Introduction to Computer Science, Shimon Schocken, IDC Herzliya. Lecture Recursion. Introduction to Computer Science, Shimon Schocken slide 1

Transcription

1 Iroducio o Compuer Sciece, Shimo Schocke, IDC Herzliya Lecure Recursio Iroducio o Compuer Sciece, Shimo Schocke slide 1 Recursio Recursio: a fudameal algorihmic echique, based o divide ad coquer Recursive fucio: a mahemaical fucio defied i erms of iself. Eample:! (-1)! Recursive mehod: a mehod ha calls iself o smaller subses of he problem space. Eample: lis all he files i a give direcory Recursive daa srucure: a daa srucure whose elemes are defied usig refereces o iself. Eample: he lis "abcd" is he characer 'a followed by he lis "bcd" Learig o hik recursively ad desig recursive programs is a fudameal programmig skill Bu, maserig i akes pracice ad isigh. Recursive image Iroducio o Compuer Sciece, Shimo Schocke slide 2

2 Oulie Recursive algorihms Recursive fucios Facorial Fiboacci Power Recursive procedures File lisig Reverse Permuaios Ieger.oSrig Tower of Haoi Permuaios Fracals Iroducio o Compuer Sciece, Shimo Schocke slide 3 The buildig blocks of a recursive desig Every recursive algorihm is based o hree desig elemes: Reducio: i mus be possible o reduce he origial problem io subproblems ha are simpler isaces of he same problem Base case: A some poi of he reducio we mus arrive o a sub-problem ha ca be solved direcly Assembly: Oce he sub-problems have bee solved, i mus be possible o combie he sub-soluios io a soluio of he origial problem. Merge Sor: a recursive algorihm eample Iroducio o Compuer Sciece, Shimo Schocke slide 4

3 Facorial Ieraive defiiio 1 facorial( ) ( 1) ( 2) K 1 1or 0 oherwise Eample: 5! facorial(5) Ieraive implemeaio class class FacorialDemo1 mai(srig[] args) args) Sysem.ou.pril("5! " facorial(5)); Reurs Reurs he he facorial facorial (!) (!) of of a give give.. log log facorial(log ) ) log log fac fac 1; 1; for for (i (i i1; i1; i<; i<; i) i) fac fac i; i; fac; fac; Eample of a o-recursive soluio Iroducio o Compuer Sciece, Shimo Schocke slide 5 Facorial: a recursive soluio Recursive defiiio 1 facorial( ) facorial( 1) 1or 0 oherwise facorial(5) 5 facorial(4) 5 4 facorial(3) facorial(2) facorial(1) facorial( Recursive implemeaio log log facorial(log ) ) ( ( 1; 1; facorial(-1); Iroducio o Compuer Sciece, Shimo Schocke slide 6

4 Elemes of he recursive approach log log facorial(log ) ) ( ( 1; 1; facorial(-1); Base case Assembly Reducio A oe-lier log log facorial(log ) ) ((1) ((1)? 1 : facorial(-1)); Iroducio o Compuer Sciece, Shimo Schocke slide 7 Ru-ime aaomy 2 Recursive call Reurig a value o he caller facorial(4) 4 facorial(3) facorial (2) facorial(1) facorial( log log facorial(log facorial(log ) ) ( ( 1; 1; facorial(-1); facorial(-1); Base case: 1 Iroducio o Compuer Sciece, Shimo Schocke slide 8

5 Sum The problem saed sum sum (a (a,, b) b)?? sum sum (a (a,, o) o) aa sum sum (a (a,, b) b) sum(a sum(a,,(b (b--1)) 1)) 1 for for b > 0 sum sum (a,b) (a,b) sum(a,b--) i i pseudo pseudo code code Recursive algorihm 11 7 The challege: defiig sum wihou acually summig ayhig. Isead, we reduce he problem io a series of add 1 operaios. sum(a, sum(a, b) b) (b (b a s sum(a,--b) s s sum (8, 3) sum (8, 2) sum (8, 1) sum (8, Iroducio o Compuer Sciece, Shimo Schocke slide 9 I Java Algorihm sum(a, sum(a, b) b) (b (b a s sum(a,--b) s s Implemeaio class class SumDemo SumDemo mai(srig args[]) args[]) Sysem.ou.pril(sum(5,3)); log log sum(log sum(log a,log a,log b) b) (b (b a; a; log log s sum(a,--b); s; s; Iroducio o Compuer Sciece, Shimo Schocke slide 10

