1/3

InternationalOlympiadinInformatics201526thJuly-2ndAugust2015Almaty,KazakhstanDay1

boxesLanguage:en-ISC

BoxeswithsouvenirsThelastactoftheIOI2015openingceremonyisinprogress.Duringtheopeningceremony,eachteamwassupposedtoreceiveaboxwithasouvenirfromthehost.However,allvolunteersaresofascinatedbytheceremonythattheycompletelyforgotaboutthesouvenirs.TheonlypersonwhoremembersaboutthesouvenirsisAman.HeisanenthusiasticvolunteerandhewantstheIOItobeperfect,sohewantstodeliverallthesouvenirsintheleastamountoftime.

Thevenueoftheopeningceremonyisacircledividedinto identicalsections.Thesectionsaroundthecirclearenumberedconsecutivelyfromto .Thatis,for ,sections and areadjacent,andalsosections and areadjacent.Thereare teamsatthevenue.Eachteamissittinginoneofthesections.Eachsectionmaycontainarbitrarilymanyteams.Somesectionsmayevenbeempty.

Thereare identicalsouvenirs.Initially,bothAmanandallofthesouvenirsareinsection .Amanshouldgiveonesouvenirtoeachteam,andafterdeliveringthelastsouvenirhemustreturntosection .Notethatsometeamsmaybesittinginsection0.

Atanymoment,Amancanonlycarryatmost souvenirs.Amanmustpickupsouvenirsinsection ,andthistakeshimnotime.Eachsouvenirmustbecarrieduntilitisdeliveredtooneoftheteams.WheneverAmancarriesoneormoresouvenirsandreachesasectionwithateamthathasnotreceivedasouveniryet,hemaygivethatteamoneofthesouvenirshecarries.Thisalsohappensinstantly.Theonlythingthattakestimeismovement.Amancanmovearoundthecircularvenueinbothdirections.Movingtoanadjacentsection(eitherclockwiseorcounterclockwise)takeshimexactlyonesecond,regardlessofhowmanysouvenirshecarries.

YourtaskistofindthesmallestnumberofsecondsAmanneedstodeliverallsouvenirsandthenreturntohisinitialposition.

ExampleInthisexamplewehave teams,Aman’scarryingcapacityis ,andthenumberofsectionsis .Theteamsarelocatedinsections1,2,and5.

2/3

Oneoftheoptimalsolutionsisshowninthepictureabove.InhisfirsttripAmantakestwosouvenirs,deliversonetotheteaminsection2,thentheothertotheteaminsection5,andfinallyhereturnstosection0.Thistriptakes8seconds.InhissecondtripAmanbringstheremainingsouvenirtotheteaminsection1andthenreturnstosection0.Heneedsanother2secondstodothis.Thus,thetotaltimeis10seconds.

TaskYouaregiven , , ,andthepositionsofallteams.ComputethesmallestnumberofsecondsAmanneedstodeliverallthesouvenirsandtoreturntosection .Youneedtoimplementthefunctiondelivery:

delivery(N,K,L,positions) —Thisfunctionwillbecalledbythegraderexactlyonce.

N:thenumberofteams.K:themaximumnumberofsouvenirsAmancancarryatthesametime.L:thenumberofsectionsinthevenueoftheopeningceremony.

positions:anarrayoflength .positions[0],...,positions[N-1]givethesectionnumberofallteams.Theelementsofpositionsareinnon-decreasingorder.ThefunctionshouldreturnthesmallestnumberofsecondsinwhichAmancancompletehistask.

Subtasks

subtask points1 102 103 154 155 20

3/3

6 30subtask points

SamplegraderThesamplegraderreadstheinputinthefollowingformat:

line1:NKLline2:positions[0]…positions[N-1]

Thesamplegraderprintsthereturnvalueofdelivery.

1/3

InternationalOlympiadinInformatics201526thJuly-2ndAugust2015Almaty,KazakhstanDay1

scalesLanguage:en-ISC

ScalesAminahassixcoins,numberedfrom to .Sheknowsthatthecoinsallhavedifferentweights.Shewouldliketoorderthemaccordingtotheirweight.Forthispurposeshehasdevelopedanewkindofbalancescale.

Atraditionalbalancescalehastwopans.Tousesuchascale,youplaceacoinintoeachpanandthescalewilldeterminewhichcoinisheavier.

