2016 Algorithmics Written examination · C. Exhaustive testing of the function requires six test...

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ALGORITHMICS (HESS)Written examination

Monday 31 October 2016 Reading time: 3.00 pm to 3.15 pm (15 minutes) Writing time: 3.15 pm to 5.15 pm (2 hours)

QUESTION AND ANSWER BOOK

Structure of bookSection Number of

questionsNumber of questions

to be answeredNumber of

A 20 20 20B 17 17 80

Total 100

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulersandonescientificcalculator.

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionandanswerbookof30pages.• Answersheetformultiple-choicequestions.

Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice

questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• AllwrittenresponsesmustbeinEnglish.

At the end of the examination• Placetheanswersheetformultiple-choicequestionsinsidethefrontcoverofthisbook.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2016

STUDENT NUMBER

Letter

2016ALGORITHMICSEXAM 2

SECTION A – continued

Question 1Considerthefollowingpseudocode,whereaandnarepositiveintegers.

b = -1c = ai = 0while(i is less than or equal to n) c = c * b i = i + 1endwhile

Whichoneofthefollowingstatementscorrectlydescribesthevalueofcafterthealgorithmisexecuted?A. cisundefinedB. cisequalto–1C. cisequalto–aifnisoddD. cisequalto–aifniseven

Question 2WhichoneofthefollowingstatementsaboutFloyd-Warshall’salgorithmrunningonagraphwithVnodesandEedgesiscorrect?A. TherecursiveversionfindsthetransitiveclosureofagraphinO(3V)time.B. Theiterative(dynamicprogramming)versionfindstheshortestpathbetweenallpairsofnodesin

O(V 3)time.C. Theiterative(dynamicprogramming)versionfindstheshortestpathbetweenallpairsofnodesin

O(3E)time.D. Theiterative(dynamicprogramming)versionalwaysfindsaminimalspanningtreerootedatevery

nodeinO(V 3)time.

Question 3Considerthefollowingthreeparametersofafunction:• row,whichcantakethevalues1or2• column,whichcantakethevalues‘a’or‘b’or‘c’• sound,whichcantakethevaluesTrueorFalse

Whichoneofthefollowingstatementsistrue?A. Pair-wisetestingofthefunctionrequiressixtestcases.B. Pair-wisetestingofthefunctionrequires12testcases.C. Exhaustivetestingofthefunctionrequiressixtestcases.D. Exhaustiveblack-boxtestingofthefunctionrequires7!testcases.

SECTION A – Multiple-choice questions

Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrectorthatbest answersthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.

3 2016ALGORITHMICSEXAM

SECTION A – continuedTURN OVER

Question 4Considerthefollowingfunctionf(x,y)thattakestwointegersasinput.

f(x,y)begin if (x < 0 or y < 0) if (x > y) a = y else a = x endif else if (x > y) a = x else a = y endif endif

return aend

Whichoneofthefollowingsetsoftestcasescoversallpathsofthefunction?A. {(x=–3,y=4), (x=–2,y=6), (x=3,y=6), (x=3,y=1)}B. {(x=–3,y=–4), (x=–2,y=6), (x=3,y=6), (x=3,y=1)}C. {(x=–3,y=–4), (x=–2,y=6), (x=3,y=6), (x=3,y=3)}D. {(x=–3,y=–4), (x=–2,y=6), (x=3,y=–6), (x=3,y=1)}

Question 5Whenconsideringdifferenttypesofalgorithms,whichoneofthefollowingstatementsistrue?A. Divideandconqueralgorithmsarealwaysfasterthangreedyalgorithmsforthesameproblem.B. Greedyalgorithmsarealwaysfasterthandivideandconqueralgorithmsforthesameproblem.C. Greedyalgorithmsgivegoodapproximateanswerstoproblems,butneverthebestpossibleanswer.D. Brute-forcealgorithmscanneverbefasterthanawell-designedgreedyalgorithmforthesameproblem.

