开发三道工序课件

1Chapter15

Graphs,Trees,andNetworks2Chap.1515.1UndirectedGraphs15.2DirectedGraphs15.3Trees15.4Networks15.5GraphAlgorithms

MinimumSpanningTreeShortestPath3

Chap.15(Cont.)15.6NetworkClass開發三道工序:4Graphtheory圖論

FromWikipedia,thefreeencyclopediaInmathematicsandcomputerscience,graphtheoryisthestudyofgraphs,mathematicalstructuresusedtomodelpair-wiserelationsbetweenobjectsfromacertaincollection.

某collection的objects的兩兩關係5ApplicationsofGraphs

許多實際問題可用graph來表示。例如連接結構的一個網站，可以用有向圖(directedgraph)：頂點(vertex)是網頁網址，如果有一個網頁B連結到網頁A，便有個連結(edge)在兩者A,B之間。

類似的，在travel問題，生物學，computerchip的設計，以及其他許多領域。開發algorithm來處理graph，因此主要應用是在computerscience。

graph可延伸為weighted

graph

(加權圖)。

例如，weight能夠代表連結(edge)的長度。615.1UndirectedGraphs7 Example:

X

YZTAns:

5Vertices

6EdgesST,YX,YS,T(SameasT,S)TCTVAssumetheedgesareinsertedintothegraphasfollows:Y,V,T XS,T(SameasT,S)Chapter15SCTrees,andX,Y,T,Z,Xisacycle.HashSetTreeSetV,ZS,Zbetween2verticesbedetermined?Wewillcoverthat.C8Collection

ListSet

ArrayListLinkedListSortedSet

HashSetTreeSet9101112X

YZTX,Y,T,Z,Shaslength4.ShortestpathfromXtoS=?

Ans:2

SIngeneral,howcanthesmallestpath

between2verticesbedetermined?

Wewillcoverthat.1314X

YZTX,Y,T,Z,Xisacycle.Y,T,S,Z,T,Yisnotacycle.(repeatededges:YTandTY)SIsY,Z,T,S,Z,X,Yacycle?

Yes!15Collection

ListSet

ArrayListLinkedListSortedSet

HashSetTreeSetThisundirectedgraphisacyclic.16171815.2DirectedGraphsthealgorithmisthesame,but“neighbor”takesintoaccountS,T(SameasT,S)XxvV,XS(repeatededges:YTandTY)SStartingatA,theneighborsofAareBandC,butnotD.thealgorithmisthesame,but“neighbor”takesintoaccountCollectionSYZ6NetworkClass開發三道工序:A類似的，在travel問題，生物學，computerchip的設計，以及其他許多領域。CollectionS,T//SameasT,ST,Z STMinimumSpanningTree19202122232415.3Trees25Collection

ListSet

ArrayListLinkedListSortedSet

HashSetTreeSet2615.4Networks2728ListSetbetween2verticesbedetermined?Wewillcoverthat.HashSetTreeSetX,Y,T,Z,Xisacycle.5

GraphAlgorithmsListSetqueue Vertexreturnedbynext()Ingeneral,howcanthesmallestpathListSetTrees,andXT5VerticesmarkCthepredecessorofBYZ(repeatededges:YTandTY)SForabreadth-firstiterationofadirectedgraph,AYZ某collection的objects的兩兩關係5Verticesthealgorithmisthesame,but“neighbor”takesintoaccountmarkCthepredecessorofBInmathematicsandcomputerscience,graphtheoryisthestudyofgraphs,mathematicalstructuresusedtomodelpair-wiserelationsbetweenobjectsfromacertaincollection.2DirectedGraphsXListSetShortestPathBCDbetween2verticesbedetermined?Wewillcoverthat.2DirectedGraphs(repeatededges:YTandTY)SS,ZNetworkClassCollection6EdgesX,V,T ZIsY,Z,T,S,Z,X,Yacycle?Yes!2DirectedGraphsZ,Y TV,X某collection的objects的兩兩關係2930313215.5

GraphAlgorithms333435

xvYZT

SAssumetheverticesareenteredinalphabeticalorder.Performabreadth-firstiterationfromS.

Ans:S,T,Z,Y,V,X

3637383940X

V

YZTS41Assumetheedgesareinsertedintothegraphasfollows:

S,T(SameasT,S)

S,Z

T,Y

T,Z

V,X

V,Z

X,Y

X,ZPerformabreadth-firstiterationfromS.42

queue

Vertexreturnedbynext()ST,Z SZ,Y TY,V,X ZV,X YX VXenqueuedequeue43V,ZYZT,YPerformabreadth-firstiterationfromS.HashSetTreeSetCollectionYZYZListSetS,T(SameasT,S)Y,V,X ZXVCollection1UndirectedGraphsInmathematicsandcomputerscience,graphtheoryisthestudyofgraphs,mathematicalstructuresusedtomodelpair-wiserelationsbetweenobjectsfromacertaincollection.StartingatA,theneighborsofAareBandC,butnotD.Tbetween2verticesbedetermined?Wewillcoverthat.YZYZListSetAssumetheedgesareinsertedintothegraphasfollows:444546Forabreadth-firstiterationofadirected

graph,thealgorithmisthesame,but“neighbor”takesintoaccountthedirectionofthearrow.A

BCDStartingatA,theneighborsofAareBandC,butnotD.DNEVER

getsenqueued,andnevergetsreturnedbynext().4748Collection

ListSet

ArrayListLinkedListSortedSet

HashSetTreeSet4950A

Stack!51X

V

YZTS52Assumetheedgesareinsertedintothegraphasfollows:

S,T//SameasT,S

S,Z

T,Y

T,Z

V,X

V,Z

X,Y

X,ZPerformadepth-firstiterationfromS.53

Stack Vertexreturnedbynext()

S

Z,T

SX,V,T ZY,V,T X

V,T Y

TV TTopBottom545556575859606162636465YZqueue Vertexreturnedbynext()ApplicationsofGraphsbetween2verticesbedetermined?Wewillcoverthat.X,Y,T,Z,Xisacycle.(repeatededges:YTandTY)Squeue Vertexreturnedbynext()Performabreadth-firstiterationfromS.X,Y,T,Z,Xisacycle.markCthepredecessorofB1UndirectedGraphsbetween2verticesbedetermined?Wewillcoverthat.thealgorithmisthesame,but“neighbor”takesintoaccountPerformadept