Professor Donald J. Patterson - Call Pattern of Recursive Fib() …djp3.westmont.edu/classes/2016_09_CS010/Lectures/Lecture... · 2016-11-18 · ch11.pptx Author: Donald Patterson

Call Pattern of Recursive Fib() Function

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Efficiency of Recursion •  Thefunc)ontakessolongbecauseitcomputesthesamevaluesover

andover.

•  Computa)onoffib(6)callsfib(3)three)mes.

•  Imitatethepencil-and-paperprocesstoavoidcompu)ngthevaluesmorethanonce.

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Efficiency of Recursion •  Occasionally,arecursivesolu)onrunsmuchslowerthanitsitera)vecounterpart.

•  Inmostcases,therecursivesolu)onisonlyslightlyslower.

•  Theitera)veisPalindrome()performsonlyslightlybe@erthanrecursivesolu)on.

•  Eachrecursivefunc.oncalltakesacertainamountofprocessor.me

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Loopfib.py(1)

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Loopfib.py(2)

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Memoized Fibonacci

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Efficiency of Recursion •  Smartcompilerscanavoidrecursivefunc)oncallsiftheyfollowsimplepa@erns.

•  Mostcompilersdon’tdothat

•  Inmanycases,arecursivesolu)oniseasiertounderstandandimplementcorrectlythananitera)vesolu)on.

•  ‘Toiterateishuman,torecursedivine.’-L.PeterDeutsch

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Iterative IsPalindrome()Function

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defisPalindrome(text):start=0end=len(text)-1whilestart<end:first=text[start].lower()last=text[end].lower()iffirst.isalpha()andlast.isalpha():#Bothareletters.iffirst==last:start=start+1end=end-1else:returnFalseifnotlast.isalpha()end=end-1

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11.5 Permutations •  Designaclassthatwilllistallpermuta)onsofstring,whereapermuta)onisarearrangementofthele@ers

•  Thestring"eat"hassixpermuta)ons:•  "eat"•  "eta"•  "aet"•  "ate"•  "tea"•  "tae"

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Generate All Permutations (1) •  Generateallpermuta)onsthatstartwith'e',then'a',then't'

•  Thestring"eat"hassixpermuta)ons:•  "eat"•  "eta"•  "aet"•  "ate"•  "tea"•  "tae"

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Generate All Permutations (2) •  Generateallpermuta)onsthatstartwith'e',then'a',then't'

•  Togeneratepermuta)onsstar)ngwith'e',weneedtofindallpermuta)onsof"at"

•  Thisisthesameproblemwithsimplerinputs

•  Userecursion

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Implementing Permutations() Function

•  Loopthroughallposi)onsinthewordtobepermuted

•  Foreachofthem,computetheshorterwordobtainedbyremovingtheithle@er:

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shorter=word[:i]+word[i+1:]

shorterPermutations=permutations(shorter)

•  Computethepermuta)onsoftheshorterword:

Implementing Permutations() Function •  Addtheremovedle@ertothefrontofallpermuta)onsoftheshorterword:

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forsinshorterPermutations:result.append(word[i]+s)

•  Specialcaseforthesimpleststring,theemptystring,whichhasasinglepermuta)on-itself

Permutations.py (1)

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Permutations.py (2)

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Backtracking •  Backtrackingexaminespar)alsolu)ons,abandoningunsuitableonesandreturningtoconsiderothercandidates

•  Canbeusedto•  solvecrosswordpuzzles

•  escapefrommazes

•  findsolu)onstosystemsthatareconstrainedbyrules

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Backtracking Characteristic Properties 1.  Aproceduretoexamineapar)alsolu)onanddeterminewhether

to:

I.  acceptitasanactualsolu)onor,

II.  abandonit(becauseiteitherviolatessomerulesorcanneverleadtoavalidsolu)on)

2.  Aproceduretoextendapar)alsolu)on,genera)ngoneormoresolu)onsthatcomeclosertothegoal

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Recursive Backtracking Algorithm Solve(partialSolution)

Examine(partialSolution).

Ifaccepted

AddpartialSolutiontothelistofsolutions.

Elseifnotabandoned

Foreachpinextend(partialSolution)

Solve(p)

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Eight Queens Problem •  Problem:posi)oneightqueensonachessboardsothatnoneofthema@acksanotheraccordingtotherulesofchess

•  Asolu)on:

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Eight Queens Problem •  Easytoexamineapar)alsolu)on:

•  Iftwoqueensa@ackoneanother,rejectit•  Otherwise,ifithaseightqueens,acceptit•  Otherwise,con)nue

•  Easytoextendapar)alsolu)on:•  Addanotherqueenonanemptysquare

•  Systema)cextensions:

•  Placefirstqueenonrow1•  Placethenextonrow2•  Etc.

