ELEMENTARY SORTING ALGORITHMS

Transcription

1 BBM ALGORITHMS DEPT. OF COMPUTER ENGINEERING ELEMENTARY SORTING ALGORITHMS Feb. 20, 2017 Acknowledgement: The course sldes are adapted from the sldes prepared by R. Sedgewck and K. Wayne of Prnceton Unversty.

2 ELEMENTARY SORTING ALGORITHMS Sortng revew Rules of the game Selecton sort Inserton sort Shellsort

3 ELEMENTARY SORTING ALGORITHMS Sortng revew Rules of the game Selecton sort Inserton sort Shellsort

4 Sortng problem Ex. Student records n a unversty. Chen 3 A Blar Rohde 2 A Forbes Gazs 4 B Brown tem Fura 1 A Brown Kanaga 3 B Brown Andrews 3 A Lttle key Battle 4 C Whtman Sort. Rearrange array of N tems nto ascendng order. Andrews 3 A Lttle Battle 4 C Whtman Chen 3 A Blar Fura 1 A Brown Gazs 4 B Brown Kanaga 3 B Brown Rohde 2 A Forbes 4

5 Sample sort clent Goal. Sort any type of data. Ex 1. Sort random real numbers n ascendng order. seems artfcal, but stay tuned for an applcaton publc class Experment { publc statc vod man(strng[] args) { nt N = Integer.parseInt(args[0]); Double[] a = new Double[N]; for (nt = 0; < N; ++) a[] = StdRandom.unform(); Inserton.sort(a); for (nt = 0; < N; ++) StdOut.prntln(a[]); } } % java Experment

6 Sample sort clent Goal. Sort any type of data. Ex 2. Sort strngs from fle n alphabetcal order. publc class StrngSorter { publc statc vod man(strng[] args) { Strng[] a = In.readStrngs(args[0]); Inserton.sort(a); for (nt = 0; < a.length; ++) StdOut.prntln(a[]); } } % more words3.txt bed bug dad yet zoo... all bad yes % java StrngSorter words3.txt all bad bed bug dad... yes yet zoo 6

7 Sample sort clent Goal. Sort any type of data. Ex 3. Sort the fles n a gven drectory by flename. mport java.o.fle; publc class FleSorter { publc statc vod man(strng[] args) { Fle drectory = new Fle(args[0]); Fle[] fles = drectory.lstfles(); Inserton.sort(fles); for (nt = 0; < fles.length; ++) StdOut.prntln(fles[].getName()); } } % java FleSorter. Inserton.class Inserton.java InsertonX.class InsertonX.java Selecton.class Selecton.java Shell.class Shell.java ShellX.class ShellX.java 7

8 Callbacks Goal. Sort any type of data. Q. How can sort() know how to compare data of type Double, Strng, and java.o.fle wthout any nformaton about the type of an tem's key? Callback = reference to executable code. Clent passes array of objects to sort() functon. The sort() functon calls back object's compareto() method as needed. Implementng callbacks. Java: nterfaces. C: functon ponters. C++: class-type functors. C#: delegates. Python, Perl, ML, Javascrpt: frst-class functons. 8

9 Callbacks: roadmap clent mport java.o.fle; publc class FleSorter { publc statc vod man(strng[] args) { Fle drectory = new Fle(args[0]); Fle[] fles = drectory.lstfles(); Inserton.sort(fles); for (nt = 0; < fles.length; ++) StdOut.prntln(fles[].getName()); } } object mplementaton publc class Fle mplements Comparable<Fle> {... publc nt compareto(fle b) {... return -1;... return +1;... return 0; } } Comparable nterface (bult n to Java) publc nterface Comparable<Item> { publc nt compareto(item that); } key pont: no dependence on Fle data type sort mplementaton publc statc vod sort(comparable[] a) { nt N = a.length; for (nt = 0; < N; ++) for (nt j = ; j > 0; j--) f (a[j].compareto(a[j-1]) < 0) exch(a, j, j-1); else break; } 9

10 Total order A total order s a bnary relaton that satsfes Antsymmetry: f v w and w v, then v = w. Transtvty: f v w and w x, then v x. Totalty: ether v w or w v or both. Ex. Standard order for natural and real numbers. Alphabetcal order for strngs. Chronologcal order for dates.... an ntranstve relaton 10