Question 6Alargenumberofpeoplewillbecompletinganonlinesurvey.Eachsurveyresponsewillbestoredintheorderitissubmitted.Whichabstractdatatype(ADT)wouldbethemostappropriateforstoringthesurveyresponses?A. graphB. queueC. stackD. dictionary

Use the following information to answer Questions 7 and 8.

Question 7Adepth-firstsearchtraversalofthegraphabovewillvisitthenodesinwhichoneofthefollowingorders?(Alphabeticalorderisusedwhenthereismorethanoneoption.)A. ABDGEFCHB. ABDEFCHGC. ABEGFCHDD. ABDFCHEG

Question 8Abreadth-firstsearchtraversalofthegraphabovewillvisitthenodesinwhichoneofthefollowingorders?(Alphabeticalorderisusedwhenthereismorethanoneoption.)A. ABEGFCHDB. ABDGFECHC. ABDGEFCHD. AGBDEFCH

Question 9Aconnected,undirectedgraphwithdistinctedgeweightshasmaximumedgeweightemaxandminimumedgeweightemin.Whichoneofthefollowingstatementsisfalse?A. emaxisnotinanyminimalspanningtree.B. Everyminimalspanningtreeofthegraphmustcontainemin.C. Prim’salgorithmwillgenerateauniqueminimalspanningtree.D. Ifemaxisinaminimalspanningtree,itsremovalwilldisconnectthegraph.

Question 10Dijkstra’ssingle-sourceshortestpathalgorithminanundirectedgraphreportsdistancesfromthesourcetoeachnode.ThesedistancesA. aretheshortestpossibledistancestoeverydestinationnode.B. arenevertheshortestpossibledistanceswhennegativeedgeweightsarepresent.C. maybetheshortestpossibledistanceswhennegativeedgeweightsarepresent.D. maynotalwaysbetheshortestpossibledistanceswhenalledgeweightsarepositive.

Question 11Considerthefollowingfouralgorithms,operatingonagraphwithV nodesandE edges:1. Floyd-Warshall’salgorithmfortransitiveclosure2. Bellman-Ford’salgorithmforthesingle-sourceshortestpathproblem3. depth-firsttraversalalgorithm4. Dijkstra’salgorithmforthesingle-sourceshortestpathproblem

Thetimecomplexitiesofthesealgorithms,inorder,areA. O(V 3),O(VE),O(V 2),O(V+E)B. O(V+E),O(V 3),O(VE),O(V 2)C. O(V 3),O(VE),O(V+E),O(V 2)D. O(VE),O(V+E),O(V 3),O(V 2)

Question 12

Algorithm AInput: two nodes X, YOutput: “yes” if successful, “no” otherwise

Let L = an empty listAppend X to L

while(L is not empty) Let Z = first node in L if(Z = Y)then return “yes” else Remove Z from L for every neighbour N connected to Z if(N.visited = False and N is not in L) Append N to the end of L endif endfor Z.visited = True endifendwhile

return “no”

ThealgorithmaboveusesA. breadth-firstsearch.B. depth-firstsearch.C. mergesort.D. minimax.

Question 13AfterkiterationsofBellman-Ford’salgorithminagraphwithnnodes(1<k<n–1),whichstatementistrueabouttheshortestpathsfromthesourcenodetoeveryothernode?A. Theshortestpathswithatmostkedgeshavebeenfound.B. Theshortestpathswithatmostk–1edgeshavebeenfound.C. Theshortestpathswithatmostk+1edgeshavebeenfound.D. Theshortestpathsforanynumberofedgescanchangeasthealgorithmhasnotcompleted.

