算法2018考试.docx

算法2018考试.docx

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1.DivideandConquer（14）

GivenabinarytreeT,pleasegiveanO（n）algorithmtoinvertbinarytree.

Forexamplebelow,invertingtheleftbinarytree,wegettherightbinarytree.

2.DynamicProgramming

Giventwosequences,Findthelengthofthecommonsubsequense（LCS）.

Note：

Asubsequenceisdifferentfromasubstring,fortheformerneedNOTbeconsecutiveoftheoriginalsequence.Forexample,for“ABCD”and“AEFC”,thelongestcommonsubesequenceis“AC”,andthelengthofitis2.

PleasedescribetheoptimalsubstructureandDPequation.

3.Greedy

SupposeyouweretodrivefromBeijingtoXizang.Withafullgastank,youcantravelatmostmkm.Andyouhaveamapthatgivesdistancesbetweengasstationsalongtheroute.Letd1

Givethemostefficientalgorithmyoucanfindtodeterminatwhichgasstationsyoushouldstopsothatthenumberoftimeyoustopisminimized,andprovethatyourstrategyyieldsanoptimalsolution.Besuretogivethetimecomplexityofyouralgorithmrespectton.

4.LinearProgramming

Anexpresscompanyneedstodelivergoodstoitsfiveusers.Thecompanywantstoupload1,2,3,4and8unitsofgoodsatfiveuserswhoseuserIDare1,2,3,4and5respectively.Thecompanyhaxatotaloffourtrucks,andthetrucks1to4havealoadof2,6,8and11units.ThecostoftruckidispatchingonceisCi.Duetodeliverytimerestrictions,onetruckcanneitherdelivergoodstoboothusersAandCattheasametime,nordelivergoodstobothBandDatthesametime.Pleasegiveadeliveringplantominimizethecost.

（1）Formulatethisproblemasanintergerlinearprogrammingproblemandexplaineveryvariableandconstraints.

（2）IftruckichargestheusersjfortheadditionalchargeKij,thenhowshouldtheproblembeformulated?

Notece:

Absolutevalue（|*|）shallneverappearinaLinearProgramming（LP）formulation,sinceitisnotalinearoperation.Nerthershallothernon-linearoperations,suchasmax（）,etc.Ifyouwanttouseyourownnotation,Pleaseexplainitfirst.

5.NetworkFlow

Givennnumbers[a1,a2...an],numberscanbenegativeorpositive.

Youcanperformthefollowingoperationanynumberoftimes（possiblezero）

·Selectapositiveintergerxiassign0toALLaisatisfiediisdivisiblebyx.（ax=0,a2x=0,a3x=0...）

Byoptimallyperformingtheoperation,whatisthemaximumsumofNnumberscanbeobtained?

（Exanple,6numbers[1,2,-6,4,5,3],answeris12）.

6.SubarraySum

Giveranarrayofpositiveintergersandanintergerk,

1.Pleasefindwhethertherearetwonumbersinthearraysuchthattheyadduptok,andthetimecomplexityofyouralgorithmshouldbebetterthanO（n2）;

2.Pleasefindwhetherthereisacontinuoussubarraywhosesumequalstok,andthetimecomplexityofyouralgorithmshouldbebetterthanO（n2）;

3.Pleasefindwhetherthereisasubsequence（Notnecessarilycontinuous）whosesumequalstok;

4.Pleaseconstructaarrayofshortestlengththatthesetofsumsofallsubsequencescoversalltheintergersbeteweenlandk.

Examplethatifk=4,theshortestarraycanbe[1,2,3]becausethatsumofsubsequence[1]equals1,sumofsubsequence[2]equals2,sumofsubsequence[1,2]queals3,andsumofsubsequence[1,3]equals4.Thesubsequencesumscancoverintergersfrom1to4.

7.Granph

Youaregivenanundiretedweighredgraphconsistingofnverticesandmedgeas,eachofedgeshasaweight.Youarealsogiventwoverticessandtinthegraph.

Forexample,agraphGconsistinfof6verticesand7edgeswithdifferentweightsisshownmsistheverticelabelled0andtistheverticelabelled5.

1.PleasefindwhetherthereisapathfromstotthatthemaximumweightofedgesinthepathisatmostW.Examplethatgivenω=6inG,thepath0->1->4->5issatisfiedbecausethemaximumedgeweightis5（1->4）.

2.YouneedtofindapathfromstotthatthemaximumweightWofedgesinthepathisminimized.PleasecalculatettheminimumvalueofW.

Examplethatthereare3pathsfromstotinG:

0->1->3->5,0->1->4->5and0->2->4->5,themaximumedgeweightsinthesepathsis10,5and7,sotheminimumvalueofWis5.

3.Pleasefindtheshortestpathfromstotandthepathgoesthroughatmostkpoints（Notincludingsandt）.