2017 Algorithmics Written examination

ALGORITHMICS (HESS)Written examination

Monday 6 November 2017 Reading time: 11.45 am to 12.00 noon (15 minutes) Writing time: 12.00 noon to 2.00 pm (2 hours)

QUESTION AND ANSWER BOOK

Structure of bookSection Number of

questionsNumber of questions

to be answeredNumber of

marks

A 20 20 20B 16 16 80

Total 100

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

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionandanswerbookof26pages• 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2017

STUDENT NUMBER

Letter

2017ALGORITHMICSEXAM 2

SECTION A – continued

Question 1Forextremelylargevaluesofn,whichoneofthefollowingstatementsinrelationtothetimecomplexityofanalgorithmiscorrect?A. ThebestcaserunningtimeofanalgorithmthatrunsinO(n)timecanbefasterthantheworstcase

runningtimeofanalgorithmthatrunsinO(logn)time.B. TheworstcaserunningtimeofanalgorithmthatrunsinO(n)timecanbefasterthanthebestcase

runningtimeofanalgorithmthatrunsinO(logn)time.C. AnalgorithmthatrunsinO(n)timeisalwaysslowerthananalgorithmthatrunsinO(logn)time.D. AnalgorithmthatrunsinΩ(n)timeisalwaysslowerthananalgorithmthatrunsinO(n2)time.

Question 2OnedifferencebetweentheBellman-FordandDijkstra’sshortestpathalgorithmsisthatA. Dijkstra’salgorithmrevisitsedgesofnodes.B. theBellman-Fordalgorithmrevisitsedgesofnodes.C. Dijkstra’salgorithmwillnotunderestimateshortestpathlengths,whereastheBellman-Fordalgorithm

will.D. theBellman-Fordalgorithmfailswhencomputedongraphswithnegativecycles,whereasDijkstra’s

algorithmdoesnot.

SECTION A – Multiple-choice questions

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

T naT n

bkn

d

n

n

c( ) =

+

>

=

if

if

1

1 wherea>0,b>1,c≥0,d≥0,k > 0

anditssolutionT nO nO n nO n

a ca c

c

cb

bba

( )( )( log )( )

loglogloglog

=

<=

ififif bb a c>

3 2017ALGORITHMICSEXAM

SECTION A – continuedTURN OVER

Question 3Considerthefollowingpseudocodeforthealgorithmf(a, b).

Algorithm f (a, b)

Inputs: a, b; both integers ≥ 0

Output: an integer ≥ 0

if (a < b)

return a

else

return f(a – b, a)

endif

Whatdoesthealgorithmfreturnwhena=2andb=3?A. 0B. 1C. 2D. 3

Use the following information to answer Questions 4 and 5.Alarge,straightshippingcanalusesaspecialsystemtoallowshipstopassthroughsafely.Forthesystemtowork,onlyoneshipcanpassthroughthecanalatatime,regardlessofwhichdirectionthatshipcomesfrom.Ifashipiscurrentlypassingthroughthecanalandanothershiparrives,thesecondshipmustwaituntilthefirstshiphasreachedtheothersidebeforeitcanproceed.Ifthereareshipswaitingoneithersideofthecanalatthesametime,theshipthatarrivedearliestisthenexttoproceed.

Question 4Whichabstractdatatype(ADT)bestmodelsthebehaviourofshipsthatwanttopassthroughthecanal?A. stackB. queueC. dictionaryD. associativearray

Question 5Themethodcurrentlyusedtomanageshipswhentheyarriveatthecanaldoesnotaccountforthecasewhentwoshipsarrive,oneoneachside,atexactlythesametime.Thiscouldbesolvedbyimplementingaheuristictoselectwhichshipwillgofirst.Tominimisethetimeittakesforshipstopassthroughthecanal,themostrelevantdataeachshipcouldprovideasinputtoaheuristicisitsA. width.B. speed.C. length.D. weight.

2017ALGORITHMICSEXAM 4

SECTION A – continued

Question 6Considerthefollowinggraphrepresentationofpseudocode.

