Post on 30-Jan-2020
ALGORITHMICS (HESS)Written examination
Tuesday 20 November 2018 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
marks
A 20 20 20B 18 18 80
Total 100
• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulersandonescientificcalculator.
• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.
Materials supplied• Questionandanswerbookof23pages• 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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2018
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2018
STUDENT NUMBER
Letter
2018ALGORITHMICSEXAM 2
SECTION A – continued
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Question 1Whichmethodormethodscanbeusedtocheckforthecorrectnessofalgorithms?A. inductiononlyB. contradictiononlyC. inductionandcontradictionD. oncewritten,analgorithmisalreadyknowntobecorrect
Question 2ArequirementofCobham’sthesisisthatitA. assumesallPproblemsareeasyandallNPproblemsarehard.B. onlyconsidersalgorithmsthataredeterminedtobepracticallyfeasible.C. considersthesizeoftheinputtobethemostimportantcomponentforsolvability.D. considersconstantfactorsandlower-ordertermsasimportantcomponentsforsolvability.
Question 3Amanagerisattemptingtocoordinateseveralimpatientcustomerswaitinginline.Tobefairesttothecustomers,whichabstractdatatype(ADT)shouldthemanagernot use?A. decisiontreeB. directedlistC. queueD. stack
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 2018ALGORITHMICSEXAM
SECTION A – continuedTURN OVER
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Question 4Considerthefollowinggraph.
Whichofthefollowingpropertiesdoesthisgraphhave?A. directedandacyclicB. cyclicandweightedC. cyclicandfullyconnectedD. non-directedandnon-connected
Question 5WhichoneofthefollowingmustexistinthedefinitionofanADT?A. auniqueidentifierB. asetofoperationsC. aconcreterepresentationofdataD. theimplementationofoperations
Question 6WhichoneofthefollowingbeststatesthedifferencebetweenalistADTandadictionaryADT?A. Findingtheminimumiteminasortedlistisusuallyfasterthanfindingtheminimumitemina
dictionary.B. Findingthemaximumiteminadictionaryisusuallyfasterthanfindingthemaximumiteminasorted
list.C. Alistcanhaveitsvaluesaccessedviaanintegerindex,whileadictionarycannot.D. PrioritiesaredefinedwithinalistADT,whileadictionaryhasnoprioritiesintheADT.
Question 7Whentestingtofindoutwhetheragraphhasmultipleconnectedcomponents,whichoneofthefollowingmodificationstoabasicimplementationofPrim’salgorithmisthemostappropriate?A. Choosethemaximumweightededgetoconnectanodetothegrowingtree.B. RunPrim’salgorithmoneverynodeandkeeptrackofuniquesetsofnodes.C. Deleteallexistingedgesoftheexistinggraphandcreatenewedgestoconnecteachnode.D. NomodificationsneedtotakeplaceasPrim’salgorithmcanalreadytestformultipledisconnected
components.
2018ALGORITHMICSEXAM 4
SECTION A – continued
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Question 8ForwhichoneofthefollowinggraphswouldtheFloyd-Warshallalgorithmreturnall-pairshortestpaths?
A
CB
–2–1
–3
A
CB
2–4
1
A B
C D
–2
2
3 –2
A. B.
C. D.A B
C D
–2
2
–1 –41 1
Question 9Assumequicksortalwayschoosestherightmostelementasapivotwhencreatingpartitions.Thesortingmaintainstherelativeorderofelementswhilepartitioningandstopstherecursionattheemptyarray.Giventhearray{1,2,3,4,5,6,7,8},howmanypivotswillbechosenbeforethealgorithmreturnsasortedarray?A. 0B. 1C. 3D. 8
Question 10Considerthefollowingpseudocodeforthealgorithmunknown(x),wherexisanodeofagraph.
Algorithm unknown(x) Let x.discovered = True
For eachNode in x.adjacentNodes
If eachNode.discovered is not True Then
unknown(eachNode)
EndIf
EndFor
Whichalgorithmdoesthepseudocodebestrepresent?A. best-firstsearchB. depth-firstsearchC. bruteforcesearchD. breadth-firstsearch
5 2018ALGORITHMICSEXAM
SECTION A – continuedTURN OVER
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Question 11
cloud
10010 1
1
1100
10
WhichoneofthefollowingADTswouldbebestsuitedtomodelthenetworktopologydiagramshownabove?A. aweighted,directedgraphB. astackwithkeysC. adirectedgraphD. anarray
Use the following information to answer Questions 12 and 13.Considerthefollowingthreealgorithms,whichalloperateoninputsofsizenelements,andtheirBig-OorBig-Θresourcerequirements:• AlgorithmA:O(n)worstcasetime• AlgorithmB:Θ(2n)bestcasetime• AlgorithmC:Θ(nlogn + n)bestcasetimeandworstcasetime
Question 12WhichoneofthefollowingstatementsaboutalgorithmsA,BandCiscorrect?A. AlgorithmsAandBareincomplexityclassP.B. AlgorithmsBandCareincomplexityclassP.C. AlgorithmsAandCareincomplexityclassP.D. AlgorithmsAandCarenotincomplexityclassP.
