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Chapter4GreedyalgorithmsTopic:HuffmancodesHuffmancodesHuffmancodesHuffmancodesHuffmancodesHuffmancodesHuffmancodesHuffmancodesHuffmancodesSingle-SourceShortestPathsGivenaweighted,directedgraphG=(V,E),withweightfunctionw:E→Rmappingedgestoreal-valued-weights.GivenavertexinV,calledthesourcevertex.Nowweneedtocalculateallotherverticesfromthesourcetothelengthoftheshortestpath.Theweightofpathp=v0,v1,...,vkisthesumoftheweightsofitsconstituentedges.ThisproblemisusuallyreferredtoasSingle-SourceShortestPaths.Dijkstraalgorithmisthegreedyalgorithmforthesolutionofsingle-sourceshortestpathproblem.ThebasicideaistosetthevertexsetSandexpandthecollectionbycontinuallymakinggreedychoices.AvertexbelongingtothesetSifandonlyiftheknownlengthoftheshortestpathfromthesourcetothevertex.Theinitial,Scontainsonlysourcevertex.AssumeuisavertexinG,theroadthatfromsourcetouandthroughtheSverticesiscalledspecialpathfromsourcetou,andusethearraydisttorecordthecorrespondingshortestspecialpathlengthofeachvertex.DijkstraalgorithmeverytimetakesoutvertexuthathastheshortestspecialpathlengthfromtheV-S,putuaddtoS,andatthesametimedosomenecessarymodificationsonthearraydist.OnceScontainsallofVvertices,thearraydisthasrecordedthelengthoftheshortestpathfromthesourcetoallothervertices.forinstance:adirectedgraph,useDijkstraalgorithmtocalculatetheShortestpathfromthesourcevertexatoothervertices.Destinationmin←selectedtheshorestpathlength//UpdateotherverticesshortestpathIf(!final[w]&&(min+G.arcs[v][w]<Dist[w]))Dist[w]←min+G.arcs[v][w]Path[w]←Path[v]+<v,w>MinimumSpanningTreesPrimealgorithmvoidPrim(intn,Type*c){T←ØS←{1}while(S!=V){(i,j)←theminimumedgeofi∈Sandj∈V-ST←T{(i,j)};S←S{j};}}AssumeG=(V,E)isaconnected,undirectedgraphwithareal-valuedweight.V={1,2,…,n}.accordingtotheweightofGfromsmalltolargeorder;MinimumSpanningTrees