C

B

A

OR

AND

StartingatC,inwhichorderwouldeachofthenodesbefirstexaminediftheyweretraversedusingdepth-firstsearch?(Alphabeticalorderisusedwhenthereismorethanoneoption.)A. A,AND,B,OR,CB. A,AND,OR,B,CC. C,OR,B,AND,AD. C,B,A,AND,OR

Question 7AspecialtypeofTuringmachine,M,terminatescorrectlywithoutputOwhenrunwithafiniteinput tape,T.WhichoneofthefollowingchangescanbemadetoTtoguaranteethatwhenMisrunwithT,itstillterminateswiththeidenticalanswerO?A. shorteningthetapeB. lengtheningthetapeC. addingnewsymbolstonon-blankcellsonthetapeD. addingexistingsymbolstonon-blankcellsonthetape

Question 8Givenfinitetime,aproblemthatisclassedasundecidableisknowntohaveA. analgorithmthatreturnsavalidsolution.B. analgorithmthatchecksifasolutionisvalid.C. nosinglealgorithmthatwillreturnatrueorfalsesolution.D. morethanonealgorithmthatwillreturnatrueorfalsesolution.

Question 9ChurchandTuringdefinedthecomputabilityofafunctionbyahumanandmadetheassumptionthatA. afiniteamountofspaceandtimewasavailabletothehuman.B. thealgorithmcanrecoverfromanyerrorahumanwouldmake.C. thehumanperformingthecalculationwouldmakenomistakes.D. thehumanwouldonlyneedinfinitespacetocomputethefunction,butnotinfinitetime.

5 2017ALGORITHMICSEXAM

SECTION A – continuedTURN OVER

Question 10AprofessorisworkingonaproblemandhasthoughtofanalgorithmtoverifyhersolutioninO(n)time.Withoutknowinganythingfurtherabouttheprofessor’salgorithm,whichoneofthefollowingstatementsisdefinitelytrue?A. Theproblemcanbesolvedbyanalgorithminquadratictime.B. Theproblemcanbesolvedbyanalgorithminlineartime.C. TheproblemisinclassNP.D. TheproblemisinclassP.

Question 11Claireisthinkingofawaytosolveaproblemthatisdemonstratingcombinatorialcharacteristics.Sheischoosingbetweenabrute-forceoragreedyapproach.Tofindasolutionwithlargeinputinreasonabletime,whichapproachshouldClairechooseandforwhichreason?A. brute-force,becauseitwillalwaysfindanoptimalsolutionB. greedy,becauseitwillalwaysfindanoptimalsolutionC. brute-force,becauseitmayfindanoptimalsolutionD. greedy,becauseitmayfindanoptimalsolution

Question 12Agraphalgorithmwillneedtoworkinconditionswheretheedgesandnodesalreadytraversedcannotbechecked.Cyclescanexist,alledgeshaveaweightof0andnopre-computationcanbeundertakenonthegraph.Whichoneofthefollowingwillguaranteethatasolutionwillbefound?A. best-firstsearchB. depth-firstsearchC. breadth-firstsearchD. meta-heuristicsearch

Question 13AfarmerisbreedingrabbitsthatreproduceatamonthlyratethatfollowsaFibonacciseries. Assumethatallrabbitsthatarebornwilllive.Ifthefarmerwantstoknowhowmanyrabbitsshewillhaveintwoyears’time,thesolvingstrategythatgivesthesmallestguaranteedBig-OtimecomplexityisA. simulatedannealing.B. dynamicprogramming.C. treesearchbybacktracking.D. recursivedivide-and-conquer.

2017ALGORITHMICSEXAM 6

SECTION A – continued

Question 14Considerthefollowingfunctions:

f (n)=4n2 + 6g(n)=4n2+lognh(n) = f (n) + g(n)

j(n) = f ng n

( )( )

WhatisthebestBig-Otimecomplexitydescriptionoftheasymptoticgrowthrateofh(n)andj(n),respectively?A. O(n),O(logn)

B. O(n2),O(logn)

C. O(n),O1log n

D. O(n2),O(1)

Question 15AstudentmakesthefollowingstatementsabouttheFloyd-Warshallalgorithmonagraphwithnnodes: I ItsrunningtimeisinthePcomplexityclass. II Inthebestcase,therunningtimeisΘ(n2).III Intheworstcase,therunningtimeisO(n3).IV Intheaveragecase,therunningtimeisO(nlogn). V ItmustruninΩ(n)timebecauseeverynodemustbeinspectedatleastonce.