11 Comparable API Implement compareto() so that v.compareto(w) Is a total order. Returns a negatve nteger, zero, or postve nteger f v s less than, equal to, or greater than w, respectvely. Throws an excepton f ncompatble types (or ether s null). v w v w v w less than (return -1) equal to (return 0) greater than (return +1) Bult-n comparable types. Integer, Double, Strng, Date, Fle,... User-defned comparable types. Implement the Comparable nterface. 11

12 Implementng the Comparable nterface Date data type. Smplfed verson of java.utl.date. publc class Date mplements Comparable<Date> { prvate fnal nt month, day, year; publc Date(nt m, nt d, nt y) { month = m; day = d; year = y; } only compare dates to other dates } publc nt compareto(date that) { f (ths.year < that.year ) return -1; f (ths.year > that.year ) return +1; f (ths.month < that.month) return -1; f (ths.month > that.month) return +1; f (ths.day < that.day ) return -1; f (ths.day > that.day ) return +1; return 0; } 12

13 Two useful sortng abstractons Helper functons. Refer to data through compares and exchanges. Less. Is tem v less than w? prvate statc boolean less(comparable v, Comparable w) { return v.compareto(w) < 0; } Exchange. Swap tem n array a[] at ndex wth the one at ndex j. prvate statc vod exch(comparable[] a, nt, nt j) { Comparable swap = a[]; a[] = a[j]; a[j] = swap; } 13

14 ELEMENTARY SORTING ALGORITHMS Sortng revew Rules of the game Selecton sort Inserton sort Shellsort

15 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. remanng entres 15

16 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn remanng entres 16

17 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn remanng entres 17

18 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. n fnal order remanng entres 18

19 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 19

20 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 20

21 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. n fnal order remanng entres 21

22 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 22

23 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 23

24 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. n fnal order remanng entres 24

25 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 25

26 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 26

27 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. n fnal order remanng entres 27

28 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 28

29 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 29

30 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. n fnal order remanng entres 30

31 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 31

32 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 32

33 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. n fnal order remanng entres 33

34 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 34

35 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 35

36 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. n fnal order remanng entres 36

37 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 37

38 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 38

39 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. n fnal order remanng entres 39

40 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 40

41 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. mn n fnal order remanng entres 41

42 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. n fnal order 42

43 Selecton sort In teraton, fnd ndex mn of smallest remanng entry. Swap a[] and a[mn]. sorted 43

44 Selecton sort: Java mplementaton publc class Selecton { publc statc vod sort(comparable[] a) { nt N = a.length; for (nt = 0; < N; ++) { nt mn = ; for (nt j = +1; j < N; j++) f (less(a[j], a[mn])) mn = j; exch(a,, mn); } } prvate statc boolean less(comparable v, Comparable w) { /* as before */ } } prvate statc vod exch(comparable[] a, nt, nt j) { /* as before */ } 44

45 Selecton sort: mathematcal analyss Proposton. Selecton sort uses (N 1) + (N 2) ~ N 2 / 2 compares and N exchanges. a[] mn S O R T E X A M P L E 0 6 S O R T E X A M P L E 1 4 A O R T E X S M P L E 2 10 A E R T O X S M P L E 3 9 A E E T O X S M P L R 4 7 A E E L O X S M P T R 5 7 A E E L M X S O P T R 6 8 A E E L M O S X P T R 7 10 A E E L M O P X S T R 8 8 A E E L M O P R S T X 9 9 A E E L M O P R S T X A E E L M O P R S T X entres n black are examned to fnd the mnmum entres n red are a[mn] entres n gray are n fnal poston A E E L M O P R S T X Trace of selecton sort (array contents just after each exchange) Runnng tme nsenstve to nput. Quadratc tme, even f nput array s sorted. Data movement s mnmal. Lnear number of exchanges. 45

46 Selecton sort: anmatons 20 random tems algorthm poston n fnal order not n fnal order 46

47 Selecton sort: anmatons 20 partally-sorted tems algorthm poston n fnal order not n fnal order 47

48 ELEMENTARY SORTING ALGORITHMS Sortng revew Rules of the game Selecton sort Inserton sort Shellsort

49 Inserton sort In teraton, swap a[] wth each larger entry to ts left. 49

50 Inserton sort In teraton, swap a[] wth each larger entry to ts left. not yet seen 50

51 Selecton sort In teraton, swap a[] wth each larger entry to ts left. j n ascendng order not yet seen 51

52 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 52

53 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j n ascendng order not yet seen 53