Amina’snewscaleismorecomplex.Ithasfourpans,labeled , , ,and .Thescalehasfourdifferentsettings,eachofwhichanswersadifferentquestionregardingthecoins.Tousethescale,Aminamustplaceexactlyonecoinintoeachofthepans , ,and .Additionally,inthefourthsettingshemustalsoplaceexactlyonecoinintopan .

Thefoursettingswillinstructthescaletoanswerthefollowingfourquestions:

1. Whichofthecoinsinpans , ,and istheheaviest?2. Whichofthecoinsinpans , ,and isthelightest?3. Whichofthecoinsinpans , ,and isthemedian?(Thisisthecointhatisneitherthe

heaviestnorthelightestofthethree.)4. Amongthecoinsinpans , ,and ,consideronlythecoinsthatareheavierthanthecoinon

pan .Ifthereareanysuchcoins,whichofthesecoinsisthelightest?Otherwise,iftherearenosuchcoins,whichofthecoinsinpans , ,and isthelightest?

TaskWriteaprogramthatwillorderAmina’ssixcoinsaccordingtotheirweight.TheprogramcanqueryAmina’sscaletocompareweightsofcoins.Yourprogramwillbegivenseveraltestcasestosolve,eachcorrespondingtoanewsetofsixcoins.

YourprogramshouldimplementthefunctionsinitandorderCoins.Duringeachrunofyourprogram,thegraderwillfirstcallinitexactlyonce.Thisgivesyouthenumberoftestcasesandallowsyoutoinitializeanyvariables.ThegraderwillthencallorderCoins()oncepertestcase.

init(T)T:Thenumberoftestcasesyourprogramwillhavetosolveduringthisrun.Tisanintegerfromtherange .

Thisfunctionhasnoreturnvalue.

orderCoins()Thisfunctioniscalledexactlyoncepertestcase.

ThefunctionshoulddeterminethecorrectorderofAmina’scoinsbycallingthegrader

2/3

functionsgetHeaviest(),getLightest(),getMedian(),and/orgetNextLightest().Oncethefunctionknowsthecorrectorder,itshouldreportitbycallingthegraderfunctionanswer().Aftercallinganswer(),thefunctionorderCoins()shouldreturn.Ithasnoreturnvalue.

Youmayusethefollowinggraderfunctionsinyourprogram:

answer(W)—yourprogramshouldusethisfunctiontoreporttheanswerthatithasfound.

W:Anarrayoflength6containingthecorrectorderofcoins.W[0]throughW[5]shouldbethecoinnumbers(i.e.,numbersfrom to )inorderfromthelightesttotheheaviestcoin.

YourprogramshouldonlycallthisfunctionfromorderCoins(),oncepertestcase.Thisfunctionhasnoreturnvalue.

getHeaviest(A,B,C),getLightest(A,B,C),getMedian(A,B,C) —thesecorrespondtosettings1,2and3respectivelyforAmina’sscale.

A,B,C:Thecoinsthatareputinpans , ,and ,respectively.A,B,andCshouldbethreedistinctintegers,eachbetween and inclusive.

EachfunctionreturnsoneofthenumbersA,B,andC:thenumberoftheappropriatecoin.Forexample,getHeaviest(A,B,C)returnsthenumberoftheheaviestofthethreegivencoins.

getNextLightest(A,B,C,D) —thiscorrespondstosetting4forAmina’sscale

A,B,C,D:Thecoinsthatareputinpans , , ,and ,respectively.A,B,C,andDshouldbefourdistinctintegers,eachbetween and inclusive.

ThefunctionreturnsoneofthenumbersA,B,andC:thenumberofthecoinselectedbythescaleasdescribedaboveforsetting4.Thatis,thereturnedcoinisthelightestamongstthosecoinsonpans , ,and thatareheavierthanthecoininpan ;or,ifnoneofthemisheavierthanthecoinonpan ,thereturnedcoinissimplythelightestofallthreecoinsonpans , ,and .

ScoringTherearenosubtasksinthisproblem.Instead,yourscorewillbebasedonhowmanyweighings(totalnumberofcallstograderfunctionsgetLightest(),getHeaviest(),getMedian()and/orgetNextLightest())yourprogrammakes.