Question 14ThePageRankalgorithmusestwoattributestocalculatethepopularityofawebpage.Whichoneofthefollowingstatementsbestdescribestheseattributesforcalculatingawebpage’sPageRank?A. thenumberofoutboundandinboundlinksofawebpageB. theprobabilityofarandomdirectlandingonawebpageandthesizeofthewebpageC. thenumberofoutboundlinksofawebpageandtheprobabilityofarandomdirectlandingonthe

webpageD. thenumberofinboundlinksofawebpageandtheprobabilityofarandomdirectlandingonthe

webpage

Use the following information to answer Questions 15 and 16.BlackboxInc.istryingtosellnewtimetablingsoftwaretoaschool.Thefollowingdatashowsrunningtimes(inmicroseconds)forhowthenewtimetablingsoftware’salgorithmcompareswithastandardbaseline,wheren isameasureofthesizeofthetimetablingproblem.

n Baseline New algorithm

10 105 53

100 1998 103

1000 29 902 152

10 000 398636 202

Question 15WhichBig-Oexpressionmostcloselydescribestherunningtimeofthebaseline?A. O(n)B. O(n2)C. O(logn)D. O(nlogn)

Question 16WhichBig-Oexpressionmostcloselydescribestherunningtimeofthenewalgorithm?A. O(n)B. O(n2)C. O(logn)D. O(nlogn)

Question 17InthecontextofAlgorithmics,adecisionproblemisA. aproblemwithonlytwopossiblesolutions.B. aproblemwithmorethantwopossiblesolutions.C. aproblemforwhichallknownalgorithmsdonotterminate.D. undecidableaccordingtotheChurch-TuringthesisbecauseitdoesnotterminateonaTuringmachine.

Use the following information to answer Questions 18 and 19.ATuringmachineisconfiguredwiththeinstructionsrepresentedinthestatediagrambelow.

blank/blank:R blank/blank:R

blank/1:L

haltq0

Eachedgeislabelledi / j : k,where:• iistheinput• jistheoutput• kisthedirectiontheheadmoves(L=left,R=right)aftertheoutput.

Themachinebeginsinstateq0.Themachineisgiventhefollowingtape.Forthismachine,thetaperemainsstationarywhiletheheadmoves.

0 1 0 0 1 0 1

Question 18TheTuringmachineisrunwiththetape.WhentheTuringmachinehalts,theappearanceofthetapeandthepositionoftheheadareasfollows.Thearrowshowstheendingpointofthehead.

1 1 0 0 1 0 1

WhichoneofthefollowingbestrepresentsthepositionoftheheaddirectlybeforetheTuringmachinestarted?

0 1 0 0 1 0 1

Question 19Thearrowbelowshowsthestartingpointoftheheadbeforethemachineisrunagain.

0 1 0 0 1 0 1

WhentheTuringmachinehalts,thenumberofstepsthattheheadwillhavemovedisA. 6B. 7C. 9D. 10

END OF SECTION A

Question 20AstudentbeginstowritethefollowingsignatureforaqueueADT.

name queueimport element, booleanoperations empty : → queue isEmpty : queue → boolean

peek : queue → element

enqueue : queue × element → queue

dequeue :

Whichoneofthefollowingisthecorrectrepresentationofthedequeueoperation?A. dequeue : queue → queue

B. dequeue : queue → boolean

C. dequeue : element → queue

D. dequeue : queue → element

SECTION B – continuedTURN OVER

Question 1 (3marks)Explainhowrandomisedheuristicscanhelpovercomethesoftlimitsofcomputation.Useanexampleaspartofyourexplanation.

Question 2 (3marks)Alaboratoryisworkingona3D-printedstructurethatusesanewexperimentalmaterial.Thematerialismadeupofsixseparatecomponentsandneedstobestructuredinsuchawaythatnocomponentconnectstoanothercomponentofthesametype.Ifcomponentsofthesametypedoconnect,thestructurewillcollapse.Theconnectionsbetweenthecomponentscanbetreatedasagraph.

Describeanapproachthatcouldbeusedinreasonabletimetoensurethatstructuresofanysizewouldnotcollapse.

SECTION B

Instructions for Section BAnswerallquestionsinthespacesprovided.