6 Fiboacci series Fiboacci series defiiio fib( fib( 1 fib(1) fib(1) 1 fib() fib() fib(-1) fib(-2) for for all all >1 >1 The Fiboacci series: 1, 1, 2, 3, 5, 8, 13,... Recursive implemeaio class class FiboacciDemo mai(srig args[]) args[]) Sysem.ou.pril(fiboacci(5)); Reurs Reurs he he umber umber i i fiboacci fiboacci series series i i fiboacci(i ) ) ( ( < < 1) 1) 1; 1; fiboacci(-1) fiboacci(-2); The recursive implemeaio follows direcly from he recursive defiiio: Base case: 0, 1 Reducio: from o (-1) ad (-2) Assembly: usig Iroducio o Compuer Sciece, Shimo Schocke slide 11 Fiboacci series: Ru-ime aaomy f(4) f(3) f(2) f(2) f(1) f(1) f( f(1) f( Buil-i redudacy: he algorihm repeas he same compuaios (e.g. f(2)) Ruig ime of his algorihm: ~ O(2 ) Q: Ca we do beer? A: A boom-up, ieraive soluio will ru i O() Coclusio: Naïve recursive soluios ca be epesive! Iroducio o Compuer Sciece, Shimo Schocke slide 12

7 Oulie Recursive algorihms Recursive fucios Facorial Fiboacci Power Recursive procedures File lisig Reverse Permuaios Ieger.oSrig Tower of Haoi Permuaios Fracals Iroducio o Compuer Sciece, Shimo Schocke slide 13 Power fucio: recursive implemeaio Recursive defiiio: power(,, 1 Power(,, ) ) power(,, -1) -1) for for all all >0 >0 1 Power(3, 4) 81 8 Ru-ime simulaio Recursive implemeaio: 3 power (3, 3) Compues Compues raised raised o o he he power power of of ) ) ) -1) power(3, 2) Ruig ime of recursive programs: Simply add up he ruig ime of all he recursive calls ) requires 1 recursive calls: ), -1),, 1), Toal ruig ime (1) O(1) O() 3 power(3, 1) power(3, Iroducio o Compuer Sciece, Shimo Schocke slide 14

8 Power fucio: recursive implemeaio, Take 2 Compues Compues raised raised o o he he power power of of Power(3, 5) Ru-ime simulaio ) ) ( ( 1 1 (%2 (%2 /2) /2) -1) -1) Ruig-ime: O(log ) Afer wo recursive calls decreases i a facor of a leas 2 (Eiher or -1 is eve, hus i wo recursive calls he algorihm divides by 2 a leas oce) Thus he oal umber of seps (recursive calls) is a mos 2 log Thus he ruig ime is O(log ). 3 power(3, 4) power(3, 2) power(3, 1) power(3, Iroducio o Compuer Sciece, Shimo Schocke slide 15 Recursive power, ake 2 ) ) ( ( 1; 1; (%2 (%2 /2); /2); ; ; -1); -1); Theorem: For ay ad posiive ieger, he algorihm s 1 Power(5, 4) Ru-ime simulaio power(5, 2) power(5, 1) Proof: By srog iducio o. 5 power(5, Base case: 0 he algorihm s 1. Iducive hypohesis: Assume ha for all k < he algorihm s k Iducive sep: If is eve, he algorihm s /2) /2) By he iducio hypohesis, /2) s ½, hus he algorihm s ( ½ ) ( ½ ) If is odd, he algorihm s -1) By he iducio hypohesis, -1) s -1, hus he algorihm s ( -1 ) Iroducio o Compuer Sciece, Shimo Schocke slide 16

9 Oulie Recursive algorihms Recursive fucios Facorial Fiboacci Power Recursive procedures File lisig Reverse Permuaios Ieger.oSrig Tower of Haoi Permuaios Fracals Iroducio o Compuer Sciece, Shimo Schocke slide 17 Recursive procedures May fucios such as facorial, Fiboacci, ec. have ihere recursive defiiios. Therefore, implemeig hem usig recursive mehods is aural Procedures, however, are desiged o do various higs wihou ig values (i Java, implemeed as mehods) I may cases, procedures ca also be described i recursive erms Eample: files lisig uiliy Iroducio o Compuer Sciece, Shimo Schocke slide 18