54 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 54

55 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 55

56 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 56

57 Inserton sort In teraton, swap a[] wth each larger entry to ts left. n ascendng order not yet seen 57

58 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 58

59 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 59

60 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 60

61 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 61

62 Inserton sort In teraton, swap a[] wth each larger entry to ts left. n ascendng order not yet seen 62

63 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 63

64 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 64

65 Inserton sort In teraton, swap a[] wth each larger entry to ts left. n ascendng order not yet seen 65

66 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 66

67 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 67

68 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 68

69 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 69

70 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 70

71 Inserton sort In teraton, swap a[] wth each larger entry to ts left. n ascendng order not yet seen 71

72 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 72

73 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 73

74 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 74

75 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 75

76 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 76

77 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 77

78 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 78

79 Inserton sort In teraton, swap a[] wth each larger entry to ts left. n ascendng order not yet seen 79

80 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 80

81 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j not yet seen 81

82 Inserton sort In teraton, swap a[] wth each larger entry to ts left. n ascendng order not yet seen 82

83 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j 83

84 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j 84

85 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j 85

86 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j 86

87 Inserton sort In teraton, swap a[] wth each larger entry to ts left. j 87

88 Inserton sort In teraton, swap a[] wth each larger entry to ts left. sorted 88

89 Inserton sort: Java mplementaton publc class Inserton { publc statc vod sort(comparable[] a) { nt N = a.length; for (nt = 0; < N; ++) for (nt j = ; j > 0; j--) f (less(a[j], a[j-1])) exch(a, j, j-1); else break; } prvate statc boolean less(comparable v, Comparable w) { /* as before */ } } prvate statc vod exch(comparable[] a, nt, nt j) { /* as before */ } 89

90 Inserton sort: mathematcal analyss Proposton. To sort a randomly-ordered array wth dstnct keys, nserton sort uses ~ ¼ N 2 compares and ~ ¼ N 2 exchanges on average. Pf. Expect each entry to move halfway back. a[] j S O R T E X A M P L E 1 0 O S R T E X A M P L E 2 1 O R S T E X A M P L E 3 3 O R S T E X A M P L E 4 0 E O R S T X A M P L E 5 5 E O R S T X A M P L E 6 0 A E O R S T X M P L E 7 2 A E M O R S T X P L E 8 4 A E M O P R S T X L E 9 2 A E L M O P R S T X E 10 2 A E E L M O P R S T X entres n gray do not move entry n red s a[j] entres n black moved one poston rght for nserton A E E L M O P R S T X Trace of nserton sort (array contents just after each nserton) 90

91 Inserton sort: anmaton 40 random tems algorthm poston n order not yet seen 91

92 Inserton sort: best and worst case Best case. If the array s n ascendng order, nserton sort makes N - 1 compares and 0 exchanges. A E E L M O P R S T X Worst case. If the array s n descendng order (and no duplcates), nserton sort makes ~ ½ N 2 compares and ~ ½ N 2 exchanges. X T S R P O M L E E A 92

93 Inserton sort: anmaton 40 reverse-sorted tems algorthm poston n order not yet seen 93

94 Inserton sort: partally-sorted arrays Def. An nverson s a par of keys that are out of order. A E E L M O T R X P S T-R T-P T-S R-P X-P X-S (6 nversons) Def. An array s partally sorted f the number of nversons s c N. Ex 1. A subarray of sze 10 appended to a sorted subarray of sze N. Ex 2. An array of sze N wth only 10 entres out of place. Proposton. For partally-sorted arrays, nserton sort runs n lnear tme. Pf. Number of exchanges equals the number of nversons. number of compares = exchanges + (N 1) 94

95 Inserton sort: anmaton 40 partally-sorted tems algorthm poston n order not yet seen 95

96 ELEMENTARY SORTING ALGORITHMS Sortng revew Rules of the game Selecton sort Inserton sort Shellsort

97 Shellsort overvew Idea. Move entres more than one poston at a tme by h-sortng the array. an h-sorted array s h nterleaved sorted subsequences h = 4 L E E A M H L E P S O L T S X R L M P T E H S S E L O X A E L R Shellsort. [Shell 1959] h-sort the array for decreasng seq. of values of h. nput 13-sort 4-sort 1-sort S H E L L S O R T E X A M P L E P H E L L S O R T E X A M S L E L E E A M H L E P S O L T S X R A E E E H L L L M O P R S S T X 97