Question 13WhichoneofthefollowingstatementsaboutalgorithmsA,BandCiscorrect?A. NoalgorithmwillterminateinO(1)timeonanylargeninputs.B. AlgorithmAcouldterminateinO(1)timeonsomelargeninputs.C. AlgorithmBcouldterminateinO(1)timeonsomelargeninputs.D. AlgorithmCcouldterminateinO(1)timeonsomelargeninputs.
2018ALGORITHMICSEXAM 6
SECTION A – continued
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Question 14SameerarunsanalgorithmonthegraphbelowtocomputeapathfromAtoB.Thealgorithmbecomestrappedinacycle.
B C
A
WhichoneofthefollowingalgorithmshasSameeraused?A. PageRankalgorithmB. Dijkstra’salgorithmC. breadth-firstsearchwithoutmarkingpreviousnodesD. depth-firstsearchwithoutmarkingpreviousnodes
Question 15CyclistSergeihasacleverplantowineveryracethatheenters.Hewillstartatthefrontoftheracepedallingslowlyat5km/h.Everytimesomeonecatchesuptohim,Sergeiwilldoublehisspeed.Assumingthatthemaximumspeedforanycyclistis60km/h,howmanypeoplewillcatchSergeibeforehecannotgoanyfaster?A. 2B. 3C. 4D. 5
Question 16Analgorithmtakesasetofinputsandperformscalculationsonit.Theoutputfromthealgorithmconsistsofasetofnumbersandlettersthatarerelatedtotheinitialinputs.TheoutputfromthealgorithmcanbeverifiedinworstcaseO(n2)time.Giventhisinformation,itcanbesaidthatthealgorithmisA. decidableandincomplexityclassNP.B. undecidableandincomplexityclassNP.C. decidableandincomplexityclassPandnotincomplexityclassNP.D. undecidableandincomplexityclassPandnotincomplexityclassNP.
Question 17AneuralnetworkcanbeusedtoA. findexactsolutionstocomputationallyundecidableproblems.B. demonstratethatSearle’sChineseRoomArgumenthasmerit.C. processcomplexinputswithnoknownrelationships.D. alwayssolveproblemsfasterthanDNAcomputing.
7 2018ALGORITHMICSEXAM
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END OF SECTION ATURN OVER
Use the following information to answer Questions 18 and 19.ThefollowingisaproofbycontradictionthatPrim’salgorithmgeneratesaminimalspanningtree(MST),withsomedetailsomitted:• LetGbeagraphwithvertices,V,andaminimalspanningtree,T*.• LetTbeaspanningtreegeneratedbyPrim’salgorithmongraphG.• AssumeT≠T*.Lettheedge(u,v)beinTbutnotinT*.• When(u,v)wasaddedtoTbyPrim’salgorithm,itwastheminimumedgecrossingfromthesetof
vertices,S,thathadalreadybeenaddedtoT,tothesetofvertices,V – S,thathadnotbeenaddedtoT.• Let(x,y)beanedgeonthepathfromu to vinT*suchthatxisinSandyisinV – S.• Formanewtree,T**,byadding(u,v)toT*andremoving(x,y)fromT*.
Question 18Whichoneofthefollowingstatementsisnot correct?A. Theedge(u,v)existsbecauseT≠T*.B. Thereisapathfromu to vinT*becauseitisaspanningtree.C. T**isnotatreebecauseadding(u,v)toT*andremoving(x,y)fromT*createsacycle.D. T**isatreebecauseadding(u,v)toT*andremoving(x, y)fromT*doesnotcreateacycle.
Question 19Whichoneofthefollowingstatementsgeneratesthecontradictionneededtocompletetheproof?A. T**hasahigherweightthanT*,thusT*couldnotbetheMSTofG.B. T**hasalowerweightthanT*,thusT*couldnotbetheMSTofG.C. T**isnotatree,thusT*couldnotbetheMSTofG.D. T**isnotatree,thusT*istheMSTofG.
