (! "#$&'.) (+,-./0,1))! "#$&'.) (Π 45$67,"#$ ( +,-./0,1)) Are these logiclly equivlent? 25 Cross-Product ( ) Ech tuple in R1 with ech tuple in R2 Nottion: R1 R2 Exmple: Employee Deprtments Minly used to express joins Students(sid,snme,gp) People(ssn,pnme,ddress) SQL: SELECT * FROM Students, People; RA: +,-./0,1 :/;<=/ 26

14 Another exmple: People ssn pnme ddress John 216 Rosse Bob 217 Rosse Students sid snm e gp 001 John Bob 1.3 +,-./0,1 :/;<=/ ssn pnme ddress sid snme gp John 216 Rosse 001 John Bob 217 Rosse 001 John John 216 Rosse 002 Bob Bob 216 Rosse 002 Bob Renming Chnges the schem, not the instnce A specil opertor- neither bsic nor derived Nottion: r B1,,Bn (R) Students(sid,snme,gp) SQL: SELECT sid AS studid, snme AS nme, gp AS grdeptavg FROM Students; Note: this is shorthnd for the proper form (since nmes, not order mtters!): r A1àB1,,AnàBn (R) RA: > 4BCDED,5$67,"F$D7GBHI" (+,-./0,1) We cre bout this opertor becuse we re working in nmed perspective 28

15 Another exmple: Students sid snme gp > 4BCDED,5$67,"F$D7GBHI" (+,-./0,1) 001 John Bob 1.3 Students studid nme grdeptavg 001 John Bob Nturl Join ( ) Nottion: R 1 R 2 Joins R 1 nd R 2 on equlity of ll shred ttributes If R 1 hs ttribute set A, nd R 2 hs ttribute set B, nd they shre ttributes A B = C, cn lso be written: R 1 L R 2 Our first exmple of derived RA opertor: Mening: R 1 R 2 = P A U B (s C=D (> M O (R 1 ) R 2 )) Where: The renme > M O renmes the shred ttributes in one of the reltions The selection s C=D checks equlity of the shred ttributes The projection P A U B elimintes the duplicte common ttributes Students(sid,nme,gp) People(ssn,nme,ddress) SQL: SELECT DISTINCT ssid, S.nme, gp, ssn, ddress FROM Students S, People P WHERE S.nme = P.nme; RA: +,-./0,1 :/;<=/ 30

16 Another exmple: Students S People P sid S.nme gp 001 John Bob 1.3 ssn P.nme ddress John 216 Rosse Bob 217 Rosse +,-./0,1 :/;<=/ sid S.nme gp ssn ddress 001 John Rosse 002 Bob Rosse 31 Nturl Join Given schems R(A, B, C, D), S(A, C, E), wht is the schem of R S? Given R(A, B, C), S(D, E), wht is R S? Given R(A, B), S(A, B), wht is R S? 32

17 Exmple: Converting SFW Query -> RA Students(sid,snme,gp) People(ssn,snme,ddress) SELECT DISTINCT gp, ddress FROM Students S, People P WHERE gp > 3.5 AND S.snme = P.snme; Π "#$,$DDF744 (! "#$&'.) (+ :)) How do we represent this query in RA? 33 Division Nottion: r s It hs nothing to do with rithmetic division. Let r nd s be reltions on schems R nd S respectively where R = (A 1,, A m, B 1,, B n ) S = (B 1,, B n ) The result of r s is reltion on schem R S = (A 1,, A m ) r s = { t t Î Õ R-S (r) Ù " u Î s ( tu Î r ) } Where tu mens the conctention of tuples t nd u to produce single tuple Dtbse System Concepts, Silberschtz, Korth nd Sudrshn 34

18 Division Opertion - Exmple 35 Reltions r, s r s A B b 1 2 A B b g d d d Î Î b r s Dtbse System Concepts, Silberschtz, Korth nd Sudrshn Division Opertion - Exmple 36 A B b b g g g C D g g g g g g b b b b b E Reltions r, s r s D b E 1 1 A B g C g g r s Dtbse System Concepts, Silberschtz, Korth nd Sudrshn

19 Division Opertion - Exmple Deprtment (did, dnme) Project (pid, pnme, did) Supplier (suppid, snme, saddress) Supply (scode, snme, mountavilble) Production (scode, did, mount) Consumption (scode, pid, mount) Purchsed (scode, suppid, mount) Find projects tht consume ll different kinds of purchsed supplies. Õ pid,scode (Consumption) Purchsed Find deprtments tht produce ll kinds of supplies. Õ did,scode (Production) Supply Veritbnı Sistemleri, Ünl Yrımğn 37 Logicl Equivlence of RA Plns Given reltions R(A,B): Here, projection & selection commute:! HP) (Π H (@)) = Π H (! HP) (@)) Wht bout here?! HP) (Π R (@)) = Π R (! HP) (@))??? 38

20 RDBMS Architecture How does SQL engine work? SQL Query Reltionl Algebr (RA) Pln Optimized RA Pln Execution We sw how we cn trnsform declrtive SQL queries into precise, compositionl RA plns 39 RDBMS Architecture How does SQL engine work? SQL Query Reltionl Algebr (RA) Pln Optimized RA Pln Execution We ll look t how to then optimize these plns lter! 40

21 RDBMS Architecture How is the RA pln executed? SQL Query Reltionl Algebr (RA) Pln Optimized RA Pln Execution We will see lter how to execute ll the bsic opertors! Advnced Reltionl Algebr