Yourprogramwillberunmultipletimeswithmultipletestcasesineachrun.Let bethenumberofrunsofyourprogram.Thisnumberisfixedbythetestdata.Ifyourprogramdoesnotorderthecoinscorrectlyinanytestcaseofanyrun,itwillget0points.Otherwise,therunsarescoredindividuallyasfollows.

Let bethesmallestnumbersuchthatitispossibletosortanysequenceofsixcoinsusingweighingsonAmina’sscale.Tomakethetaskmorechallenging,wedonotrevealthevalueof

3/3

here.

Supposethelargestnumberofweighingsamongstalltestcasesofallrunsis forsomeinteger.Then,considerasinglerunofyourprogram.Letthelargestnumberofweighingsamongstall

testcasesinthisrunbe forsomenon-negativeinteger .(Ifyouusefewerthan weighingsforeverytestcase,then .)Then,thescoreforthisrunwillbe ,roundeddownto

twodigitsafterthedecimalpoint.

Inparticular,ifyourprogrammakesatmost weighingsineachtestcaseofeveryrun,youwillget100points.

ExampleSupposethecoinsareordered fromthelightesttotheheaviest.

Functioncall Returns ExplanationgetMedian(4,5,6) 6 Coin isthemedianamongcoins , ,and .getHeaviest(3,1,2) 1 Coin istheheaviestamongcoins , ,and .getNextLightest(2,3,4,5) 3 Coins , ,and arealllighterthancoin ,sothelightestamong

them( )isreturned.getNextLightest(1,6,3,4) 6 Coins and arebothheavierthancoin .Amongcoins and ,

coin isthelightestone.getHeaviest(3,5,6) 5 Coin istheheaviestamongcoins , and .getMedian(1,5,6) 1 Coin isthemedianamongcoins , and .getMedian(2,4,6) 6 Coin isthemedianamongcoins , and .answer([3,4,6,2,1,5]) Theprogramfoundtherightanswerforthistestcase.

SamplegraderThesamplegraderreadsinputinthefollowingformat:

line : —-thenumberoftestcases

eachofthelinesfrom to :asequenceof distinctnumbersfrom to :theorderofthecoinsfromthelightesttotheheaviest.

Forinstance,aninputthatconsistsoftwotestcaseswherethecoinsareordered andlooksasfollows:

2123456346215

Thesamplegraderprintsthearraythatwaspassedasaparametertotheanswer()function.

1/2

InternationalOlympiadinInformatics201526thJuly-2ndAugust2015Almaty,KazakhstanDay1

teamsLanguage:en-ISC

TeamsThereisaclassof students,numbered through .Everydaytheteacheroftheclasshassomeprojectsforthestudents.Eachprojecthastobecompletedbyateamofstudentswithinthesameday.Theprojectsmayhavevariousdifficulty.Foreachproject,theteacherknowstheexactsizeofateamthatshouldworkonit.

Differentstudentsmaypreferdifferentteamsizes.Moreprecisely,student canonlybeassignedtoateamofsizebetween and inclusive.Oneachday,astudentmaybeassignedtoatmostoneteam.Somestudentsmightnotbeassignedtoanyteams.Eachteamwillworkonasingleproject.

Theteacherhasalreadychosentheprojectsforeachofthenext days.Foreachofthesedays,determinewhetheritispossibletoassignstudentstoteamssothatthereisoneteamworkingoneachproject.

ExampleSupposethereare studentsand days.Thestudents’constraintsonteamsizesaregiveninthetablebelow.

student 0 1 2 31 2 2 22 3 3 4

Onthefirstdaythereare projects.Therequiredteamsizesare and .Thesetwoteamscanbeformedbyassigningstudent0toateamofsize1andtheremainingthreestudentstoateamofsize3.

Ontheseconddaythereare projectsagain,butthistimetherequiredteamsizesareand .Inthiscaseitisnotpossibletoformtheteams,asthereisonlyonestudent

whocanbeinateamofsize1.

TaskYouaregiventhedescriptionofallstudents: , ,and ,aswellasasequenceof questions—oneabouteachday.Eachquestionconsistsofthenumber ofprojectsonthatdayandasequenceoflength containingtherequiredteamsizes.Foreachquestion,yourprogrammustreturn

whetheritispossibletoformalltheteams.

Youneedtoimplementthefunctionsinitandcan:init(N,A,B)—Thegraderwillcallthisfunctionfirstandexactlyonce.