10 File lisig Lis Lis all all files files uder uder a a give give direcory direcory privae privae lisfiles(file lisfiles(file direcory) direcory) Creaes Creaes a a array array of of all all file file ames ames Srig[] Srig[] iemnames iemnames direcory.lis(); direcory.lis(); for for (i (i i0 i0 ; ; i<filenames.legh i<filenames.legh ; ; i) i) File File file file ew ew File(direcory, File(direcory, filenames[i]); filenames[i]); (file.isdirecory()) (file.isdirecory()) pridirecorydeails(file); pridirecorydeails(file); lisfiles(file); lisfiles(file); pridirecorydeails(file); pridirecorydeails(file); Pris Pris direcory direcory ame ame ad ad deails deails privae privae pridirecorydeails(file pridirecorydeails(file direcory) direcory) Pris Pris file file ame ame ad ad deails deails privae privae prifiledeails(file prifiledeails(file file) file) Iroducio o Compuer Sciece, Shimo Schocke slide 19 Reverse prireverse() prireverse() pri("eer pri("eer a a umber, umber, or or 0 0 o o ed: ed: ") ") read() read() ( (!! prireverse(); prireverse(); pri(); pri(); Read() prireverse() wrie() 2 '51' 5 Read() prireverse() wrie() 3 '5' 4 Read() prireverse() wrie() User s ipu: 5 User s ipu: 4 User s ipu: 0 Mehod callig is a comple process, maaged behid he scee Whe a mehod calls aoher, he caller s variables ad address are saved Whe he called mehod s, he saved values are re-isaiaed As far as he caller is cocered, everyhig is back o ormal. Iroducio o Compuer Sciece, Shimo Schocke slide 20

11 Ieger.oSrig() The ask: Wrie Wrie a a mehod mehod ha ha akes akes a a o-egaive ieger ieger ad ad oupus oupus is is decimal decimal represeaio usig usig characer characer oupu oupu Eample: Eample: Give Give he he ieger ieger ipu ipu 513, 513, oupu oupu he he srig srig "513" "513" The Uicode values of '0', '1', '2', '3',, '9' are 48, 49, 50, 51,, 57 Therefore, < 10, oe way o ge is Uicode value is (char)('0' ) Recursive hikig: Base case: If < 10 we oupu (char)('0' ) Reducio: a ieger of he form yyy y (each ad y beig a sigle digi) ca be reduced usig /10 ad yyy y %10 Assembly: The cocaeaio operaor ca be used o combie parial resuls io he fial value. Iroducio o Compuer Sciece, Shimo Schocke slide 21 Recursive implemeaio The ask: Wrie Wrie a a mehod mehod ha ha akes akes a a o-egaive ieger ieger ad ad oupus oupus is is decimal decimal represeaio usig usig characer characer oupu oupu Eample: Eample: Give Give he he ieger ieger ipu ipu 513, 513, oupu oupu he he srig srig "513" "513" class class PriIegerDemo mai(srig[] args) args) Sysem.ou.pri(oSrig(513)); Srig Srig osrig(i ) ) ( ( < 1 1 "" "" ((char) ((char) (48 (48 )); )); osrig( / 1 1 (char) (char) (48 (48 % 1; 1; Iroducio o Compuer Sciece, Shimo Schocke slide 22

12 Ru-ime aaomy osrig(5137) osrig(513) '7' 1 '513' 6 osrig(51) '3' Srig Srig osrig(i osrig(i ) ) ( ( < < 1 1 " " (char) (char) (48 (48 ); ); osrig( osrig( / / 1 1 (char) (char) (48 (48 % % 1; 1; 2 '51' 5 osrig(5) '1' 3 '5' 4 Base case: 5 < 10 Iroducio o Compuer Sciece, Shimo Schocke slide 23 Tower of Haoi The rules of he game: Move Move all all he he disks disks o o aoher spidle, so so ha: ha: (1) (1) oly oly oe oe disk disk moves a a a ime ime (2) (2) The The smaller disks disks are are always a a he he op op Iroducio o Compuer Sciece, Shimo Schocke slide 24

13 Tower of Haoi Sar: Fiish: The rules of he game: Move Move all all he he disks disks o o aoher spidle, so so ha: ha: (1) (1) oly oly oe oe disk disk moves a a a ime ime (2) (2) The The smaller disks disks are are always a a he he op op Iroducio o Compuer Sciece, Shimo Schocke slide 25 The recursive sraegy Sar: Move he op -1 disks from A o C Move he remaiig disk from A o B Move all he disks from C o B Fiish: Doe! Iroducio o Compuer Sciece, Shimo Schocke slide 26