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Function: Examine()

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defexamine(partialSolution):foriinrange(0,len(partialSolution)):forjinrange(i+1,len(partialSolution)):ifattacks(partialSolution[i],partialSolution[j]):returnABANDONiflen(partialSolution)==NQUEENS:returnACCEPTelse:returnCONTINUE

Function: Extend()

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defextend(partialSolution):results=[]row=len(partialSolution)+1forcolumnin"abcdefgh":newSolution=list(partialSolution)newSolution.append(column+str(row))results.append(newSolution)returnresults

Diagonal Attack •  Todeterminewhethertwoqueensa@ackeachotherdiagonally:

•  Checkwhetherslopeis±1

(row2–row1)/(column2–column1)=±1

row2–row1=±(column2–column1)

row2–row1|=|column2–column1|

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Backtracking in the Four Queens Problem (1)

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Backtracking in the Four Queens Problem (2)

•  Star)ngwithablankboard,fourpar)alsolu)onswithaqueeninrow1

•  Whenthequeenisincolumn1,fourpar)alsolu)onswithaqueeninrow2

•  Twoareabandonedimmediately

•  Othertwoleadtopar)alsolu)onswiththreequeensand,allbutoneofwhichareabandoned

•  Onepar)alsolu)onisextendedtofourqueens,butallofthoseareabandonedaswell

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Queens.py

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Queens.py

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Queens.py

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11.7 Mutual Recursion •  Problem:Computethevalueofarithme)cexpressionssuchas:

3 + 4 * 5 (3 + 4) * 5 1 - (2 - (3 - (4 - 5)))

•  Compu)ngtheexpressioniscomplicated

•  *and/ bindmorestronglythan+ and–

•  Parenthesescanbeusedtogroupsub-expressions

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Syntax Diagrams for Evaluating an Expression

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Mutual Recursion •  Anexpressioncanbebrokendownintoasequenceofterms,separatedby+or–

•  Eachtermisbrokendownintoasequenceoffactors,separatedby*or/

•  Eachfactoriseitheraparenthesizedexpressionoranumber

•  Thesyntaxtreesrepresentwhichopera)onsshouldbecarriedoutfirst

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Syntax Trees for Two Expressions

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Mutual Recursion •  Inamutualrecursion,asetofcoopera)ngfunc)onscallseachotherrepeatedly

•  Tocomputethevalueofanexpression,implement3func)onsthatcalleachotherrecursively:

•  expression()

•  term()

•  factor()

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Function: Expression()

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defexpression(tokens):value=term(tokens)done=Falsewhilenotdoneandlen(tokens)>0:next=tokens[0]ifnext=="+"ornext=="-":tokens.pop(0)#Discard"+"or"-"value2=term(tokens)ifnext=="+":value=value+value2else:value=value-value2else:done=Truereturnvalue

Function: Term()•  Theterm()func)oncallsfactor()inthesameway,mul)plyingordividingthefactorvalues

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defterm(tokens):value=factor(tokens)done=Falsewhilenotdoneandlen(tokens)>0:next=tokens[0]ifnext=="*"ornext=="/":tokens.pop(0)value2=factor(tokens)ifnext=="*":value=value*value2else:value=value/value2else:done=Truereturnvalue

Function: Factor()

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deffactor(tokens):next=tokens.pop(0)ifnext=="(":value=expression(tokens)tokens.pop(0)#Discard")"else:value=nextreturnvalue

Trace (3+4)*5

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To see the mutual recursion clearly, trace through the expression (3+4)*5:

•  expression()callsterm()

•  term()callsfactor()

•  factor()consumesthe(input

•  factor()callsexpression()

•  expression()returnseventuallywiththevalueof7,havingconsumed3+4.Thisistherecursivecall.

•  factor()consumesthe)input

•  factor()returns7

•  term()consumestheinputs*and5andreturns35

•  expression()returns35

Evaluator.py (1)

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Evaluator.py

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Evaluator.py

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Evaluator.py

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Summary •  Understandthecontrolflowinarecursivecomputa)on.

•  Arecursivecomputa)onsolvesaproblembyusingthesolu)ontothesameproblemwithsimplerinputs.

•  Forarecursiontoterminate,theremustbespecialcasesforthesimplestvalues.

•  Designarecursivesolu)ontoaproblem.

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Summary •  Iden)fyrecursivehelperfunc)onsforsolvingaproblem.

•  Some)mesitiseasiertofindarecursivesolu)onifyoumakeaslightchangetotheoriginalproblem.

•  Contrasttheefficiencyofrecursiveandnon-recursivealgorithms.•  Occasionally,arecursivesolu)onrunsmuchslowerthanitsitera)vecounterpart.However,inmostcases,therecursivesolu)onisonlyslightlyslower.

•  Inmanycases,arecursivesolu)oniseasiertounderstandandimplementcorrectlythananitera)vesolu)on.

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Summary •  Reviewacomplexrecursionexamplethatcannotbesolvedwithasimpleloop.•  Thepermuta)onsofastringcanbeobtainedmorenaturallythroughrecursionthanwithaloop.

•  Usebacktrackingtosolveproblemsthatrequiretryingoutmul)plepaths.•  Backtrackingexaminespar)alsolu)ons,abandoningunsuitableonesandreturningtoconsiderothercandidates.

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Summary •  Recognizethephenomenonofmutualrecursioninanexpressionevaluator.•  Inamutualrecursion,coopera)ngfunc)onsormethodscalleachotherrepeatedly.

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