N:thenumberofstudents.

2/2

A:anarrayoflengthN:A[i]istheminimumteamsizeforstudent .B:anarrayoflengthN:B[i]isthemaximumteamsizeforstudent .Thefunctionhasnoreturnvalue.

Youmayassumethat A[i] B[i] Nforeach N .

can(M,K)—Aftercallinginitonce,thegraderwillcallthisfunction timesinarow,onceforeachday.

M:thenumberofprojectsforthisday.K:anarrayoflengthMcontainingtherequiredteamsizeforeachoftheseprojects.Thefunctionshouldreturn1ifitispossibletoformalltherequiredteamsand0otherwise.

Youmayassumethat M ,andthatforeach M wehave K[i].NotethatthesumofallK[i]mayexceed .

SubtasksLetusdenoteby thesumofvaluesofMinallcallstocan(M,K).

subtask points AdditionalConstraints1 21 none2 13 none3 434 23

SamplegraderThesamplegraderreadstheinputinthefollowingformat:

line1:Nlines2,…,N+1:A[i]B[i]lineN+2:QlinesN+3,…,N+Q+2:MK[0]K[1]…K[M-1]

Foreachquestion,thesamplegraderprintsthereturnvalueofcan.

1/3

InternationalOlympiadinInformatics201526thJuly-2ndAugust2015Almaty,KazakhstanDay2

horsesLanguage:en-ISC

HorsesMansurlovestobreedhorses,justlikehisancientancestorsdid.HenowhasthelargestherdinKazakhstan.Butthiswasnotalwaysthecase. yearsago,Mansurwasjustadzhigit(Kazakhforayoungman)andheonlyhadasinglehorse.Hedreamedtomakealotofmoneyandtofinallybecomeabai(Kazakhforaveryrichperson).

Letusnumbertheyearsfrom to inchronologicalorder(i.e.,year isthemostrecentone).Theweatherofeachyearinfluencedthegrowthoftheherd.Foreachyear ,Mansurremembersapositiveintegergrowthcoefficient .Ifyoustartedyear with horses,youendedtheyearwith horsesinyourherd.

Horsescouldonlybesoldattheendofayear.Foreachyear ,Mansurremembersapositiveinteger:thepriceforwhichhecouldsellahorseattheendofyear .Aftereachyear,itwaspossibleto

sellarbitrarilymanyhorses,eachatthesameprice .

Mansurwonderswhatisthelargestamountofmoneyhecouldhavenowifhehadchosenthebestmomentstosellhishorsesduringthe years.YouhavethehonorofbeingaguestonMansur’stoi(Kazakhforholiday),andheaskedyoutoanswerthisquestion.

Mansur’smemoryimprovesthroughouttheevening,andsohemakesasequenceof updates.Eachupdatewillchangeeitheroneofthevalues oroneofthevalues .Aftereachupdateheagainasksyouthelargestamountofmoneyhecouldhaveearnedbysellinghishorses.Mansur’supdatesarecumulative:eachofyouranswersshouldtakeintoaccountallofthepreviousupdates.Notethatasingle or maybeupdatedmultipletimes.

TheactualanswerstoMansur’squestionscanbehuge.Inordertoavoidworkingwithlargenumbers,youareonlyrequiredtoreporttheanswersmodulo .

ExampleSupposethatthereare years,withthefollowinginformation:

0 1 2

X 2 1 3

Y 3 4 1

Fortheseinitialvalues,Mansurcanearnthemostifhesellsbothhishorsesattheendofyear1.Theentireprocesswilllookasfollows:

Initially,Mansurhas1horse.

Afteryear0hewillhave horses.

2/3

Afteryear1hewillhave horses.

Hecannowsellthosetwohorses.Thetotalprofitwillbe .

Then,supposethatthereis update:changing to .

Aftertheupdatewewillhave:

0 1 2

X 2 1 3

Y 3 2 1

Inthiscase,oneoftheoptimalsolutionsistosellonehorseafteryear0andthenthreehorsesafteryear2.Theentireprocesswilllookasfollows:

Initially,Mansurhas1horse.

Afteryear0hewillhave horses.

Hecannowselloneofthosehorsesfor ,andhaveonehorseleft.

Afteryear1hewillhave horse.

Afteryear2hewillhave horses.