14 Tower of Haoi implemeaio sar fiish emp 11 :: move move his his sigle sigle disk disk from from sar saro o fiish fiish > > 11 :: Move Move he he op op -1-1 disks disks from from sar saro o emp, emp, usig usig fiish fiish Move Move he he boom boom disk disk from from sar saro o fiish fiish Move Move he he op op -1-1 disks disks from from emp empo o fiish, fiish, usig usig sar sar class class TowerOfHaoi TowerOfHaoi mai(srig[] mai(srig[] args) args) movetower(3, movetower(3, "A", "A", "B", "B", "C"); "C"); Moves Moves a a ower ower of of disks disks from from A A o o B B usig usig C C movetower(i movetower(i,, Srig Srig sar, sar, Srig Srig fiish, fiish, Srig Srig emp) emp) (1) (1) Sysem.ou.pril(sar Sysem.ou.pril(sar "->" "->" fiish); fiish); movetower(-1, movetower(-1, sar, sar, emp, emp, fiish); fiish); Sysem.ou.pril(sar Sysem.ou.pril(sar "->" "->" fiish); fiish); movetower(-1, movetower(-1, emp, emp, fiish fiish,, sar); sar); Iroducio o Compuer Sciece, Shimo Schocke slide 27 Simulaio movetower(3, A, B, C ) Goal: Goal: move move a a ower ower of of size size 3 from from A o o B usig usig C: C: Goal: sar fiish emp Goal: move move a a ower ower of of size size 2 from from A o o C usig usig B: B: 3 "A" B" C" sar fiish emp Goal: Goal: move 2 move a a ower ower of "A" of size size 1 C" 1 from from A o o B B" usig usig C: C: Thigs Thigs o o do: do: sar fiish emp movetower(-1, sar, sar, emp, emp, fiish); fiish); Thigs Sysem.ou.pril(sar Thigs o o do: do: 1 "A" B" C" "->" "->" fiish); fiish); movetower(-1, movetower(-1, emp, sar, emp, sar, fiish emp, fiish emp,, sar); fiish); sar); fiish); Sysem.ou.pril(sar Thigs "->" "->" fiish); Thigs o o do: do: fiish); movetower(-1, emp, emp, fiish fiish, sar); movetower(-1, sar); sar, sar, emp, emp, fiish); fiish); Sysem.ou.pril(sar "->" "->" fiish); fiish); movetower(-1, emp, emp, fiish fiish, sar); sar); Iroducio o Compuer Sciece, Shimo Schocke slide 28

15 Simulaio movetower(3, A, B, C ) Goal: Goal: move move a a ower ower of of size size 3 from from A o o B usig usig C: C: Goal: sar fiish emp Goal: move move a a ower ower of of size size 2 from from A o o C usig usig B: B: 3 "A" B" C" sar fiish emp 2 "A" C" B" Thigs Thigs o o do: do: movetower(-1, sar, sar, emp, emp, fiish); fiish); Thigs Sysem.ou.pril(sar Thigs o o do: do: "->" "->" fiish); fiish); movetower(-1, movetower(-1, emp, sar, emp, sar, fiish emp, fiish emp,, sar); fiish); sar); fiish); Sysem.ou.pril(sar "->" "->" fiish); fiish); movetower(-1, emp, emp, fiish fiish, sar); sar); Iroducio o Compuer Sciece, Shimo Schocke slide 29 Simulaio movetower(3, A, B, C ) Goal: Goal: move move a a ower ower of of size size 3 from from A o o B usig usig C: C: Goal: sar fiish emp Goal: move move a a ower ower of of size size 2 from from A o o C usig usig B: B: 3 "A" B" C" sar fiish emp 2 "A" C" B" Thigs Thigs o o do: do: movetower(-1, sar, sar, emp, emp, fiish); fiish); Thigs Sysem.ou.pril(sar Thigs o o do: do: "->" "->" fiish); fiish); movetower(-1, movetower(-1, emp, sar, emp, sar, fiish emp, fiish emp,, sar); fiish); sar); fiish); Sysem.ou.pril(sar "->" "->" fiish); fiish); movetower(-1, emp, emp, fiish fiish, sar); sar); Iroducio o Compuer Sciece, Shimo Schocke slide 30

