Algoritmalar, Tamsayılar ve Matrisler

Transcription

1 Algoitml, Tmsyıl ve Mtisle CSC-59 Ayık Ypıl Kostti Busch - LSU f Foksiyolı Büyümesi : R R g : R R f ( is O( g( Big-Oh: E kötü duum Big-Omeg: f ( is ( g( E iyi duum Big-Thet: f ( is ( g( Ayı duum Kostti Busch - LSU

2 Big-Oh: f ( is O( g( (Nottio buse: f ( O( g( C, k Thee e costts such tht fo ll : k f ( C g( (clled witesses Kostti Busch - LSU f ( g( f ( O( g( O ( Fo : f ( g( Witesses: C, k Kostti Busch - LSU 4

3 Kostti Busch - LSU 5 ( O 4 Fo : ( 4 ( f g, 4 C k ( g ( f ( ( ( f O g Witesses: Kostti Busch - LSU 6 ( ( ( g O f ( ( ( f O g ve f ve yı deecededi g Emple: ve yı deecededi

4 4 Kostti Busch - LSU 7 ( ( ( g O f ( ( h g ve Emple: ( O ( ( ( h O f ( O Kostti Busch - LSU 8 ( O ( O Suppose C The fo ll : k C Impossible fo, m( k C

5 5 Kostti Busch - LSU 9 Theoem: If ( f the ( ( O f Poof: ( ( f fo, k C Ed of Poof Witesses: Kostti Busch - LSU ( O, C k Witesses:

6 6 Kostti Busch - LSU (! O!, C k Witesses: Kostti Busch - LSU! ( O! ( (, C k Witesses:

7 log! O( log log! log log Witesses: C, k Kostti Busch - LSU O( log O( Fo : log log log Witesses: C, k Kostti Busch - LSU 4 7

8 log O(log log log log Witesses: C, log k Kostti Busch - LSU 5 costt ( O Fo : Witesses: C, k Kostti Busch - LSU 6 8

9 Iteestig fuctios log log! Highe gowth Kostti Busch - LSU 7 Theoem: If f O( g (, f O( g ( ( ( ( the f f( O(m( g(, g ( Poof: k f( C g( k f( C g( m( k, k ( f ( C f ( f ( f ( f ( C g ( C g ( C m( g (, g ( f ( Witesses: C C C k, k m( k, Ed of Poof Kostti Busch - LSU 8 9

10 Coolly: If f ( O( g(, f ( O( g( the ( f f( O( g( Theoem: If f O( g (, f O( g ( ( the f f ( O( g ( g ( ( ( Kostti Busch - LSU 9 log(! ( log O( log O( log(! O( log O( log O(log Multiplictio log(! O( log O( ( O( log log( log log(! ( O( Additio log log Kostti Busch - LSU

11 Big-Omeg: f ( is ( g( (Nottio buse: f ( ( g( C, k Thee e costts such tht fo ll : k f ( C g( (clled witesses Kostti Busch - LSU ( Witesses: C 8, k Kostti Busch - LSU

12 Sme ode Big-Thet: f ( is ( g( (Nottio buse: f ( ( g( f ( O( g( d f ( ( g( f Altetive defiitio: ( O( g( d g( O( f ( Kostti Busch - LSU 8 log ( 8 log 8 8 log O( Witesses: C, k 8 log 8 log ( Witesses: C, k Kostti Busch - LSU 4

13 Theoem: If f ( the f ( ( Poof: We hve show: f ( O( We oly eed to show f ( ( Tke d emie two cses Cse : Cse : Kostti Busch - LSU 5 Cse : ( b m(,,, f ( b Fo Cse is simil d b ( Ed of Poof Kostti Busch - LSU 6

14 Compleity of Algoithms Time compleity Numbe of opetios pefomed Spce compleity Size of memoy used Kostti Busch - LSU 7 Lie sech lgoithm dsil(it Dizi[], it N, it { it k; fo (k=; k<n; k++ { if (Dizi[k]== etu k; } etu -; } Kostti Busch - LSU 8 4

15 Time compleity Compisos Item ot foud i list: ( Item foud i positio : i i Wost cse pefomce: ( O( Kostti Busch - LSU 9 Biy sech lgoithm Biy-Sech(,,,, { i j while( i j { m ( i j / if ( m i m else j m } if ( i etu else etu } //left edpoit of sech e //ight edpoit of sech e i //item is i ight hlf //item is i left hlf //item foud //item ot foud Kostti Busch - LSU 5