Hecannowsellthosethreehorsesfor .Thetotalamountofmoneyis .

TaskYouaregiven , , ,andthelistofupdates.Beforethefirstupdate,andaftereveryupdate,computethemaximalamountofmoneythatMansurcouldgetforhishorses,modulo .Youneedtoimplementthefunctionsinit,updateX,andupdateY.

init(N,X,Y)—Thegraderwillcallthisfunctionfirstandexactlyonce.N:thenumberofyears.X:anarrayoflength .For , givesthegrowthcoefficientforyear.

Y:anarrayoflength .For , givesthepriceofahorseafteryear.

NotethatbothXandYspecifytheinitialvaluesgivenbyMansur(beforeanyupdates).Afterinitterminates,thearraysXandYremainvalid,andyoumaymodifytheircontentsifyouwish.

ThefunctionshouldreturnthemaximalamountofmoneyMansurcouldgetfortheseinitialvaluesof and ,modulo .

updateX(pos,val)pos:anintegerfromtherange .

3/3

val:thenewvaluefor pos .ThefunctionshouldreturnthemaximalamountofmoneyMansurcouldgetafterthisupdate,modulo .

updateY(pos,val)pos:anintegerfromtherange .

val:thenewvaluefor pos .ThefunctionshouldreturnthemaximalamountofmoneyMansurcouldgetafterthisupdate,modulo .

Youmayassumethatalltheinitial,aswellasupdatedvaluesof and arebetween andinclusive.

Aftercallinginit,thegraderwillcallupdateXandupdateYseveraltimes.ThetotalnumberofcallstoupdateXandupdateYwillbe .

Subtasks

subtask points additionalconstraints

1 17 ,

2 17 none

3 20 and forinitandupdateXcorrespondingly

4 23 none5 23 none

SamplegraderThesamplegraderreadstheinputfromthefilehorses.ininthefollowingformat:

line1:Nline2:X[0]…X[N-1]line3:Y[0]…Y[N-1]line4:Mlines5,…,M+4:threenumberstypeposval(type=1forupdateXandtype=2forupdateY).

ThesamplegraderprintsthereturnvalueofinitfollowedbythereturnvaluesofallcallstoupdateXandupdateY.

1/3

InternationalOlympiadinInformatics201526thJuly-2ndAugust2015Almaty,KazakhstanDay2

sortingLanguage:en-ISC

SortingAizhanhasasequenceof integers .Thesequenceconsistsofdistinctnumbersfrom to .Sheistryingtosortthissequenceinascendingorderbyswappingsomepairsofelements.HerfriendErmekisalsogoingtoswapsomepairsofelements—notnecessarilyinahelpfulway.

ErmekandAizhanaregoingtomodifythesequenceinaseriesofrounds.Ineachround,firstErmekmakesaswapandthenAizhanmakesanotherswap.Moreprecisely,thepersonmakingaswapchoosestwovalidindicesandswapstheelementsatthoseindices.Notethatthetwoindicesdonothavetobedistinct.Iftheyareequal,thecurrentpersonswapsanelementwithitself,whichdoesnotchangethesequence.

AizhanknowsthatErmekdoesnotactuallycareaboutsortingthesequence .ShealsoknowstheexactindicesErmekisgoingtochoose.Ermekplanstotakepartin roundsofswapping.Wenumbertheseroundsfrom to .Foreach between and inclusive,Ermekwillchoosetheindices and inround .

Aizhanwantstosortthesequence .Beforeeachround,ifAizhanseesthatthesequenceisalreadysortedinascendingorder,shewillterminatetheentireprocess.Giventheoriginalsequence andtheindicesErmekisgoingtochoose,yourtaskistofindasequenceofswaps,whichAizhancanusetosortthesequence .Inaddition,insomesubtasksyouarerequiredtofindasequenceofswapsthatisasshortaspossible.Youmayassumethatitispossibletosortthesequence in orfewerrounds.

NotethatifAizhanseesthatthesequence issortedafterErmek’sswap,shecanchoosetoswaptwoequalindices(e.g., and ).Asaresultthesequence isalsosortedaftertheentireround,soAizhanreacheshergoal.Alsonotethatiftheinitialsequence isalreadysorted,theminimalnumberofroundsneededtosortitis .

Example1

Supposethat:

Theinitialsequenceis .

