SolutionsHW5.docx

SolutionsHW5.docx

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SolutionsHW5

参考答案

HomeworkFive

DesignandAnalysisofAlgorithms

2010,北京

1LetG=（V,E）beaconnectedweightedgraph,where|V|=n,|E|=m.Everyedgehasaweightwhichisanintegerbetween1andW,whereWisaconstant.DesignanO（m）algorithmtocomputetheminimumspanningtree（MST）.

Solution:

ThekeyideaishowcanweefficientlyimplementthepriorityQtosupporttheExtract-Min（Q）andupdatethevaluekey[v]intheQ.

Sincethekey[v]isin[0,W]wedefineasetforeachvalue:

L[k]={u|key[u]=k},0kW,and

L[W+1]={u|key[u]=}.

Weusealinkedlisttochainallverticesineachset.Obviously,uL[key[u]].

WeinitializeL[0]={s},L[W+1]=V–{s},andL[k]=fork{1,2,…,W}.Theseindexedsets,L[k]0kW+1,formthepriorityqueueQ.

EachtimeweneedtoExtract-Min（Q）,wesearchthelistsfromL[0]untilanon-emptylistisfound.Avertexinthefirstnon-emptylististhevertexweneed.

Whenavertexisupdated,itwillbemovedtoacorrespondingset.Usinglinkedlists,deletingandinsertinganelementinalistcanbedoneinO

（1）time.

Now,wearereadytopresentthealgorithm.

ModifiedPrim’salgorithm（G,W,r）

1foreachvV[G]

2do{key[v]

3[v]nil

4}

5key[r]0

6S

7fork=1toW

8doL[k].

9L[0]{s}

10L[W+1]=V–{s}

11whileSV

12do{i0

13whileL[i]=

14doii+1

15uExtractanelementfromL[i]

16SS{u}

17foreachvAdj[u]

18doifvSandw（u,v）

19then{[v]u

20L[key[v]]L[key[v]]-{v}

21key[v]w（u,v）

22L[key[v]]L[key[v]]{v}

23}

24}

25End

ThetimecomplexityisO（m）becausetheExtractanelementoperationneedsatmostO（W）=O

（1）timeandtherearensuchoperations.Moreover,thereatotalofO（m）updatesonthevalueofkey（v）andeachneedsO

（1）time.

2UseDijkstra’salgorithmtofindtheshortestpathtreeforthefollowinggraph.Pleasetakethevertexsasthesource.Youmustshowdetailedsteps,onefigureforeachstep.（Initializationcouldbeomitted.）

Solution:

Weusecircletoindicatethatcorrespondingvertexismarked.

Step0.Initialization

Step1

Step2

Step3

Step4

Step5

Step6

Step7

Step8

Step9

Step10

Theshortestpathtreeisasfollows.

3（a）LetTandT’betwospanningtreesofaconnectedgraphG.SupposethatanedgexisinTbutnotinT’.ProvethatthereisanedgeyinT’butnotinTsuchthat（T-{x}）{y}and（T’-{y}）{x}arespanningtreesofG.Note,TandT’maynotbetheminimumspanningtrees.

Proof.

Supposex=（u,v）.RemovingxfromTwillpartitionTintotwosubtreesT1andT2asshownbythefollowingfigure（a）.

LetV1bethesetofverticesinT1andV2bethesetofverticesinT2.IfwefollowtheuniquepathfromutovinT’,wewillmeetanedgey=（a,b）suchthataV1andbV2.（Notethattheremaybemorethanonesuchedge.Edgeyisthefirstonewemeet.）

Now,（T-{x}）{y}isobviouslyaspanningtree.

WeobservethatremovingyfromT’partitionsT’intotwosubtreessuchthatverticesaandbbelongtodifferentsubtrees.Obviously,verticesuandabelongtothesamesubtreeandverticesvandbbelongtotheother.Therefore,（T’-{y}）{x}isaspanningtreealso.

3（b）SupposealledgesinaconnectedgraphGhavedistinctweights.ProvethatGhasauniqueminimumspanningtree.Thatis,youcannotfindtwodifferentminimumspanningtrees.

Proof.

Weprovethisbycontradiction.Suppose,forthesakeofcontradiction,therearetwodifferentminimumspanningtrees,TandT’.BecauseTT’,thereisanedgexTbutxT’.Fromproblem1（a）,thereisanedgeyT’butyTsuchthat（T-{x}）{y}and（T’-{y}）{x}arespanningtreesofG.

Becausetheweightsofxandyaredifferent,wemayassumew（x）>w（y）.Then,theweightofspanningtree（T-{x}）{y}isW（T）-w（x）+w（y）,andW（T）-w（x）+w（y）