Chapter 11 · Chapter Goals • To learn to “think recursively” • To be able to use recursive...

RECURSION

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Chapter 11

Chapter Goals •  Tolearnto“thinkrecursively”•  Tobeabletouserecursivehelperfunc5ons

•  Tounderstandtherela5onshipbetweenrecursionanditera5on•  Tounderstandwhentheuseofrecursionaffectstheefficiencyofanalgorithm

•  Toanalyzeproblemsthataremucheasiertosolvebyrecursionthanbyitera5on

•  Toprocessdatawithrecursivestructuresusingmutualrecursion

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Contents •  TriangleNumbersRevisited

•  ProblemSolving:ThinkingRecursively

•  RecursiveHelperFunc5ons•  TheEfficiencyofRecursion

•  Permuta5ons

•  Backtracking•  MutualRecursion

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11.1 Triangle Numbers Revisited •  Triangleshapeofsidelength4:[]

[][]

[][][] [][][][]

•  Willuserecursiontocomputetheareaofatriangleofwidthn,assumingeach[] squarehasanareaof1

•  Alsocalledthenthtrianglenumber

•  Thethirdtrianglenumberis6,thefourthis10

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Handling Triangle of Width 1 •  Thetriangleconsistsofasinglesquare•  Itsareais1•  Takecareofthiscasefirst:

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deftriangleArea(sideLength):ifsideLength==1:return1...

Handling The General Case •  Assumeweknowtheareaofthesmaller,coloredtriangle:

[] [][] [][][] [][][][]

•  Areaoflargertrianglecanbecalculatedas

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area=smallerArea+sideLength

smallerSideLength=sideLength-1smallerArea=triangleArea(smallerSideLength)

•  Togettheareaofthesmallertriangle

•  CallthetriangleArea()func5on:

Computing the Area of a Triangle With Width 4

•  triangleArea()func5onmakesasmallertriangleofwidth3

•  ItcallstriangleArea()onthattriangle

•  Thatfunc5onmakesasmallertriangleofwidth2


•  Thatfunc5onmakesasmallertriangleofwidth1


•  Thatfunc5onreturns1

•  Thefunc5onreturnssmallerArea+sideLength =1+2=3

•  Thefunc5onreturnssmallerArea+sideLength=3+3=6

•  Thefunc5onreturnssmallerArea+sideLength=6+4=10

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Recursive Computation •  Arecursivecomputa.onsolvesaproblembyusingthesolu5ontothesameproblemwithsimplerinputs

•  CallpaWernofarecursivefunc.oniscomplicated

•  Key:Don’tthinkaboutit

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Successful Recursion •  Everyrecursivecallmustsimplifythecomputa5oninsomeway

•  Theremustbespecialcasestohandlethesimplestcomputa5onsdirectly

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Other Ways to Compute Triangle Numbers

•  Theareaofatriangleequalsthesum:

1+2+3+...+sideLength

•  Usingasimpleloop:

area=0;foriinrange(1,(sideLength+1),1):area=area+i

•  Usingmath:

1+2+...+n=n×(n+1)/2=>n*(n+1)/2

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Trianglenumbers.py

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Special Topic 11.1 •  AnObject-Orientedversionofthetrianglesprogram

•  Generalcase:computetheareaofthelargertriangleassmallerArea+self._sideLength.

•  Togetthesmallerarea:

classTriangledef__init__(self,sideLength):self._sideLength=sideLengthdefgetArea(self):ifself._sideLength==1:return1...

smallerTriangle=Triangle(self._sideLength-1)smallerArea=smallerTriangle.getArea()area=smallerArea+self._sideLength

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Common Error 11.1 •  Infiniterecursion:

•  Afunc5oncallingitselfoverandoverwithnoendinsight.•  Thecomputerneedssomeamountofmemoryforbookkeepingduringeachcall.

•  Adersomenumberofcalls,allmemorythatisavailableforthispurposeisexhausted.

•  Yourprogramshutsdownandreportsa“stackoverflow”.

•  Causes:•  Theargumentsdon’tgetsimplerorbecauseaspecialtermina5ngcaseismissing.

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Thinking Recursively •  Problem:Testwhetherasentenceisapalindrome

•  Palindrome:Astringthatisequaltoitselfwhenyoureverseallcharacters•  Aman,aplan,acanal–Panama!•  Gohangasalami,I’malasagnahog•  Madam,I’mAdam

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Implement IsPalindrome() Function

##Testswhetherastringisapalindrome.#@paramtextastringthatisbeingchecked#@returnTrueiftextisapalindrome,Falseotherwise#defisPalindrome(text):...