16 Sech Kostti Busch - LSU Time compleity Size of sech list t itetio : Size of sech list t itetio : Size of sech list t itetio k: k Kostti Busch - LSU 6

17 Size of sech list t itetio : k k Smllest list size: m i lst itetio : m m log Kostti Busch - LSU Totl compisos: ( log (log #itetios Lst compiso Compisos pe itetio Kostti Busch - LSU 4 7

18 Bubble sot lgoithm Bubble-Sot(,,, { fo ( to { fo ( to if ( j j swp } i i j j j, Kostti Busch - LSU Fist itetio Secod itetio Lst itetio 5 4 Kostti Busch - LSU 6 8

19 Time compleity Compisos i itetio : Compisos i itetio : Compisos i itetio : ( Totl: ( ( Kostti Busch - LSU 7 P Clss : Hsss poblemle Poblems with lgoithms whose time compleity is polyomil O( b Emples: Sech, Sotig, Shotest pth Kostti Busch - LSU 8 9

20 NP Clss : Itctble poblems Solutio c be veified i polyomil time but o polyomil time lgoithm is kow Emples: Stisfibility, TSP (gezgi stıcı poblemi, Vete coloig Impott compute sciece questio P NP? Kostti Busch - LSU 9 Usolvble poblems Thee eist usolvble poblems which do ot hve y lgoithm Emple: Hltig poblem i Tuig Mchies Tuig mkisıd ksızlık poblemlei Kostti Busch - LSU 4

21 b Iteges d Algoithms Bse epsio of itege : b Iteges: b b k k k k ( k k b k i b Emple: ( Kostti Busch - LSU 4 Biy epsio Digits:, ( Kostti Busch - LSU 4

22 Hedeciml epsio Digits:,,,,9, A, B, C, D, E, F (AEB Kostti Busch - LSU 4 Octl epsio Digits:,,,,7 ( Kostti Busch - LSU 44

23 Covesio betwee biy d hedeciml ( hlf byte ( 9 5 F 6 Covesio betwee biy d octl ( ( Kostti Busch - LSU 45 Biy epsio of 4 ( Kostti Busch - LSU 46

24 Octl epsio of 45 ( Kostti Busch - LSU 47 b q k While ( q { k q mod b q q / b Bse epsio( { k k } etu } ( k k b Kostti Busch - LSU 48 4

25 Cy bit: b Time compleity of biy dditio: O( (coutig bit dditios O(log Kostti Busch - LSU 49,b Biy_dditio( { ( b ( b b bb c //cy bit fo j to { d ( j bj c / //uilliy s j j bj c d //j sum bit c d //cy bit } s c //lst sum bit etu ( ss ss } Kostti Busch - LSU 5 5

26 Time compleity of multiplictio: O( (coutig shifts d bit dditios O(log Kostti Busch - LSU 5 b c c c Biy_multiplictio( { ( b ( b b bb fo j to { if ( } b j j c j j,b else c j } p c c c etu biy epsio of // shifted j positios p Kostti Busch - LSU 5 6

27 Tm syıl ve böle Tm syıl,b ( böle : b b c, b c fktö Öekle: 4 7 Kostti Busch - LSU 5 b, if, Tm syı c b the bc b s b s bc (sc Kostti Busch - LSU 54 7

28 , b, b c Tm syı if d the c ( b c b c s t b s c t b c ( s t Kostti Busch - LSU 55 b,, iteges c b b c if d the c b b c s t b s c bt c st Kostti Busch - LSU 56 8

29 b, c, m,, iteges b c if d the mb c b c mb c mb c Kostti Busch - LSU 57 Thee e uique The divisio lgoithm Z d Z q, Z d q such tht: böle bölüm kl d Kostti Busch - LSU 58 9

30 q q d div d d q mod d d d 9 Emples: 9 div mod ( 4 4 div mod Kostti Busch - LSU 59 d d d d d d d d d Numbe of positive iteges divisible by d ot eceedig : d d Kostti Busch - LSU 6

31 Divisio_lgoithm( { } q d d while ( { q q, d } if ( d { // is egtive //djust q ( q //djust q } else if ( { q q } etu q ( div d, ( mod d d Kostti Busch - LSU 6 d q mod 4 q 5 div 4 Time compleity of divisio lg.: O( q log Thee is bette lgoithm: O(log log d (bsed o biy sech Kostti Busch - LSU 6