98 h-sortng How to h-sort an array? Inserton sort, wth strde length h. 3-sortng an array M O L E E X A S P R T E O L M E X A S P R T E E L M O X A S P R T E E L M O X A S P R T A E L E O X M S P R T A E L E O X M S P R T A E L E O P M S X R T A E L E O P M S X R T A E L E O P M S X R T A E L E O P M S X R T Why nserton sort? Bg ncrements small subarray. Small ncrements nearly n order. [stay tuned] 98

99 Shellsort example: ncrements 7, 3, 1 nput S O R T E X A M P L E 7-sort S O R T E X A M P L E M O R T E X A S P L E M O R T E X A S P L E M O L T E X A S P R E M O L E E X A S P R T 3-sort M O L E E X A S P R T E O L M E X A S P R T E E L M O X A S P R T E E L M O X A S P R T A E L E O X M S P R T A E L E O X M S P R T A E L E O P M S X R T A E L E O P M S X R T A E L E O P M S X R T 1-sort A E L E O P M S X R T A E L E O P M S X R T A E L E O P M S X R T A E E L O P M S X R T A E E L O P M S X R T A E E L O P M S X R T A E E L M O P S X R T A E E L M O P S X R T A E E L M O P S X R T A E E L M O P R S X T A E E L M O P R S T X result A E E L M O P R S T X 99

100 Shellsort: ntuton Proposton. A g-sorted array remans g-sorted after h-sortng t. 7-sort M O R T E X A S P L E M O R T E X A S P L E M O L T E X A S P R E M O L E E X A S P R T M O L E E X A S P R T 3-sort M O L E E X A S P R T E O L M E X A S P R T E E L M O X A S P R T E E L M O X A S P R T A E L E O X M S P R T A E L E O X M S P R T A E L E O P M S X R T A E L E O P M S X R T A E L E O P M S X R T A E L E O P M S X R T stll 7-sorted 100

101 Shellsort: whch ncrement sequence to use? Powers of two. 1, 2, 4, 8, 16, 32,... No. Powers of two mnus one. 1, 3, 7, 15, 31, 63,... Maybe. 3x , 4, 13, 40, 121, 364,... OK. Easy to compute. mergng of (9 4 ) (9 2 ) + 1 and 4 (3 2 ) + 1 Sedgewck. 1, 5, 19, 41, 109, 209, 505, 929, 2161, 3905,... Good. Tough to beat n emprcal studes. = Interested n learnng more? See Secton 6.8 of Algs, 3 rd edton or Volume 3 of Knuth for detals. Do a JP on the topc. 101

102 Shellsort: Java mplementaton publc class Shell { publc statc vod sort(comparable[] a) { nt N = a.length; 3x+1 ncrement sequence nt h = 1; whle (h < N/3) h = 3*h + 1; // 1, 4, 13, 40, 121, 364, 1093,... whle (h >= 1) { // h-sort the array. for (nt = h; < N; ++) { for (nt j = ; j >= h && less(a[j], a[j-h]); j -= h) exch(a, j, j-h); } nserton sort move to next ncrement } } h = h/3; } prvate statc boolean less(comparable v, Comparable w) { /* as before */ } prvate statc boolean vod(comparable[] a, nt, nt j) { /* as before */ } 102

103 Shellsort: vsual trace nput 40-sorted 13-sorted 4-sorted result 103

104 Shellsort: anmaton 50 random tems algorthm poston h-sorted current subsequence other elements 104

105 Shellsort: anmaton 50 partally-sorted tems algorthm poston h-sorted current subsequence other elements 105

106 Shellsort: analyss Proposton. The worst-case number of compares used by shellsort wth the 3x+1 ncrements s O(N 3/2 ). Property. The number of compares used by shellsort wth the 3x+1 ncrements s at most by a small multple of N tmes the # of ncrements used. N compares N N lg N measured n thousands Remark. Accurate model has not yet been dscovered (!) 106

107 Why are we nterested n shellsort? Example of smple dea leadng to substantal performance gans. Useful n practce. Fast unless array sze s huge. Tny, fxed footprnt for code (used n embedded systems). Hardware sort prototype. Smple algorthm, nontrval performance, nterestng questons. Asymptotc growth rate? Best sequence of ncrements? Average-case performance? open problem: fnd a better ncrement sequence Lesson. Some good algorthms are stll watng dscovery. 107