University of Illinois at Urbana Champaign Spring

A B C D A D 3
B C 2
A
CB
D
A B C 2 D
3
One must remove 2 edges Removing three edges is needed
to disconnect this graph in order to disconnect this graph.
b.23.1.(a)ossible,devDraalwwneedaIllinoigraphofwitheacvumerticesedes,er.Fn,t,en.andgraphsymineinedgeswhicrhoftheavvalence(explain).ofisvecauseerticesbridadd-vanderticesFysisFCorrection.ofandelotheevminimalalenceerofhvbertmoicedisconnectse:alenceandand1hasisalenceMidterm?.AnswF1ItGroupimp181bMaththe(b)umIseritopalenossiblevtoisdraaweva2.gorrhathephbonw,thetsamer2007theSpringnibnofwhicthataignto,eUrbana-Champeandvttohait.vvUniversityertices2 211 4 4
2 3
3
1 3
8 78 7
4 595 9
56 14
6 139 137 10 610
12
12
8 1111
This graph doesn’t have any Euler circuit; on the left you see an Eulerization This graph admits an Euler
with three added vertices (which is optimal since there are six odd−valentcircuit, as verified by the
vertices in the graph) and an Euler circuit in the Eulerized graph; one that is represented
on the right you see a circuit that covers all edges while reusing a minimalabove.
number of them, obtained by "squeezing" the Euler circuit from the
Eulerized graph.
B
1060
3020
A C
65
90 35
8030
40E D
A
60+10+30+40+80 = 220
10+35+80+90+20 = 235
5
(5−1)!/2 = 4!/2 = 12
oferbuminthetonormw,uSinceminimofaer.reusesanthathamiltonianuitfocecirgraphareplacingobtainthetoetwhiciiteHosoneuto4.methoandvgraphertheaofertices)EulerizationFtnearest-neighecienbanalgorithm.ndtimeesn't,AEBCDdooitisifmi;ygraph.a)manFworvtheingraphtheabcoherevaree,tndumthamiltonianhge(thehabmigraphsltonianeaccircuitquestion,obtainedthebbyalgorithmapplyingythesorted-edgestheAnswonThisedges.ontheobtainsorcircuitalgorithm,A,startingcostatfcircuith.EulerWhattsiadswhetherthesacosteloofc)twhyatcircuitscircuitould?haAnsweer.considerOneorderobtainsapplythebrutecircuitrABCEDeA,dthe?costthereof5whicertices,hhisnEulerbanofwcircuitsdrathises,rdophitcompleteIfwithnot.voriscircuitthe.hb)orSamenearest3.neighb.26×26×26×10×10
1/2
(9−1!)/2 = 8!/2
(1/2)×(8!/2) 10080
7 1 4
2
4 65
6 15
8 6 4
6 71 8
2 4 3
35 23
13 5 3
Thereer.Answ?er.applyinggraphdierenletterstplatespuseossibilities.toursb)requiresYthisou6.oumerals.wnstaumcbhaSinceiofnuteofenineoraalgorithmppartmenytfcomplexess(includ?itotalnergtthewoneusingyhouhalf-minlivtionebin),algorithmandeyouldouminplanmintoKruskvisitteaelocwhandofmixtureylicenseourtpropInerties.AnswIfTheitntakbesofumeralsisnyoedminfolloutethreeto.computeeacthetourtotalalengthuteocomputa-ftime,athetour,rwforceareinwillsituationlongtakwillconstructeditbtakceossible(inutes,mipnutes.utes)Applytoal'sapplytothehbew.brutemanforceHoanllettersgooriathmplatesto,ndetheaoptimalsometourho5.a)wT T1 4
10 13
T7
7
T T2 56 15
T TTT 96 83 12 5 38
T T T T T T1 4 7 3 5 7
30
79
2 79/2 = 39.5
40
79/3 = 26.333...
30
T1
3
T
7TT 52 T T T T T T5 5 3 8 33 5 86
T T T T T T T T T T T8T 12 1 4 6 7 2 4 5T 763 9 6 4
155 8 9 10 14 16 1819 5 8 9 10 1819
T Schedule obtained with the Schedule obtained with the 4 2
decreasing−time list method .critical−path scheduling
method.
T T T T T T T T3 2 5 1 4 6 8 7
T T T T T T T T3 6 2 5 8 1 7 4
toftimeer(are)isisGivequalsctobtheintotalbtasknetime,listwhiceheisouWithtminoutesarehere.from(iii)yTheredared.tewcessors,owproAnswcessorstheAnswsituationser.toTofotal:taskpriorittimeanalysisdivided?bw.yiser.obtainedistheAnswdulingcessor.theprohedulingonetheisgivTheretimate(ii)bminofutes,nsoon'ttheIfminimalproamounumtknoof(i)timefolloneededtheistimeatminimalleastanpath)..minpathsutes.er.(iv)listTherethearewcriticalermouncriticalaWhatthe,theycessor,decreasing-timeminimalproproTheseoststheohedcessors)ebrepresenapplyingabcritical-pathvhethemethoofandhdecreasing-timetsc(lengmethoutesjobminthatleasteatcanrequirew;ssinceestthistheiproserloumwtheerknothandthewlengther.ofcessors.aofcriticalbpath,nwwedon'taYgain:canwingonlythesajobynishinneededthisofcaseamounthatthetheestimatejobewill(b)requireqndathereleastcriticalwillAnswminTheutes.y8.obtainedFyorcritical-pathtishetorder-requiremenTheretAnswdpath(s)ithegisr(a)aelophwhilebprioritelolistw,thendlistthebscdigraphhtedulesorder-requiremen(onthetConsiderwthree7.processors..Flioryieldthreescproulcessors,swtedeogete.3 4
1 1
1 3475
65 22
7,6,5,5,4,4,3,3,2,2,1,1,1
12 3 4 4 1
1
257 6 5
3
1 3 4 4 11
2
257 6 5
3
nbareloywoalgorithmynext-tWhentheandUseistic(a)path4lbs.the3lbs,starting2lb,ery6lbs,algo3lbs,completiono,b(a)lnot(b)TRSameyquestidgesosalesmann,endenusingFthisctimegraph.theusingrst-tthedecreasingmaalgorithm;Answ:er.ourTheilistheuofdowpeighoptimalts(b)inducednonincreasingsorder-iswhen:v1m1lb,e5lbs,on4lbs,y1lb,2lbs,tree7lbs,m5lbs,tain:of9lbsALSEthanhedulingmorelist-pronoithm,holdrequiredcanhthatincreasebinsTRtotsolutiobtain;inttsyrepresen10.hruegfalsetseeighwTwingoro?follAther.algorithmApplyingesthenecessarilyrsrt-tducealgorithmresults.toUEitThegivproebsthetheofollotedwingepacalgorithmkingsolving:trawelingkproblemacapbthedepktpacthe(c)citspann.nALSEustAmiougtoofYgraphxtustoonpageev(e)edgewaalgorithmFer(d)morescotasksttheancessingrst-trFdecreasinger.timeWbeeacusetasktheylisthettime.fromUEquestionsee(b),e9.b(c)okAgain87.theThesameorst-tquestion,nevusingusesthebwxesorst-thdecreasingthealgorithm.algorithm.AnswALSE