32 , b Z Modul Aithmetic b (mod m m Z b m is coguet to modulo mod m b mod m Emples: (mod 5 (mod 6 m (mod m k m (mod m Kostti Busch - LSU 6 Equivlet defiitios b (mod m mod m b mod m m b k Z, b km Kostti Busch - LSU 64

33 mod Legth of lie epesets umbe Kostti Busch - LSU 65 mod Legth of heli lie epesets umbe Kostti Busch - LSU 66

34 9 mod Legth of heli lie epesets umbe Kostti Busch - LSU 67 9(mod Heli lies temite i sme umbe Kostti Busch - LSU 68 4

35 Coguece clss of modulo : S m { b b (mod m} Thee e S m coguece clsses:, S,, S m Kostti Busch - LSU 69 b c d (mod m (mod m c b d (mod m b (mod m b sm c d b ( s t m c d (mod m c d tm Kostti Busch - LSU 7 5

36 b c d (mod m (mod m c b d (mod m b (mod m b sm c ( b sm( d tm c d (mod m c d tm bd m( bt ds stm Kostti Busch - LSU 7 7 (mod 5 (mod ( (mod 5 (mod ( (mod 5 (mod 5 Kostti Busch - LSU 7 6

37 ( b mod m (( mod m ( b mod m mod m bmod m (( mod m( b mod m mod m Follows fom pevious esults by usig: mod m ( mod m mod m b mod m ( b mod m mod m Kostti Busch - LSU 7 Compute Modul epoetitio b mod m Biy epsio of efficietly usig smll umbes b b k k k k b b b b b mod m k (( b k k b k b mod m mod m ( b mod m ( b mod mmod m Kostti Busch - LSU 74 7

38 Emple: 644 mod ( (( mod mod 645 mod 645( 7 mod 645( mod 645 mod 645 Kostti Busch - LSU 75 mod 645 9mod Compute ll the powes of efficietly mod 645 mod 645 mod 645 mod 645 (( mod 645( mod 645mod 645 (99mod mod 645 (( mod 645( mod 645mod mod 645 mod 645 (( mod 645( mod 645mod Use the powes of to get esult efficietly 644 ( 9 ( 9 ( 9 ((( ((( ( 7 mod 645 mod 645 mod 645 ( 8mod mod 6458 mod 645 mod 645 ( ((968 mod 645 mod 645 ( mod mod mod mod 645 Kostti Busch - LSU 76 8

39 Modul_Epoetitio( { ( } b,, m powe b mod m fo to { if ( ( powe mod m powe ( powe powe mod m } etu ( b mod m i i k Time compleity: O(log m log bit opetios Kostti Busch - LSU 77 Coguet pplictio: Hshig fuctios h( k k mod m Emple: h( k k mod Employe id Folde# h( mod 4 h( mod 65 h( mod 4 collisio Kostti Busch - LSU 78 9

40 Applictio: Pseudodom umbes Sequece of pseudodom umbes,,, Lie coguetil method: (7 m ( c mod m seed c m m 4 mod Emple: 9,7,8,6,,,,4,5,,7,8,6,,,,4,5, seed Kostti Busch - LSU 79 Applictio: Cyptology MEET YOU IN THE PARK ecyptio f ( ( mod m f decyptio ( ( mod m PHHW BRX LQ WKH SDUN Shift ciphe: f ( ( mod m Affie tsfomtio: f ( ( b mod m Kostti Busch - LSU 8 4

41 Asl ve E Büyük Otk Böle Asl : ppositive itege gete th, oly positive fctos e, p Asl olmyl = composite (bileşik Asll:,,5,7,,,7,9,,9,,7,4, Kostti Busch - LSU 8 Aitmetiği temel teoemi He pozitif tmsyı, y sldı y d sl syılı soucudu. Asl çplı yım: m sl p k p k p k p k l l Öekle: Kostti Busch - LSU 8 4

42 Theoem: If is composite the it hs pime diviso p Poof: is composite, b,, b, b c mi(, b sice othewise b Fom fudmetl theoem of ithmetic is eithe pime o hs pime diviso c Ed of Poof Kostti Busch - LSU 8 Pime_fctoiztio( { //fist pime } p ' while ( d { if ( divides { p is fcto of p / p } else p et pime fte } etu ll pime fctos foud p Kostti Busch - LSU 84 p 4