Question 20Simulatedannealinghasbeensuggestedasausefultechniquetocomputeaglobalminimumrainfalloveralargesetofrainfalldata.Whichheuristicappliedtosimulatedannealingismorelikelytoreturnaglobalminimumrainfall?A. increasetheprobabilityofchoosingaworsesolutionatrandomasrunningtimeincreasesB. decreasetheprobabilityofchoosingaworsesolutionatrandomasrunningtimeincreasesC. allowthesystemtemperaturetocoolquicklytofindasolutioninashortamountoftimeD. keepthesystemataconstanttemperaturetofindthebestsolutionwithinashortamountoftime
2018ALGORITHMICSEXAM 8
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SECTION B – continued
Question 1 (2marks)DescribeonesimilarityandonedifferencebetweenaTuringmachineandamoderncomputer.
Question 2 (4marks)Eliseneedstouseapriorityqueueabstractdatatype(ADT).
WriteapriorityqueueADTspecificationforElise.
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>
9 2018ALGORITHMICSEXAM
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SECTION B – continuedTURN OVER
Question 3 (4marks)Thefollowingalgorithmfindsthemaximumelementinalist.
Algorithm FIND_MAX(L): Input: L, a non-empty list of elements
If L has only one element
return first element of L as the maximum
EndIf
m = FIND_MAX(L without the first element)
If m is greater than the first element in L
return m
Else
return the first element of L
EndIf
a. Writearecurrencerelationanditssolutiontodescribetheworstcaserunningtimeofthealgorithmwhenthelistcontainsnelements. 3marks
b. WouldtheBig-Oforthebestcaserunningtimeofthealgorithmbesmallerthantheworstcaserunningtimeofthealgorithm?Justifyyouranswer. 1mark
2018ALGORITHMICSEXAM 10
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SECTION B – continued
Question 4 (2marks)Considerthefollowingrecurrencerelations.
S nS n n
O
n
n
T nT n O n
O
( )( )
( )( )
(
=
+
>
<
=
+
23
3
1
3
4
22
1
if
))
>
<
if n
n
1
2
a. WhatistheBig-OsolutiontoS(n)? 1mark
b. WhatistheBig-OsolutiontoT(n)? 1mark
Question 5 (3marks)Maryaminvestsincompaniesonthestockmarket.Sheiscurrentlyconsidering50potentialcompaniesshecaninvestinandhasestimatedthepotentialprofitsheislikelytoreceivefromeachcompanyoverthenext12months.TheinvestmentineachcompanyhasaninitialcostandMaryamhasafixedbudgetthatshecannotexceed.
DescribeadynamicprogrammingalgorithmthatMaryamcanusetoselectasetofcompaniestoinvestinthatwillprovidemaximumprofitforherbudget.
11 2018ALGORITHMICSEXAM
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SECTION B – continuedTURN OVER
Question 6 (5marks)Anewshoppingcentrehasbeenbuiltandhasattractedmanycustomers.Thecentre’smanagementhasnotedthatmanyofthesecustomersaregettinglostwithintheshoppingcentre.Thecentre’smanagementwantstodesignandbuildaninteractivemapapplicationthatwillallowcustomerstofindtheirwaytoeachshopfromfixedinformationkiosksaroundtheshoppingcentre.
a. NameandjustifyagraphADTthatwouldbestsuittheinteractivemapapplication. 3marks
b. Thecentre’smanagementhasdecidedtoinstallanadditionalinformationkiosk.
Explainhowtheadditionalinformationkioskcouldbeintegratedwithintheinteractivemapapplication.WhatchangestothegraphADTnamedinpart a.wouldhavetooccur? 2marks
2018ALGORITHMICSEXAM 12
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SECTION B – continued
Question 7 (4marks)Considerthedecisiontreebelow.
weekday weekend
time > 9 amand time < 3 pm
time > 3 pmand time < 9 am
no homework homework
no homework homework
school sleep
sleep study
study
Representthisdecisiontreeusinglogicaloperationsinpseudocode.
13 2018ALGORITHMICSEXAM
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SECTION B – continuedTURN OVER
Question 8 (4marks)Proveviainductionthatthesumofthefirstnoddnumbersisthesameasn2,foralln≥1.