Whichgroupofstatementsiscorrect?A. IIandIVB. IIIandIVC. I,IIandVD. I,IIIandV

Question 16

Whatvaluesofa,bandcintherecurrenceT naT n

bkn n

n

c( ) =

+

>

=42

1

1

if

if

wouldgivetherunningtimeT(n)asO(n2logn)?A. a=9, b=3, c=2B. a=9, b=2, c=3C. a=27,b=3, c=2D. a=27,b=2, c=3

7 2017ALGORITHMICSEXAM

END OF SECTION ATURN OVER

Question 17Theownerofabusinesshastakenonanumberofcontractsthatrequirehertohirenewemployees.Theseemployeeswillworkinsmallteams,withamanagerwhowillprovidedailyreportstotheowner.Astheprojectevolves,certaintasksareaddedthathavevaryinglevelsofimportance.Theownerhasdecidedtoexpandheremployeesystemtoreflecttheinclusionofthenewteams.WhichcombinationofADTswouldbethemostappropriatetorepresentthereportingstructureandthetaskswithineachteam,respectively?A. disconnectedgraph,priorityqueueB. directedgraph,priorityqueueC. weightedgraph,stackD. decisiontree,stack

Question 18Thetimecomplexityofthebestknownalgorithmtosolvefourdifferentcomputationalproblemsisgivenbelow:Problem1:O(2n)Problem2:O(n2)Problem3:O(logn)Problem4:O(nlogn)

AccordingtoCobham’sthesis,whichproblemisnotknowntobefeasible?A. Problem1B. Problem2C. Problem3D. Problem4

Question 19WhichoneofthefollowingdidSearlenotexplicitlystateregardinghisoriginalChineseRoomArgument?A. TheChinesecharacterscanbeinterpretedandreplicated.B. Theentityinsidetheroomneedstounderstandthelanguagetoproduceanoutput.C. Thelanguageusedbytheentityisfixedandeverystatementisknowninadvanceandinitsentirety.D. Theentityinsidetheroomcontainsinstructionsthat,ifgivenaninput,canproduceanappropriate

output.

Question 20OnebenefitofusingDNAcomputingasanalternativemethodofcomputationisthatitallowsforA. largeamountsofparallelprocessing.B. theself-correctionoferrorsduringcomputation.C. resultsthatcanbeeasilyinterpretedwhencomparedtothatofabinarymachine.D. largesequentialcomputingthatcanbecomputedquickly,typicallyintheorderofmilliseconds.

2017ALGORITHMICSEXAM 8

THIS pAGE IS BLANK

9 2017ALGORITHMICSEXAM

SECTION B – continuedTURN OVER

Question 1 (3marks)Alocalartistisplanningtocreateaverylargemuralmadeoutofsmallpiecesofcolouredglassthatarejoinedtogether.Herrequirementsarethateachpieceofglasscannottouchthesideofanyotherpieceofglassiftheyarethesamecolour.Thereareninecoloursofglassandtheglasspiecesareallsix-sidedpolygons.

Outlineanalgorithmtheartistcouldusetocreateamuralthatmeetshercriteriainareasonableamountoftime.Aspartofyourdescription,identifyanappropriateabstractdatatype(ADT)thatcouldbeused,referringtoitsspecifications.

SECTION B

Instructions for Section BAnswerallquestionsinthespacesprovided.UsetheMasterTheoremtosolverecurrencerelationsoftheformshownbelow.

T naT n

bkn

d

n

n

c( ) =

+

>

=

if

if

1

1 wherea>0,b>1,c≥0,d≥0,k > 0

anditssolutionT nO nO n nO n

a ca c

c

cb

bba

( )( )( log )( )

loglogloglog

=

<=

ififif bb a c>

2017ALGORITHMICSEXAM 10

SECTION B – Question 2–continued

Question 2 (6marks)Simonisdocumentinganewlydiscoveredlanguage,wherewordsvaryinthenumberoflettersandvowels(‘a’,‘e’,‘i’,‘o’,‘u’).Thislanguagemarkseachvowelinoneofthreeways.Forexample,‘a’canbemarkedas‘ā’,‘ǎ’and‘â’.Simonwouldliketoproduceanalgorithm,generateWords,thathecanusetodiscoverpotentialnewwords.Whengivenaninputword,word,withoutmarkingsshownonanyvowels,generateWords(word)wouldproduceasoutputallofthepotentialwordsthatmightexistwiththevowelmarkingsincluded.