Ermekiswillingtomake swaps.

Thesequences and thatdescribetheindicesErmekisgoingtochooseareand .Inotherwords,thepairsofindicesthatErmek

planstochooseare , , , , ,and .

InthissettingAizhancansortthesequence intotheorder inthreerounds.Shecandosobychoosingtheindices , ,andthen .

ThefollowingtableshowshowErmekandAizhanmodifythesequence.

2/3

Round Player Pairofswappedindices Sequencebeginning

ErmekAizhanErmekAizhanErmekAizhan

Round Player Pairofswappedindices Sequence

Example2

Supposethat:

Theinitialsequenceis .

Ermekiswillingtomake swaps.

ThepairsofindicesthatErmekplanstochooseare , , , ,and .

InthissettingAizhancansortthesequence inthreerounds,forexamplebychoosingthepairsofindices , ,andthen .ThefollowingtableshowshowErmekandAizhanmodifythesequence.

Round Player Pairofswappedindices Sequencebeginning

ErmekAizhanErmekAizhanErmekAizhan

TaskYouaregiventhesequence ,thenumber ,andthesequencesofindices and .Computeasequenceofswaps,whichAizhancanusetosortthesequence .Insubtasks and thesequenceofswapsyoufindhastobetheshortestpossible.

YouneedtoimplementthefunctionfindSwapPairs:findSwapPairs(N,S,M,X,Y,P,Q) —Thisfunctionwillbecalledbythegraderexactlyonce.

N:thelengthofthesequence .

S:anarrayofintegerscontainingtheinitialsequence .

3/3

M:thenumberofswapsErmekplanstomake.

X,Y:arraysofintegersoflength .For ,inround Ermekplanstoswapnumbersatindices and .

P,Q:arraysofintegers.UsethesearraystoreportonepossiblesequenceofswapsAizhancanmaketosortthesequence .Denoteby thelengthofthesequenceofswapsthatyourprogramhasfound.Foreach between and inclusive,theindicesAizhanshouldchooseinround shouldbestoredinto and .YoumayassumethatthearraysPandQhavealreadybeenallocatedto elementseach.

Thisfunctionshouldreturnthevalueof (definedabove).

Subtasks

subtask points extraconstraintsonX,Y requirementonR1 8 forall2 12 forall3 16 forall4 18 none5 20 none minimumpossible6 26 none minimumpossible

Youmayassumethatthereexistsasolutionthatrequires orfewerrounds.

SamplegraderThesamplegraderreadstheinputfromthefilesorting.ininthefollowingformat:

line1:Nline2:S[0]…S[N-1]line3:Mlines4,…,M+3:X[i]Y[i]

Thesamplegraderprintsthefollowingoutput:

line1:thereturnvalue offindSwapPairsline2+ ,for :P[i]Q[i]

1/3

InternationalOlympiadinInformatics201526thJuly-2ndAugust2015Almaty,KazakhstanDay2

townsLanguage:en-ISC

TownsThereare smalltownsinKazakhstan,numberedfrom through .Thereisalsoanunknownnumberoflargecities.ThesmalltownsandlargecitiesofKazakhstanarejointlycalledsettlements.

AllthesettlementsofKazakhstanareconnectedbyasinglenetworkofbidirectionalhighways.Eachhighwayconnectstwodistinctsettlements,andeachpairofsettlementsisdirectlyconnectedbyatmostonehighway.Foreachpairofsettlements and thereisauniquewayinwhichonecangofrom to usingthehighways,aslongasnohighwayisusedmorethanonce.

Itisknownthateachsmalltownisdirectlyconnectedtoasingleothersettlement,andeachlargecityisdirectlyconnectedtothreeormoresettlements.

Thefollowingfigureshowsanetworkof smalltownsand largecities.Smalltownsaredepictedascirclesandlabeledbyintegers,largecitiesaredepictedassquaresandlabeledbyletters.

Everyhighwayhasapositiveintegerlength.Thedistancebetweentwosettlementsistheminimumsumofthelengthsofthehighwaysoneneedstotravelinordertogetfromonesettlementtotheother.

Foreachlargecity wecanmeasurethedistance tothesmalltownthatisthefarthestawayfromthatcity.Alargecity isahubifthedistance isthesmallestamongalllargecities.Thedistancebetweenahubandasmalltownthatisfarthestawayfromthehubwillbedenotedby .Thus, isthesmallestofallvalues .