SECTION B – continued

Question 3 (3marks)Acompanymakessyntheticdiamondstoselltocustomerswhousetheminhigh-poweredlasers.Thecompanyhasreceivedcomplaintsfromitscustomersaboutsomeofthediamondsbeingfaulty,makingthelasersunsafetouse.Oninvestigating,atechniciannoticesthatthefaultydiamondsweighafewmicrogramslessthanthediamondsthatarenotfaultyandthateachbatchofdiamondsisproducingexactlyonefaultydiamond.Asthediamondsareproducedinverylargebatches,itwouldnotbefeasibletosimplyweigheachdiamonduntilthefaultyoneisfound.Themanagerwouldliketoputinplaceamethodtofindfaultydiamonds.

Stateanappropriatealgorithmdesignpatternthatthemanagercouldusetosuccessfullyfindfaultydiamondsandexplainhowthatalgorithmwillsolvetheproblemefficiently.

Question 4 (2marks)Whendataistransferredacrosscomputernetworks,itisfirstbrokenupintopackets.Computernetworktrafficisnormallyprocessedintheorderthatpacketsarriveateachdevicealongthepathbetweencommunicatingdevices.Packetsmaybepiecesofemail,webcontent,voiceorvideo.Whilesometraffic,suchasemailorwebcontent,canwithstanddelaysindelivery,others,suchasvoiceandvideo,cannothavedelays;thesepacketscannotwaitateachdeviceforothertraffictobeprocessedaheadofthem.

Describeastandardabstractdatatype(ADT)thatcouldbeusedtomanagethepacketsarrivingatacomputer.

Question 5 (2marks)Anundirectedgraph,G,ispossiblydisconnected.

Howcanadepth-firstsearchalgorithmbemodifiedtocheckfordisconnectedness?

SECTION B – Question 6–continued

Question 6 (6marks)Thefollowinggraphrepresentslinksbetweenwebpages.

ThePageRankofPageAisgivenby

( ) = −( )+

( )( )

+( )( )

wherePR(x)isthePageRankofPagex,NisthenumberofpagesinthisnetworkandL(x)isthenumberofoutgoinglinksfromPagex.

a. ExplainthepurposeofdinthePageRank. 2marks

b. Whatdoes1−( )dN

representinthePageRank? 1mark

c. Whatdoes dBB

( )( )

representinthePageRank? 1mark

d. Anewpage,E,isaddedtothegraphasanode,shownbelow.

ExplainhowthePageRankwouldincludenodeEiftherearenooutboundlinksfromPageE. 2marks

Question 7 (3marks)AstudentrunsBellman-Ford’ssingle-sourceshortestpathalgorithmonthefollowingdirectedgraphusingnodeAasthesource.Afternineiterations,shenotesthedistancefromAtoeachoftheothernodes.ShethenrunsatenthiterationofthealgorithmandnotesthedistancefromAtoeachoftheothernodes.

E–2 3

5 –4

–2 –3

a. WhichnodeswillshowachangeindistancefromsourcenodeAbetweentheninthandtenthiterations? 1mark

b. ExplainwhysomenodeshaveremainedthesamedistancefromsourcenodeAwhileothershaveanewdistance. 2marks

Question 8 (4marks)Anartistwishestoconstructasculpture.ThesculpturewillbemadewithnumerousL-shapedblocksconsistingoffourcubesstucktogether.Theseblockscanonlybeplacedontopofeachotherusingtheirconnectors,shownbelow.

View from top

View from side

Theartistwouldliketojoinacollectionofblockstogethersothat,uponlookingdownonthesculpture, everycellofan8×8squareofcellsiscovered,whereeachblockwouldcoverfourcells.

Describetwodifferentapproachesthatcouldbeusedbytheartisttofindapossiblearrangementofblocks.

Question 9 (4marks)DescribethedecisionversionofthetravellingsalesmanproblemandexplainwhyitisaNon-deterministicPolynomial-time(NP)problem.