Example with one vowelInput:catOutput:cāt,cǎt,cât

Example with two vowels Input:cateOutput:cātē,cātê,cātě,cǎtē,cǎtě,cǎtê,câtē,câtě,câtê

Simonwillusethislisttoaskspeakersofthelanguageifanyofthegeneratedwordsarerealwordsintheirlanguage.

a. DescribeanalgorithmdesignpatternthatSimoncoulduseinthegenerateWords algorithmtosolvehisproblem. 3marks

11 2017ALGORITHMICSEXAM

SECTION B – continuedTURN OVER

b. Simonwouldalsolikeanalgorithmthattakesasinputasetofwordswithoutvowelmarkings,returningasoutputallofthepotentialwordsbasedonvowelcombinations,aswellasallofthepotentialsentencesbasedonwordorder,whereeachwordcanappearinanypositioninthesentence.

ExplainhowmodifyingthealgorithmtogenerateallpossiblesentenceswillaffectthesolvabilityofSimon’sproblem. 3marks

2017ALGORITHMICSEXAM 12

SECTION B – continued

Question 3 (6marks)Alocksmithhasforgottenthecombinationtoalockthatheowns.Thecombinationlockhasthreedials,whereeachdialhas10possiblenumbers.Allthreenumbersmustbecorrectforthelocktoopen.Anexampleisshownbelow.

0 5 9

Writeanalgorithmthatwillfindthecorrectcombination.Assumethatthefollowingfunctionsexist:• set(x, y) –setsthevalueofdialx to y

– xisthedialfromlefttoright,where0≤x≤2 – yisthenumber,where0≤y ≤9

• unlock()–willreturnTrueifthecombinationiscorrect• reset() –willresetalldialstozero

13 2017ALGORITHMICSEXAM

SECTION B – continuedTURN OVER

Question 4 (4marks)Asoftwareengineerhasdesignedaprogramthatusesanumberofdifferentalgorithmswithvaryingcomplexitytosolvealargerproblem.

Discusshowvariationincomplexitycanaffecttheoverallrunningtimeoftheprogramifthealgorithmsareinsequencecomparedtoifthealgorithmsarenested,withonecallinganother.Inyourresponse,useexamplesoftimecomplexity.

Question 5 (3marks)DescribetheHaltingProblemasdefinedbytheChurch-TuringThesis.Inyourresponse,refertothedecidabilityofaproblemandhowtheHaltingProblemaffectstheabilityofaprogramtobeverified.

2017ALGORITHMICSEXAM 14

SECTION B – continued

Question 6 (6marks)‘Hilbert’s1927programcouldbeconsideredafailure.’

ExplainwhyandwriteanargumentthatshowsthatHilbert’sworkinfluencedChurchandTuring,formingthebasisoftheChurch-TuringThesis.

15 2017ALGORITHMICSEXAM

SECTION B – continuedTURN OVER

Question 7 (8marks)Emergingtechnologiesmayhelpovercomesomeofthelimitsofcurrentcomputingsystems.

a. Describeonelimitthatexistsincurrentmodelsofcomputing. 2marks

b. Describeonewayaneuralnetworkcouldbeusedtoovercomecurrentlimitsofcomputation. 3marks

c. Drawadiagramofabasicneuralnetworkrepresentedasagraph,makingsuretoindicatethepurposeofeachedgeandnode. 3marks

2017ALGORITHMICSEXAM 16

SECTION B – Question 8–continued

Question 8 (10marks)Harryownsachildcarecentre.Heurgentlyneedstoknowthemedianageofallchildrenatthecentreinordertoqualifyfornewfunding.Thechildrenaresplitintotworooms,eachwithonechildcarespecialist.Thefirstroomcontains nchildrenandthesecondroomcontainsn+1children.Thespecialistineachroomknowstheorder,byage,ofallofthechildreninthatroomandcaninstantlyanswerthequestion,‘Whatistheageofthekthchildinyourroom?’