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Thinking Recursively: Step 1 •  Considervariouswaystosimplifyinputs.

•  Severalpossibili5es:

•  Removethefirstcharacter

•  Removethelastcharacter

•  Removeboththefirstandlastcharacters

•  Removeacharacterfromthemiddle

•  Cutthestringintotwohalves

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Thinking Recursively: Step 2 (1) •  Combinesolu5onswithsimplerinputsintoasolu5onoftheoriginal

problem.

•  Mostpromisingsimplifica5on:Removebothfirstandlastcharacters.

•  “adam,I’mAda”isapalindrometoo!

•  Thus,awordisapalindromeif

•  ThefirstandlastleGersmatch,and

•  WordobtainedbyremovingthefirstandlastleGersisapalindrome

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Thinking Recursively: Step 2 (2) •  WhatiffirstorlastcharacterisnotaleWer?Ignoreit

•  IfthefirstandlastcharactersareleGers,checkwhethertheymatch;ifso,removebothandtestshorterstring

•  Iflastcharacterisn’taleGer,removeitandtestshorterstring

•  Iffirstcharacterisn’taleGer,removeitandtestshorterstring

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Thinking Recursively: Step 3 •  Findsolu5onstothesimplestinputs.

•  Stringswithtwocharacters

•  Nospecialcaserequired;steptwosMllapplies

•  Stringswithasinglecharacter

•  Theyarepalindromes

•  Theemptystring

•  Itisapalindrome

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Thinking Recursively: Step 4 (1) •  Implementthesolu5onbycombiningthesimplecasesandthe

reduc5onstep.

Continued

defisPalindrome(text):length=len(text)#Separatecaseforshorteststrings.iflength<=1:returnTrueelse:#Getfirstandlastcharacters,convertedto#lowercase.first=text[0].lower()last=text[length-1].lower()

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Thinking Recursively: Step 4 (2)

#Nonbasecaseiffirst.isalpha()andlast.isalpha():#Bothareletters.iffirst==last:#Removebothfirstandlastcharacter.shorter=text[1:length-1]returnisPalindrome(shorter)else:returnFalseelifnotlast.isalpha():#Removelastcharacter.shorter=text[0:length-1]returnisPalindrome(shorter)else:#Removefirstcharacter.shorter=text[1:length]returnisPalindrome(shorter)

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Recursive Helper functions •  Some5mesitiseasiertofindarecursivesolu5onifyoumakeaslightchangetotheoriginalproblem.

•  Considerthepalindrometestofprevioussec5on.

•  Itisabitinefficienttoconstructnewstringobjectsineverystep.

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Substring Palindromes (1) •  Ratherthantes5ngwhetherthesentenceisapalindrome,checkwhetherasubstringisapalindrome:

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##Recursivelytestswhetherasubstringis#apalindrome.#@paramtextastringthatisbeingchecked#@paramstarttheindexofthefirstcharacterofthesubstring#@paramendtheindexofthelastcharacterofthesubstring#@returnTrueifthesubstringisapalindrome#defsubstringIsPalindrome(text,start,end):

Substring Palindromes (2) •  Then,simplycallthehelperfunc5onwithposi5onsthattesttheen5restring:

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defisPalindrome(text):returnsubstringIsPalindrome(text,0,len(text)–1)

Recursive Helper function

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Continued

defsubstringIsPalindrome(text,start,end):#Separatecaseforsubstringsoflength0and1.ifstart>=end:returnTrueelse:#Getfirstandlastcharacters,convertedtolowercase.first=text[start].lower()last=text[end].lower()

Recursive Helper Function

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iffirst.isalpha()andlast.isalpha():iffirst==last:#Testsubstringthatdoesn’tcontainthematching#letters.returnsubstringIsPalindrome(text,start+1,end-1)else:returnFalseelifnotlast.isalpha():#Testsubstringthatdoesn’tcontainthelastcharacter.returnsubstringIsPalindrome(text,start,end-1)else:#Testsubstringthatdoesn’tcontainthefirst#character.returnsubstringIsPalindrome(text,start+1,end)

Chapter 11 · Chapter Goals • To learn to “think recursively” • To be able to use recursive...

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