2018ALGORITHMICSEXAM 14
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SECTION B – Question 9–continued
Question 9 (9marks)Devicesthatarecapableofwirelesstransmissionarebecomingcheaperandeasiertointegrateintoexistingtechnology,suchasvacuumcleanersandrefrigerators.Thesedevicestypicallycommunicateusinganetwork.ThenetworkbelowshowsdevicesnamedA–O.
Grid 1 2 3 4 5 6 7 8 9
a
b
c
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e
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i
O
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F
E
I
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D
JH
KB
A
M
N
L
Thenetworkabovehasthefollowingconditions:• Eachdeviceisawareofitscoordinatesandthereisacompletelistofdevices.• Eachsquarerepresentsa50m×50marea.• Forsuccessfulconnectionoverwirelesstransmission,devicesmustbewithinrange:twosquares
verticallyorhorizontallyandonesquarediagonally.Forexample,deviceAcancommunicatewiththosesquareswithanunderscore.
• Eachdevicecanrelaydatabetweennodeswithinrange.Forexample,deviceAcancommunicatewithdeviceE,whichcanthencommunicatewithdeviceF.
• Eachdevicecanreturntemperaturevaluesmeasuredwithinitssquare.
a. Whichdevice(s)cannotsuccessfullyconnectoverwirelesstransmissionwithanyotherdevice? 1mark
15 2018ALGORITHMICSEXAM
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SECTION B – continuedTURN OVER
b. DescribehowtheFloyd-Warshallalgorithmcanbeusedtocheckifalldevicescancommunicatewithanyotherdevice. 2marks
c. Writeanalgorithmtocomputetheaveragetemperatureofalldevicesinthenetwork,givenalltemperaturesinaninputlist,temperature_list.Somedevicesmaybedefectiveandreturn–255astheirtemperaturevalue.Thealgorithmshouldnotincludethesetemperaturesinthecalculation. 6marks
2018ALGORITHMICSEXAM 16
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SECTION B – continued
Question 10 (4marks)Joedeliversgroceriestothesamefivehouseseverydayaspartofhisdeliverybusiness.Toconservefuel,hecomputedthelengthofallpossibleroutesbetweenhishomeandthosefivehousestofindtheminimumroute.
a. HowmanyroutesdidJoecompute? 1mark
b. Joe’salgorithmtodeterminetheminimumrouteisanexampleofwhatalgorithmicdesignpattern? 1mark
c. AsJoe’sdeliverybusinessgrowsandmorehousesareaddedtothedeliverylist,willJoe’salgorithmicapproachstillbeusefulforcomputingtheoptimaldeliveryroute?Justifyyouranswer. 2marks
17 2018ALGORITHMICSEXAM
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SECTION B – continuedTURN OVER
Question 11 (4marks)Considerthefollowingalgorithmforsortingalistofintegersinascendingorder,whichmakes useofanotheralgorithmcalledSORT2.AssumethatSORT2runsinΘ(n logn)timeonalistoflengthn,unlessthelistissortedindescendingorder,inwhichcaseSORT2runsinΘ(n2)time.
Algorithm SORT(L) INPUT: L, a list of integers
Let sorted = True
For each element x except the last in L
If x is greater than its right neighbour in L
Let sorted = False
EndFor
If sorted is False Then
Return the result of SORT2(L)
Else
Return L
EndIf
a. WhatisthebestcaserunningtimeofSORT?Writedownalistof10integersthatgivesthebestcaserunningtime. 2marks
b. WhatistheworstcaserunningtimeofSORT?Writedownalistof10integersthatgivestheworstcaserunningtime. 2marks
2018ALGORITHMICSEXAM 18
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SECTION B – continued
Question 12 (2marks)
–5
–2 1
–2
2
1 0
A B
C D
E F
ExplainwhythegraphaboveisnotasuitableinputtoanaiveimplementationoftheBellman-Fordalgorithm.
Question 13 (3marks)DescribeDNAcomputingandexplainhowitcanbeusedasanalternativemethodofcomputation.Provideanexampleaspartofyourexplanation.