a. Writeabrute-forcealgorithmthatwouldreturnthemedianageofallofthechildreninthechildcarecentreandwouldrequireHarrytoaskthespecialistsatotalof2n+1timesfortheageofsomechildintheirroom. 5marks

17 2017ALGORITHMICSEXAM

SECTION B – continuedTURN OVER

b. HarryhasbeengivenanunfinisheddraftofadifferentalgorithmthatwillalsogivehimthemedianageofallofthechildrenwhencalledasSearch(0, n – 1, 0, n).BeforeHarrycompletesthealgorithm,hewantsanestimationofthetotalnumberofquestionshewouldaskthespecialists.

Algorithm Search(L1, R1, L2, R2)

Input L1: lowest rank of child in room 1 being considered

Input R1: highest rank of child in room 1 being considered

Input L2: lowest rank of child in room 2 being considered

Input R2: highest rank of child in room 2 being considered

Let n1 = R1 - L1 + 1

Let n2 = R2 - L2 + 1

Ask specialist in Room 1 for age of child L1 + n1/2 and call that m1.

Ask specialist in Room 2 for age of child L2 + n2/2 and call that m2.

if n1 equals 0

return m2

if n2 equals 0

return m1

if m1 > m2

return Search(L1, L1 + n1/2, L2 + n2/2, R2)

else

return Search(L1 + n1/2, R1, L2, L2 + n2/2)

endif

Writearecurrencerelationthatreflectstherunningtimeofthealgorithmasitiswritten. 2marks

c. AssumethatthecompletedSearchalgorithmhasthesametimecomplexityasthedraft.

IfHarry’saimistoaskasfewquestionsaspossibleofhisspecialists,wouldyourecommendabrute-forcealgorithmortheSearchalgorithm?Justifyyourrecommendation. 3marks

2017ALGORITHMICSEXAM 18

SECTION B – continued

Question 9 (5marks)Consideraconnected,weighted,undirectedgraph,G,whosesmallestedgeweightisw.Thereisonlyoneedge(u,v)inGthathasweightw.

Outlineaproofbycontradictionthatshowsthat(u,v)mustbeinanyminimalspanningtreeofG.

19 2017ALGORITHMICSEXAM

SECTION B – continuedTURN OVER

Question 10 (4marks)Josephishavingabirthdaypartyforhislittlesisterandhewantstomakesureshewinstheprizeinagamethatthechildrenwillplay.Inthegame,allthechildrenattheparty,N,willsitinacircleandpasstheprizetotheirleftMtimes.AfterMpasses,thechildholdingtheprizepassesittothechildtotheleftifN>1,thenleavesthecircle,andtheprizeispassedMtimesagain.JosephwantstochooseMsothathissisteristhelastchildleftinthecircleandwinstheprize.Josephdoesnotknowyethowmanychildrenwillbecomingtotheparty.HedecidestowriteanalgorithmthatwilllethiminputM, N, P andS,whereSisthestartinglocationofhissisterandPisthestartinglocationoftheprize(0≤P < N).Thealgorithmwillreturntrueifhissisterwillwinandfalseifanyotherchildwins.

Algorithm GetWinner (M,N,P,S)

Input: M, integer, number of prize passes to the left

Input: N, integer, number of children at the start of the game

Input: P, integer, starting location of the prize

Input: S, integer, starting location of the sister

Output: true if S is the only player left, false otherwise

Let children = [0,1,...,N-1]

Let current = P

repeat N - 1 times

Let current = remainder of (current + M)/(number of children)

Remove current from children

end repeat

if children contains only S

return true

else

return false

endif

a. Josephwritesabrute-forcesearchalgorithmthatcallsalgorithmGetWinneronallpossiblecombinationsofM,N,PandStofindwinningvaluesforhissister.