43 77 p,,5 p 7 p 7 p 7 p p do ot divide does ot divide 4 4 ( 77 7 Kostti Busch - LSU 85 Theoem: Sosuz syıd sl syı vdı Poof: Solu syıd sl syı olduğuu vsylım p, p,, pk Let q If some pime Sice p p p p pk p i q p i p k p i q p p pk impossible No pime divides q (Fom fudmetl theoem of ithmetic q is pime Cotdictio! Ed of Poof Kostti Busch - LSU 86 4

44 Bilie e büyük sl (s of 6 Mesee sl syılıı gösteimi: Kostti Busch - LSU 87 Asl syı teoemi The umbe of pimes less o equl to ppoches to: l log e Kostti Busch - LSU 88 44

45 Goldbch ı vsyımı: He tm syı iki sl syıı toplmıdı İkili sl syıl vsyımı: Sosuz syıd ikiz sl vdı. İki fklı ikili sl syı vdı:,5 5,7, 7,9 Kostti Busch - LSU 89 gcd(, b, b Z b E büyük otk böle lgest itege such tht d d d d b Emples: gcd( 4,6 Commo divisos of 4, 6:,,, 4, 6, gcd( 7, Commo divisos of 7, : Kostti Busch - LSU 9 45

46 Öemsiz duuml: Kostti Busch - LSU 9 Theoem: b q ( / b If b the gcd(, b gcd( b, Poof: d ds d( s tq d d b b dt b dt d b Thus, (, b d ( b, hve the sme set of commo divisos Ed of poof Kostti Busch - LSU 9 46

47 divisios / / / / b q q q q fist zeo emide esult Kostti Busch - LSU 9 66 b esult 4 8 gcd(66,44 gcd(44,48 gcd(48,66 gcd(66,8 gcd(8, gcd(, Kostti Busch - LSU 94 47

48 b 4 mod mod mod 4 mod mod Kostti Busch - LSU 95 gcd(, b 66 b mod mod mod mod 8 8 mod gcd(, b Kostti Busch - LSU 96 48

49 49 Kostti Busch - LSU 97 i i i mod i i i b Özellik: i i Azl sılm i i i i i Duum : Kostti Busch - LSU 98 i i i mod i i i b Özellik: i i Azl sılm i i i i i i Duum :

50 b Azl sılm i i i mod i i i Özellik: i i log Kostti Busch - LSU 99 Euclidi Algoithm gcd( { },b y b while ( y { mod y y y } etu Time compleity: O(log divisios Kostti Busch - LSU 5

51 Alıd sl syıl:,b If gcd (, b the e eltively pime b d hve o commo fctos i thei pime fctoiztio Öek:, lıd sl syıldı gcd(, 7 Kostti Busch - LSU E küçük otk kt lcm(, b,b Z smllest positive itege such tht d d b d d Emples: lcm(,4 lcm(5, Kostti Busch - LSU 5

52 Syıl teoisii uygulmlı Doğusl kombisyo:, Z b Z s, t if the thee e such tht gcd(, b s tb Öek: gcd( 6,4 ( 6 4 Kostti Busch - LSU Doğusl kombisyo, Öklid lgoitmsıı dımlıı tesie çevieek bulubili. gcd( 5, gcd(5, ( ( Kostti Busch - LSU 4 5

53 Doğusl uyguluk We wt to solve the equtio fo b(mod m? (mod m Kostti Busch - LSU 5 (mod m Ivese of : b(mod m mod m b(mod m (mod m (mod m (mod m b(mod m Kostti Busch - LSU 6 5

54 Theoem: If d m e eltively pime the the ivese modulo m eists Poof: gcd(, m s tm s (mod m s Ed of poof Kostti Busch - LSU 7 4(mod 7 Emple: solve equtio, b 4, m 7 ü tesi: gcd(,7 7 (mod m b(mod m 4(mod 7 8(mod 7 6mod 7 Kostti Busch - LSU 8 54

55 Chiese emide poblem :piwise eltively pime (mod m (mod m Hs uique solutio fo (mod m modulo: (mod m Kostti Busch - LSU 9 Uique solutio modulo : M y M y M y M k m m k y :ivese of M modulo k k mk Kostti Busch - LSU 55