19 2018ALGORITHMICSEXAM
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SECTION B – continuedTURN OVER
Question 14 (9marks)Thestrategyusedforclimbingarockwallisveryimportant.Foragivenrockwall,thereisasetofnumberedhandholdsandfootholdsthatcanbechoseninsequencetoallowtheclimbertogettothetopofthewall.Longrouteswithhandholdsandfootholdsthatarespacedfarapartcancausetheclimbertotirequicklyandincreasetheirriskoffalling.Shortrouteswithcloselyspacedhandholdsandfootholdsallowclimberstopreserveenergyandmakeittothetop.YouhavebeenselectedasthestrategycoachoftheAustralianteamfortheClimbingWorldChampionships.Inparticular,youhavetoplotacourseforeachclimberoneachwall.
a. Whatinformationwouldyouneedabouteachwallandeachclimber?WhatADTwouldbebesttousetomodelthisproblem?Why? 3marks
b. WhatalgorithmwouldyouusetodeterminetheoptimalclimbingrouteusingtheADTfrompart a.?Inyouranswer,describehowyouwouldtranslatetheoutputfromyouralgorithmintomeaningfulinformationthattheclimbercoulduse.Alsoincludeanestimationofthetimecomplexityofyourapproachandhowitcomparestoabruteforceapproachtotheproblem. 6marks
2018ALGORITHMICSEXAM 20
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SECTION B – continued
Question 15 (3marks)Socialmediaprofileshavebeenreceivedbyacompanythatplanstousethedatatoinferwhichusersitshouldtargetfirstinordertoinfluencelargecommunities.Thedatacontainsuserprofiles,theirlistsoffriendsandtheirpubliccomments.
Describeanalgorithmthatcouldbeusedtoidentifythemostinfluentialprofilesinthedata.
Question 16 (4marks)Bernieisdevelopingamobilerobotforanationalpark.Therobotwilltravelthroughtheparkusingexistingpathsanddepositanimalfoodatfeedingstations.Thesefeedingstationswillbechangedeveryseasontomatchthemovementofanimalsduringtheyear.Bernieiscurrentlydesigningthealgorithmthattherobotwillusetoselectthemostefficientpathitshouldtaketovisitallofthefeedingstationswhileminimisingthedistanceittravels.
WhatisthebestalgorithmforBernietousetodecidetheorderinwhichtherobotwillvisiteachfeedingstation?Willitguaranteeanoptimalsolution?Explainwhyorwhynot.
21 2018ALGORITHMICSEXAM
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SECTION B – continuedTURN OVER
Question 17 (4marks)Afarmersharesafieldwithanotherfarmer.Everyyeareachfarmercansell10cowsandreducetheirincomeby10%orbuy10cowsandincreasetheirincomeby10%ordonothing.Atthebeginningofeachyear,theavailableresourcesinthefield,asapercentage,changeby100minusthetotalnumberofcows.Forexample,assumethatthefieldhas100%resourcesand,atthebeginningoftheyear,Farmer1has50cowsandFarmer2has70cows,thustheresourceschangeto80%.Ifnextyearthetotalnumberofcowsis110,theresourceschangeto70%.Ifnextyearthenumberofcowsis90,theresourceschangeto80%.Iftheavailableresourcesatthebeginningofayeardropbelow50%,allcowsmustberemovedandbothfarmerslosetheirbusinesses.
Describeanalgorithmthatwillselectastrategythatmaximisesonefarmer’sincomeattheendoftwoyears,withoutlosingallofthisfarmer’scows.
2018ALGORITHMICSEXAM 22
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SECTION B – Question 18–continued
Question 18 (10marks)Twopirates,MontyandBlackbeard,havedecidedtodividetheirfortunes.MontymakesthefollowingpropositiontoBlackbeard:Choosebetweentakingthesquaremadeofgoldwhosesideisthehypotenuse(sidec)oftheright-angletriangleorthetwosquaresmadeofgoldwhosesidesareadjacenttotheright-angletriangle(sidesaandb),asshowninthediagrambelow.
c b
a
IfBlackbearddecidestotakethesquarewiththehypotenuse(sidec),hemustgiveacertainpercentageofthegoldtoMonty.
a. Writeanalgorithm,square_area,inpseudocodethatwillreturntheareaofanygoldensquare. 2marks
23 2018ALGORITHMICSEXAM
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END OF QUESTION AND ANSWER BOOK
b. Assumesquare_areaexists.WriteanalgorithminpseudocodethatwilldeterminethebestdecisionforBlackbeardtomakeinordertomaximisetheamountofgoldhereceives.IfBlackbeardshouldchoosethesquarewiththehypotenuse(sidec)thenthealgorithmshouldreturnTrue,otherwiseitshouldreturnFalse. 4marks
c. WhatisthebestdecisionforBlackbeardtomaketomaximisetheamountofgoldhereceives?Justifyyouranswer. 2marks
d. Insteadofsquaresmadeofgold,MontyandBlackbeardmustsharecubesmadeofgold.
WhatisthebestdecisionforBlackbeardtomakeinthisinstance?Justifyyouranswer. 2marks