16 Simulaio (TBD) movetower(3, A, B, C ) Goal: Goal: move move a a ower ower of of size size 3 from from A o o B usig usig C: C: Goal: sar fiish emp Goal: move move a a ower ower of of size size 2 from from A o o C usig usig B: B: 3 "A" B" C" sar fiish emp 2 "A" C" B" Thigs Thigs o o do: do: movetower(-1, sar, sar, emp, emp, fiish); fiish); Thigs Sysem.ou.pril(sar Thigs o o do: do: "->" "->" fiish); fiish); movetower(-1, movetower(-1, emp, sar, emp, sar, fiish emp, fiish emp,, sar); fiish); sar); fiish); Sysem.ou.pril(sar "->" "->" fiish); fiish); movetower(-1, emp, emp, fiish fiish, sar); sar); Iroducio o Compuer Sciece, Shimo Schocke slide 31 Permuaios The ask: Wrie a mehod lispermuaios(srig s) s) ha ha geeraes a complee se se of of all all he he permuaios of of he he characers i i s. s. For For eample, lispermuaios( abcd ) will will geerae: abcd abcd bacd bacd cabd cabd dabc dabc abdc abdc badc badc cadb cadb dacb dacb acbd acbd bcad bcad cbad cbad dbac dbac acdb acdb bcda bcda cbda cbda dbca dbca adbc adbc bdac bdac cdab cdab dcab dcab adcb adcb bdca bdca cdba cdba dcba dcba (oe (oe afer afer he he oher) Observaios: If legh(s), here are (-1) (-2) 1! permuaios he lis of permuaios sarig wih a coais he prefi a followed by all he permuaios of bcd Iroducio o Compuer Sciece, Shimo Schocke slide 32

17 Sraegy Give Give "abcd" "abcd" we we have have o o geerae: abcd abcd bacd bacd cabd cabd dabc dabc abdc abdc badc badc cadb cadb dacb dacb acbd acbd bcad bcad cbad cbad dbac dbac acdb acdb bcda bcda cbda cbda dbca dbca adbc adbc bdac bdac cdab cdab dcab dcab adcb adcb bdca bdca cdba cdba dcba dcba Pri 'a' all permuaios of "abcd" wihou chara 0 Pri 'b' all permuaios of "abcd wihou chara 1 Pri 'c' all permuaios of "abcd wihou chara 2 Pri 'd' all permuaios of "abcd wihou chara 3 Srig Srig wihou wihou chara(i): res res s.subsrig(0,i) s.subsrig(i1, s.legh()); Iroducio o Compuer Sciece, Shimo Schocke slide 33 Implemeaio class class LisPermuaiosDemo LisPermuaiosDemo mai(srig[] mai(srig[] args) args) lispermuaios("abcd"); lispermuaios("abcd"); lispermuaios(srig lispermuaios(srig s) s) lispermuaios("", lispermuaios("", s); s); privae privae lispermuaios(srig lispermuaios(srig prefi, prefi, Srig Srig s) s) (s.legh() (s.legh() Sysem.ou.pril(prefi); Sysem.ou.pril(prefi); for for (i (i i0 i0 ; ; i<s.legh() i<s.legh() ; ; i) i) char char ch ch s.chara(i); s.chara(i); Srig Srig res res s.subsrig(0,i) s.subsrig(0,i) s.subsrig(i1); s.subsrig(i1); lispermuaios(prefi lispermuaios(prefi ch, ch, res); res); Iroducio o Compuer Sciece, Shimo Schocke slide 34

18 Fracals Fracal: A geomeric shape ha ca be spli io pars, each of which is (eiher eacly or approimaely) a reduced-size copy of he whole Some well-kow fracal shapes: Sierpiski riagles Koch sowflake Typical geeraive sraegy: Iroducio o Compuer Sciece, Shimo Schocke slide 35 TurleFracal DrawFracal(500, 1) DrawFracal(500, 2) DrawFracal(500, 3) DrawFracal(500, 4) Iroducio o Compuer Sciece, Shimo Schocke slide 36

19 Implemeaio class class TurleFracal Turle Turle urle urle ew ew Turle(); Turle(); mai(srig[] args) args) urle.aildow(); drawfracal(500,2); drawfracal(i legh, legh, i i level) level) (level urle.moveforward(legh); drawfracal(legh/3, level-1); level-1); urle.urlef(6; drawfracal(legh/3, level-1); level-1); urle.urrigh(12; drawfracal(legh/3, level-1); level-1); urle.urlef(6; drawfracal(legh/3, level-1); level-1); Iroducio o Compuer Sciece, Shimo Schocke slide 37 Fracals Iroducio o Compuer Sciece, Shimo Schocke slide 38