Intheaboveexamplethefarthestsmalltownfromcity istown ,andthedistancebetweenthemis.Forcity wealsohave .(Oneofthesmalltownsthatare

farthestawayfrom istown6.)Theonlyhubintheaboveexampleiscity ,with .Hence,intheaboveexample is .

Removingahubdividesthenetworkintomultipleconnectedpieces.Ahubisbalancedifeachof

2/3

thosepiecescontainsatmost smalltowns.(Westressthatwedonotcountthelargecities.)Notethat denotesthelargestintegerwhichisnotgreaterthan .

Inourexample,city isahub.Ifweremovecity ,thenetworkwillbreakintofourconnectedpieces.Thesefourpiecesconsistofthefollowingsetsofsmalltowns:{ },{ },{ },and{ }.Noneofthesepieceshasmorethan smalltowns,hencecity isabalancedhub.

TaskInitially,theonlyinformationyouhaveaboutthenetworkofsettlementsandhighwaysisthenumberofsmalltowns.Youdonotknowthenumberoflargecities.Youalsodonotknowanythingabout

thelayoutofhighwaysinthecountry.Youcanonlyobtainnewinformationbyaskingqueriesaboutdistancesbetweenpairsofsmalltowns.

Yourtaskistodetermine:

Inallsubtasks:thedistance .

Insubtasks3to6:whetherthereisabalancedhubinthenetwork.

YouneedtoimplementthefunctionhubDistance.Thegraderwillevaluatemultipletestcasesinasinglerun.Thenumberoftestcasesperrunisatmost .ForeachtestcasethegraderwillcallyourfunctionhubDistanceexactlyonce.Makesurethatyourfunctioninitializesallnecessaryvariableseverytimeitiscalled.

hubDistance(N,sub)N:thenumberofsmalltowns.sub:thesubtasknumber(explainedintheSubtaskssection).Ifsubis1or2,thefunctioncanreturneither or

Ifsubisgreaterthan2,ifthereexistsabalancedhubthenthefunctionmustreturn ,otherwiseitmustreturn .

YourfunctionhubDistancecanobtaininformationaboutthenetworkofhighwaysbycallingthegraderfunctiongetDistance(i,j).Thisfunctionreturnsthedistancebetweenthesmalltownsand .Notethatif and areequal,thefunctionreturns .Italsoreturns whentheargumentsareinvalid.

SubtasksIneachtestcase:

isbetween and inclusive.

Thedistancebetweenanytwodistinctsmalltownsisbetween1and1,000,000inclusive.

Thenumberofqueriesyourprogrammaymakeislimited.Thelimitvariesbysubtask,asgiveninthetablebelow.Ifyourprogramtriestoexceedthelimitonthenumberofqueries,itwillbeterminatedanditwillbeassumedtohavegivenanincorrectanswer.

3/3

subtask points numberofqueries

findbalancedhub additionalconstraints

1 13 NO none

2 12 NO none

3 13 YES none

4 10 YES eachlargecityisconnectedtoexactlythreesettlements

5 13 YES none6 39 YES none

Notethat denotesthesmallestintegerwhichisgreaterthanorequalto .

SamplegraderNotethatthesubtasknumberisapartoftheinput.Thesamplegraderchangesitsbehavioraccordingtothesubtasknumber.

Thesamplegraderreadstheinputfromfiletowns.ininthefollowingformat:line1:Subtasknumberandthenumberoftestcases.

line2: ,thenumberofsmalltownsinthefirsttestcase.

following lines:The -thnumber inthe -thoftheselines isthedistancebetweensmalltowns and .

Thenexttestcasesfollow.Theyaregiveninthesameformatasthefirsttestcase.

Foreachtestcase,thesamplegraderprintsthereturnvalueofhubDistanceandthenumberofcallsmadeonseparatelines.

Theinputfilecorrespondingtotheexampleaboveis:

11110171820171220162320111702325221725212825161823012211624202724172025120231826222926191722212309211726231612171618901612211811202524262116010292619162120221712100252215232827292621292502122202524262318262221019111617191611191522190

Thisformatisquitedifferentfromspecifyingthelistofhighways.Notethatyouareallowedtomodifysamplegraders,sothattheyuseadifferentinputformat.