Question 10 (4marks)TheMasterTheoremprovidesageneralsolutiontorecurrencerelationsoftheform

T n aT nb

f n a b( ) ( ), ,=

+ > > where 1 1

If f n O nd( ) ,∈ ( ) theaboverecurrencerelationhasthesolution

( ) log

( )( )( )

Considerthefollowingversionofmergesort,wherealistisdividedintothreesub-listsofapproximatelyequalsize.Eachofthelistsissortedrecursivelyintosub-listsandthenthesortedsub-listsaremergedintoasinglesortedlist.Assumethatthesub-listscanbemergedinlineartime.

Writearecurrencerelationforthetimecomplexityofthisversionofmergesort.StatetheappropriatecaseoftheMasterTheoremforthisversionofmergesort,includingthevaluesofa,bandd,anduseittosolvethetimecomplexity.

Question 11 (5marks)Considerthefollowingalgorithmthatmightbeusedtosolveaproblemwheresolutionscanberandomlygenerated.

soln = generate random solutiontemperature = 1min_temperature = 0.01n_iterations = 100while temperature > min_temperature for i = 1 to n_iterations soln_new = generate neighbouring solution of soln if cost(soln) >= cost(soln_new) soln = soln_new else prob = (random 0 to 100)/100 if e(cost(soln)- cost(soln_new))/temperature > prob soln = soln_new endif endif endfor temperature = cooling_factor * temperatureendwhile

a. Statetherangeofvalidvaluesforcooling_factor,sothatatleast200randomsolutionsaregeneratedandthealgorithmterminates. 2marks

b. Giventhatsoln_newisgeneratedintheneighbourhoodofsoln,whyisitagoodideatosometimesreplacesolnwithsoln_newwhencost(soln) < cost(soln_new)? 1mark

c. Giveoneexampleofaproblemwhereaversionofthealgorithmaboveislikelytogiveanacceptablesolution.Describeapossiblecost(soln)forthatproblem. 2marks

Question 12 (3marks)Consideragamefortwoplayers,PlayerAandPlayerB,whichusesthreepilesofstones.Eachplayertakesaturntoremoveasmanystonesastheywishfromoneofthepiles.Theobjectofthegameistomakeanopponentpickupthelaststone.Supposethatduringthegamethethreepilesofstonesconsistofonestone,twostonesandonestone,anditisPlayerA’sturn.

Usingtheminimaxalgorithm,completethefollowinggametreeuntilitdeterminesthemovePlayerAshouldmaketoguaranteethattheywin.

(1,2,1)

After A’s move (0,2,1) (1,1,1) (1,0,1) (1,2,0)

Question 13 (7marks)Belowisagraphrepresentationofapossiblewayinwhichacollectionofcomputerscanbeconnected.Eachcomputerislabelledwithaletterandisanodeinthegraph.Cablesthatareusedtoconnectthecomputersareshownasedgesandthelengthofeachcableisgivenasanedgeweight.

Thecollectionofcomputersneedstobeconnectedwithcablessuchthatthefollowingconditionsaremet:• Condition1:Therearenocycles.• Condition2:TheshortestlengthofcablingisusedfromS,thesource,toeveryothercomputer

whilethetotalcablelengthforthewholenetworkisthesmallestpossiblelength.

a. DrawthegraphproducedbyPrim’salgorithmandindicatethecondition(s)thatthegraphmeets. 2marks

Condition(s)met

b. DrawthegraphproducedbyDijkstra’salgorithmandindicatethecondition(s)thatthegraphmeets. 2marks

Condition(s)met

c. IsthereamodificationtoDijkstra’salgorithmthatwillallowforbothCondition1andCondition2tobemet?Explainyouranswer. 3marks

Question 14 (8marks)Considerthefollowingtableofdistancesbetweenschools.

School 1 School 2 Distance

A B 10km

A C 8km

B C 5km

C E 3km

B D 1km

D E 9km

a. Drawagraphthatwouldrepresentthisdataandindicatehowtheschoolsanddistancesarerepresentedinthegraphdrawn. 3marks

Howtheschoolsarerepresented

Howthedistancesarerepresented

b. GertywantstowalkfromSchoolAtoSchoolD,beginningatSchoolA,andwillalwayschoosetowalktotheclosestschool.