AssumingthatthealgorithmGetWinnercouldbeimplementedinO(N)time,whatisthebestcaserunningtimeofJoseph’sbrute-forcesearchalgorithm?Why? 2marks

b. AssumingthatthealgorithmGetWinnercouldbeimplementedinO(N)time,istheworstcaserunningtimeofJoseph’sbrute-forcesearchalgorithmgreaterthanO(N2)?Whyorwhynot? 2marks

2017ALGORITHMICSEXAM 20

SECTION B – Question 11–continued

Question 11 (6marks)Aplanetaryspacestationispopulatedbyworkersandtheirfamilies,whereeachfamilylivesinitsownhousingdome.Domesareconnectedtooneanotherviaanextensivetunnelsystemthatcanrunformultiplekilometres,andcontainsmanyentryandexitpointstoothertunnels.Anoutbreakofavirusisrapidlyspreadingthroughthespacestation.Thevirusisspreadbetweenpeopleiftheyliveinthesamedome.Atleastonepersonfromeachdomeneedstobevaccinatedagainstthevirustohaltitsspread.Scientistshavedevelopedavaccinethatworkswhensprayedfromtheceilingofahousingdome.Theyhavealargemapoftheareawiththelocationofalldomesclearlymarked,sortedbyascendingdistancefromtheirlaboratory.Theyproposetousethefollowingalgorithmtocarryoutthevaccinations.

Algorithm Vaccinate(L)

Input: L, a list of domes sorted by distance from the

laboratory.

while L is empty

Deliver vaccinations in the first dome in L.

Remove the last element of L.

a. Theproposedalgorithmhaserrors.

Howmanyhousingdomeswillbevaccinatedusingthisalgorithmasitiswritten? 1mark

b. Writeacorrectedversionofthealgorithmthatwillallowthevaccinationtobedeliveredtoeachdomeinorderofthedome’sdistancefromthelaboratory. 2marks

21 2017ALGORITHMICSEXAM

SECTION B – continuedTURN OVER

c. Afterthescientistshavevaccinatedthepeopleinsomedomesusingacorrectalgorithm,theydiscoverthatvaccinatedpeoplecanspreadthevaccinetootherswithoutthespraybeingneeded.Thismeansthatanyvaccinatedpersoncanvaccinateeveryoneintheirnearestneighbouringdomesbysimplytouchingonememberofeachneighbouringdome.

WhatADTwouldbemostappropriatetomodelhousingdomesandneighbouringdomes?Assumingthatadomecanvaccinateexactlyallofitsneighbouringdomesinoneday,whattypeofalgorithmcouldbeusedwithyourselectedADTtocomputethemaximumnumberofdaysafterscientistsstopsprayinguntileveryoneinthespacestationisvaccinated?Brieflyexplainhowthistypeofalgorithmwouldwork. 3marks

2017ALGORITHMICSEXAM 22

SECTION B – Question 12–continued

Question 12 (5marks)BattleForFlagsisacomputergamewherepairsofplayerscompeteagainstoneanotherinvirtualcombat.Eachplayerinapairhasaparticularsetofskills,withthemosteffectivepairscontainingplayerswhohavecomplementaryskills.Skilllevelsarerepresentedbywholeintegers,wherelargerintegersindicatehigherabilityinthatskill.Acomplementaryratingofaskillcanthereforebedeterminedbysummingtheabsolutedifferencesbetweentheskillsoftwoplayers.Thecomplementaryrating(CR)canberepresentedas

CR( , ) | [ ] [ ] |,A B A k B k nk

n

= − >=

−

∑ 00

1

wherenisthetotalnumberofskills,AandBarearraysofplayerskillvaluesandkistheindexvalueinthosearrays.Assumeallplayershavethesamesetsofskills.Forexample,CaroistryingtodecideifsheshouldteamupwithMaxorNik.Thesearetheirskills.

k Caro Max Nik

Speed 0 10 6 10

Accuracy 1 4 8 9

Endurance 2 5 10 4

CarohasaCRof13betweenherselfandMax,andaCRof6withNik.Sheshould,therefore,pairupwithMax.BattleForFlagsisaverypopularcomputergame,withhundredsofthousandsofplayers.Playerscanaddeachotherasfriends,allowingthemtokeeptrackofpeopletheyenjoyplayingwith.Itwouldbeusefulforthegametohaveanalgorithmthatsuggeststoaplayerthemostcomplementaryplayerthatisavailabletoplaywiththeminagame,basedontheirfriends,thefriendsoftheirfriends,andsoon.Aplayerisunavailableiftheyarealreadycompetinginagame.Iftheselectedplayerisunavailable,thenextbestplayerwhoisnotinagameisselected.

a. Aplayerwhohasjustinstalledthegamewillhavenofriends.