56 Epltio: M k m m k y k :ivese of M k y k M k mod m modulo k mk M y (mod m (mod m M y M y M y (mod m (mod m M k m (mod Simil fo y m j Kostti Busch - LSU Emple: (mod m 57 5 (mod5 (mod 7 M M m / 5/ 5 M m / 5 5/ 5 m / 7 5/ 7 5 y y y M y M 5 5 y M (mod 57 (mod5 y Kostti Busch - LSU 56

57 A Applictio of Chiese emide poblem Pefom ithmetic with lge umbes usig ithmetic modulo smll umbes Emple: eltively pime umbes m 99, m 98, m 97, m4 95 m ,4,9,684 (, 8, 9, 89 Ay umbe smlle th hs uique epesettio m,684mod 99,684mod 98 8,684mod 97 9,684mod Kostti Busch - LSU, ,456 (, 8, 9, (, 9, 4, 6 (65 mod99, mod98, 5mod97,5 mod95 57,4 (65,, 5, We obti this by usig the Chiese emide poblem solutio Kostti Busch - LSU 4 57

58 Femt s little theoem: p Fo y pime d itege ot divisible by ( : p p (mod p gcd(, p Emple: 4 (mod4 p 4 Kostti Busch - LSU 5 Poof: Popety : p does ot divide y of: Kostti Busch - LSU 6 58

59 Epltio: p k Suppose divides, k p s Z : k sp gcd(, p k p Does ot hve s pime fcto p p hs s pime fcto Cotdicts fudmetl theoem of ithmetic Kostti Busch - LSU 7 Popety : y pi below is ot coguet modulo : p Kostti Busch - LSU 8 59

60 Epltio: y (mod p y p Suppose, s Z : y sp ( y sp p ( y divides y p Cotdicts Popety Kostti Busch - LSU 9 Popety : Kostti Busch - LSU 6

61 Epltio: Fom Popety ( p i j (mod p, (mod p, p (mod p, p ( p p p p i j p p ( p ( p (mod p Kostti Busch - LSU Popety 4: (follows diectly fom popety Kostti Busch - LSU 6

62 Popety 5: Kostti Busch - LSU Epltio: ( p! fom Popety 4: ( p ( p!(mod p p does ot divide ( p! gcd( p,( p! ( p! (mod p eists Multiply both sides with: ( p! ( p (mod p Ed of Poof Kostti Busch - LSU 4 6

63 RSA (Rivest-Shmi-Adlem cyptosystem MEET YOU IN THE PARK f ecyptio decyptio e ( mod f ( d mod p q,e e public keys Lge pimes p q, d, e pivte keys Kostti Busch - LSU 5 Ecyptio emple: p 4 q 59 e p q 57 gcd( e,( p ( q gcd(,4 58 Messge to ecypt: STOP Tslte to equivlet umbes Goup ito blocks of two umbes Kostti Busch - LSU 6 6

64 89 45 Apply ecyptio fuctio to ech block f ( e mod Ecypted messge: 8 8 mod 57 f (89 89 mod 57 8 f (45 45 mod 57 8 Kostti Busch - LSU 7 M Messge decyptio : oigil block of the messge C :espective ecypted block C M e (mod We wt to fid M by kowig C, p, q, e Kostti Busch - LSU 64

65 d :ivese of e modulo ( p ( q de (mod( p ( q by defiitio of coguet de k( p ( q Ivese eists becuse gcd( e,( p ( q gcd( e,( p ( q se t( p ( q se mod( p ( q d s Kostti Busch - LSU 9 C M e (mod C d e d M (mod de k( p ( q C d M de M k ( p( q (mod Kostti Busch - LSU 65

66 Vey likely it holds gcd( M, p p (becuse is lge pime d is smll M gcd( M, p By Femt s little theoem M p (mod p Kostti Busch - LSU M p (mod p p k ( q k ( q M (mod p M M (mod p M p k ( q M M (mod p M k ( p( q M (mod p Kostti Busch - LSU 66

67 We showed: M k ( p( q M (mod p By symmety, whe eplcig with : p q M k ( p( q M (mod q By the Chiese emide poblem: M k ( p( q M (mod pq M (mod Kostti Busch - LSU We showed: C M d M k ( p( q k ( p( q (mod M (mod C d M (mod I othe wods: M C d mod Kostti Busch - LSU 4 67

68 Decyptio emple: p 4 59 p q q 57 e gcd( e,( p ( q gcd(,4 58 We c compute: d f ( C C d mod 8 97 mod mod = STOP Kostti Busch - LSU 5 68