22 Wht you will lern bout in this section 1. Set Opertions in RA 2. Fncier RA 3. Extensions & Limittions 43 Reltionl Algebr (RA) Five bsic opertors: Selection: s Projection: P Crtesin Product: Union: È Difference: - Derived or uxiliry opertors: Intersection, complement Joins (nturl,equi-join, thet join, semi-join) Renming: r Division 44

23 Union (È) nd Difference ( ) R1 È R2 Exmple: ActiveEmployees È RetiredEmployees R 1 R 2 R1 R2 Exmple: AllEmployees RetiredEmployees R 1 R 2 45 Wht bout Intersection (Ç)? It is derived opertor R1 Ç R2 = R1 (R1 R2) Also expressed s join! Exmple UnionizedEmployees Ç RetiredEmployees R 1 R 2 46

24 Thet Join ( q ) A join tht involves predicte R1 q R2 = s q (R1 R2) Here q cn be ny condition Students(sid,snme,gp) People(ssn,snme,ddress) SQL: SELECT * FROM Students,People WHERE q; Note tht nturl join is thet join + projection. RA: +,-./0,1 T :/;<=/ 47 Equi-join ( A=B ) A thet join where q is n equlity R1 A=B R2 = s A=B (R1 R2) Exmple: Employee SSN=SSN Dependents Most common join in prctice! Students(sid,snme,gp) People(ssn,pnme,ddress) SQL: SELECT * FROM Students S, People P WHERE snme = pnme; RA: + 45$67P#5$67 : 48

25 Semijoin ( ) R S = P A1,,An (R S) Where A 1,, A n re the ttributes in R Exmple: Employee Dependents Students(sid,snme,gp) People(ssn,pnme,ddress) SQL: SELECT DISTINCT sid,snme,gp FROM Students,People WHERE snme = pnme; RA: +,-./0,1 :/;<=/ 49 Semijoins in Distributed Dtbses Semijoins re often used to compute nturl joins in distributed dtbses Employee SSN Nme network Dependents SSN Dnme Age Send less dt to reduce network bndwidth! Employee ssn=ssn (s ge>71 (Dependents)) R = Employee T T = P SSN s ge>71 (Dependents) Answer = R Dependents 50

26 RA Expressions Cn Get Complex! P nme buyer-ssn=ssn pid=pid seller-ssn=ssn P ssn P pid s nme=fred s nme=gizmo Person Purchse Person Product 51 Opertions on Multisets All RA opertions need to be defined crefully on bgs s C (R): preserve the number of occurrences P A (R): no duplicte elimintion Cross-product, join: no duplicte elimintion This is importnt- reltionl engines work on multisets, not sets! 52

27 RA hs Limittions! Cnnot compute trnsitive closure Nme1 Nme2 Reltionship Fred Mry Fther Mry Joe Cousin Mry Bill Spouse Nncy Lou Sister Find ll direct nd indirect reltives of Fred Cnnot express in RA!!! Need to write C progrm, use grph engine, or modern SQL 53 Exmple: Bnking Dtbse brnch (brnch_nme, brnch_city, ssets) customer (customer_nme, customer_street, customer_city) ccount (ccount_number, brnch_nme, blnce) lon (lon_number, brnch_nme, mount) depositor (customer_nme, ccount_number) borrower (customer_nme, lon_number) 54

28 brnch (brnch_nme, brnch_city, ssets) customer (customer_nme, customer_street, customer_city) ccount (ccount_number, brnch_nme, blnce) lon (lon_number, brnch_nme, mount) depositor (customer_nme, ccount_number) borrower (customer_nme, lon_number) 55 Exmple Queries Find ll lons of over $1200 s mount > 1200 (lon) Find the lon number for ech lon of n mount greter thn $1200 Õ lon_number (s mount > 1200 (lon)) Dtbse System Concepts, Silberschtz, Korth nd Sudrshn 56

29 Exmple Queries Find the nmes of ll customers who hve lon, n ccount, or both, from the bnk Õ customer_nme (borrower) È Õ customer_nme (depositor) Find the nmes of ll customers who hve lon nd n ccount t bnk. Õ customer_nme (borrower) Ç Õ customer_nme (depositor) Dtbse System Concepts, Silberschtz, Korth nd Sudrshn 57 Exmple Queries Find the nmes of ll customers who hve lon t the Perryridge brnch. Õ customer_nme (s brnch_nme= Perryridge (s borrower.lon_number = lon.lon_number (borrower x lon))) Find the nmes of ll customers who hve lon t the Perryridge brnch but do not hve n ccount t ny brnch of the bnk. Õ customer_nme (s brnch_nme = Perryridge (s borrower.lon_number = lon.lon_number (borrower x lon))) Õ customer_nme (depositor) Dtbse System Concepts, Silberschtz, Korth nd Sudrshn 58

30 Exmple Queries Find the nmes of ll customers who hve lon t the Perryridge brnch. Query 1 Õ customer_nme (s brnch_nme = Perryridge ( s borrower.lon_number = lon.lon_number (borrower x lon))) Query 2 Õ customer_nme (s lon.lon_number = borrower.lon_number ( (s brnch_nme = Perryridge (lon)) x borrower)) Dtbse System Concepts, Silberschtz, Korth nd Sudrshn 59 Exmple Queries Find ll customers who hve n ccount t ll brnches locted in Brooklyn city. Dtbse System Concepts, Silberschtz, Korth nd Sudrshn 60

31 Acknowledgements The course mteril used for this lecture is mostly tken nd/or dopted from the course mterils of the CS145 Introduction to Dtbses lecture given by Christopher Ré t Stnford University ( 61