Suggestanadditionalconditiontoincludeinthealgorithmthatwillallownewplayerswithoutfriendstoreceivecomplementaryplayersuggestions. 2marks

23 2017ALGORITHMICSEXAM

SECTION B – continuedTURN OVER

b. Explainhowacombinationoftreesearchandanyotheralgorithmdesignpatternscouldbeusedinthealgorithmtoreturnthebestavailablecomplementaryplayerinreasonabletime.Aspartofyourexplanation,refertoanyassumptionsthatmustbemadeabouthowplayerdataisrepresented. 3marks

Question 13 (3marks)Twofriends,JuneandDavid,arehavingaconversationaboutthepowerofalgorithmicthinkinginreal-worldscenarios.JunechallengesDavidwiththefollowing:‘Takingintoaccountthatthereareapproximately50000citiesintheworldandthattheycanbesortedinalphabeticalorder,chooseanycityintheworldanddon’ttellmeitsname.Allowmetoaskyou“yes”or“no”questionsaboutitsnameinrelationtoothercities.I’msurethatIcanfindoutwhichcityyou’vechoseninfewerthan16questions!’

StateanddescribeanalgorithmdesignpatternthatJunecanusetoguaranteethatshewillselectDavid’schosencityusingfewerthan16questions.

2017ALGORITHMICSEXAM 24

SECTION B – continued

Question 14 (4marks)Apostofficeistestinganewself-drivingdeliveryvan.Thevanstartsitsdayinadepot,whereitreceivesalistofrequeststotransportpackagesfromthedepottotheirfinaldestinations.Assumetheself-drivingvanisofasufficientlylargesizetofitallofthepackagesthatneedtobedeliveredonanygivenday,thateachlocationwillonlyhaveonepackagedeliveredtoitandthatthevanwillalwayshaveenoughtimetodeliverallofitspackages.

a. Describeanalgorithmthatwillquicklyfindaroutefortheself-drivingvanonasingleday. 2marks

b. Describeanalgorithmthatwillfindtheshortestroutefortheself-drivingvanonasingleday. 2marks

25 2017ALGORITHMICSEXAM

SECTION B – continuedTURN OVER

Question 15 (4marks)Harddrivesfromalargestoragedevicehavebeeninstalledincorrectly,asshowninFigure1.ThedatacannotberecoveredinitsentiretyunlessthedrivesarereorderedasshowninFigure2.Tofurthercomplicatetheissue,thesystemitselfcanonlydeactivateonedriveoutofeight,temporarily,tobemovedtoanadjacent,vacantdrivebay.Baysabove,below,totheleftortotherightareconfiguredtoreceiveswappeddrives.

HDD1 HDD2 HDD5 HDD1 HDD2 HDD3

HDD4 HDD3 HDD4 HDD5 HDD6

HDD6 HDD7 HDD8 HDD7 HDD8

Figure 1 Figure 2

Beforeaharddrivecanbemovedtothevacantdrivebay,themanagerwantstoconfirmthathecansuccessfullymovethedrivesinFigure1tomatchtheorderinFigure2.

ExplainamethodthemanagercouldusetocheckiftheharddrivesinFigure1canbereorderedtomatchFigure2.

2017ALGORITHMICSEXAM 26

END OF QUESTION AND ANSWER BOOK

Question 16 (3marks)Inasimulation,asharkishuntingafishtoeatfordinner.Thesharkandthefisharebothcontainedin asquare5×5gridmadeofcells.Boththesharkandthefishcanmoveup,down,leftandrighttoadjacentcellsinthegrid.Theycannotmovediagonally.Assumethatthesharkstartsinthecellinthebottom-leftcornerandthefishstartsinthecellinthe upper-rightcornerofthegrid.Assumealsothateachtimethesharkmovesintoacelladjacenttoitscurrentpositioninthegrid,thefishwillrandomlymoveintoacelladjacenttoitscurrentpositioninthegrid.

Explainhowtheminimaxalgorithmcanbeusedbythesharktomaximisethelikelihoodofitcatching thefish.