WhatdistancedoesGertycoverifsheneverrevisitsaschool? 1mark

c. ThereisashorterpathforGertytogetfromSchoolAtoSchoolD.

Whatisthatpath? 1mark

d. Nedisinterestedinconnectingalloftheseschoolswithfibre-opticcableforeasyvideoconferencing.

Assumingthateachschoolrequiresalengthofcableequaltothedistancebetweentheschools,whatistheshortesttotallengthofcablerequiredtoconnectalloftheschools? 1mark

e. SamhastowalkbetweenSchoolAandsomeotherschool.Foreveryschoolhepasses,hestopsandgivestheprincipalatthatschooloneapple.ItcostsSam$2foreachkilometretravelledand$1foreachapplethathegivesout.

Howwouldtheproblembemodelledsothatasingle-sourceshortestpathalgorithmcouldbeusedtodeterminethelowestcostforSamtotraveltoeachschoolstartingatSchoolA? 2marks

Question 15 (8marks)Wallythewasherusesaspecificapproachtowashabasketofclothes.Hebeginsbydividingtheclothesintotwosmallerbasketsandwashesthefirstbasketofclothesusinghisspecificapproach.Oncehehaswashedalloftheclothesinthatbasket,hethenwashesthesecondbasketofclothesusingthesameapproach.

a. Writethepseudocodeforanalgorithm,washClothes(basket),thatdescribesWally’srecursivewashingsystem. 2marks

Whenthewashinghasbeencompleted,Wallyhastomatchupeachpairofwashedsocks.

b. Writethepseudocodeforanalgorithm,findMatch(sock, sockList),thatwilllookthroughalistofsocks,sockList,toreturnamatchingpairforagivensock,sock.Allsocksareuniquepairs. 3marks

c. AssumethatfindMatch(sock, sockList)existsandthatthereisaremovePair(sockPair, sockList)algorithmthatwillreturnsockListwithoutthegivenpairofsocks,sockPair.

Writethepseudocodeforatail-recursivealgorithmthattakesalistofsocksasinputandreturnsallpairsofmatchingsocks. 3marks

Question 16 (9marks)Toreviseherstudies,Trudiplanstodosixtestpapersinarowwithoutsleep.Trudiisveryparticularaboutherstudyenvironmentandinsiststhattherebeonlythreestacksoftests:incompletetests,completetestsandmarkedtests.Shemayalsohaveasingletestonherdesk.Trudibeginswiththefirsttestonthestackofincompletetests,doesthetestandthenplacesitontopofthestackofcompletetests.Onceallofthetestsarecomplete,shethenmarksthembytakingatestfromthetopofthecompletestack,markingitandplacingitonthestackofmarkedtests.

a. AssumingeachstackoftestsismodelledasastackADTwiththeusualoperations,writethepseudocodeforanalgorithmthatbestrepresentsTrudi’sstudysession. 4marks

b. HowmanypopoperationswillbeexecutedforTrudi’ssixtests? 1mark

c. Ingeneral,iftherearentests,howmanypopoperationswillbeexecuted? 1mark

d. AfterTrudihasmarkedallsixtests,shewantstofindthehighest-scoringtestbymovingtestsfromthemarkedstackbacktothecompletestack.

WritethepseudocodeforanalgorithmthatwouldfindsuchatestwhileobeyingTrudi’sconstraints,asindicatedintheintroductiontothisquestion. 3marks

END OF QUESTION AND ANSWER BOOK

Question 17 (6marks)JohnSearlearguesagainstthepositionofstrongartificialintelligence(AI)byusingtheChineseRoomArgument.

DiscusstwostandardresponsestoSearle’sChineseRoomArgument.

2016 Algorithmics Written examination · C. Exhaustive testing of the function requires six test...

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