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1Chapter15
Graphs,Trees,andNetworks1Chapter15
Graphs,12Chap.1515.1UndirectedGraphs15.2DirectedGraphs15.3Trees15.4Networks15.5GraphAlgorithms
MinimumSpanningTreeShortestPath2Chap.1515.1UndirectedGraph23
Chap.15(Cont.)15.6NetworkClass開發三道工序:3Chap.15(Cont.)34Graphtheory圖論
FromWikipedia,thefreeencyclopediaInmathematicsandcomputerscience,graphtheoryisthestudyofgraphs,mathematicalstructuresusedtomodelpair-wiserelationsbetweenobjectsfromacertaincollection.
某collection的objects的兩兩關係4Graphtheory圖論
FromWikipedi45ApplicationsofGraphs
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。5ApplicationsofGraphs5615.1UndirectedGraphs615.1UndirectedGraphs67 Example:
X
YZTAns:
5Vertices
6EdgesS7 Example:78Collection
ListSet
ArrayListLinkedListSortedSet
HashSetTreeSet88999101010111111S6NetworkClass開發三道工序:Alinkedlist!X某collection的objects的兩兩關係Performadepth-firstiterationfromS.Stack Vertexreturnedbynext()Z,T SHashSetTreeSetListSetxv5
GraphAlgorithms6ANetworkClass6ANetworkClassSqueue Vertexreturnedbynext()Z,T SV,T YYZ開發algorithm來處理graph，因此主要應用是在computerscience。12X
YZTX,Y,T,Z,Shaslength4.ShortestpathfromXtoS=?
Ans:2
SIngeneral,howcanthesmallestpath
between2verticesbedetermined?
Wewillcoverthat.S121213131314X
YZTX,Y,T,Z,Xisacycle.Y,T,S,Z,T,Yisnotacycle.(repeatededges:YTandTY)SIsY,Z,T,S,Z,X,Yacycle?
Yes!141415Collection
ListSet
ArrayListLinkedListSortedSet
HashSetTreeSetThisundirectedgraphisacyclic.15151616161717171815.2DirectedGraphs1815.2DirectedGraphs18191919202020212121222222X,Zthedirectionofthearrow.Graphtheory圖論
FromWikipedia,thefreeencyclopediaV,Zorderisunimportant:HashSetTreeSet6NetworkClass開發三道工序:5
GraphAlgorithmsAssumetheedgesareinsertedintothegraphasfollows:TNetworkClassS,T//SameasT,SCollectionPerformadepth-firstiterationfromS.YZStack Vertexreturnedbynext()YZ23X,Z23232415.3Trees2415.3Trees2425Collection
ListSet
ArrayListLinkedListSortedSet
HashSetTreeSet25252615.4Networks2615.4Networks262727272828282929293030303131313215.5
GraphAlgorithms3215.5
GraphAlgorithms3233333334343435
xvYZT
SAssumetheverticesareenteredinalphabeticalorder.Performabreadth-firstiterationfromS.
Ans:S,T,Z,Y,V,X
3535363636YZS,ZAcollectioninwhichForabreadth-firstiterationofadirectedgraph,orderisunimportant:Z,T SHashSetTreeSetYZListSet6NetworkClass開發三道工序:SAssumetheedgesareinsertedintothegraphasfollows:Assumetheedgesareinsertedintothegraphasfollows:orderisunimportant:ArrayListLinkedListSortedSetpriorityqueue(pq):<C,7><B,10>;remove<C,7>S,T(SameasT,S)Athealgorithmisthesame,but“neighbor”takesintoaccountS37Y3738383839393940X
V
YZTS404041Assumetheedgesareinsertedintothegraphasfollows:
S,T(SameasT,S)
S,Z
T,Y
T,Z
V,X
V,Z
X,Y
X,ZPerformabreadth-firstiterationfromS.41Assumetheedgesareinserte4142
queue
Vertexreturnedbynext()ST,Z SZ,Y TY,V,X ZV,X YX VXenqueuedequeue42queue Vertexreturnedby4243434344444445454546Forabreadth-firstiterationofadirected
graph,thealgorithmisthesame,but“neighbor”takesintoaccountthedirectionofthearrow.A
BCDStartingatA,theneighborsofAareBandC,butnotD.DNEVER
getsenqueued,andnevergetsreturnedbynext().46Forabreadth-firstiteratio46Y,V,T XXV,XV,X YT,YHashSetTreeSetCV,ZTHashSetTreeSetZ,Y TcheckC’sneighbors,find<B,9>;andaddittopq;5VerticesT,YCollection6ANetworkClassZ,T SAcollectioninwhich6NetworkClass開發三道工序:Performadepth-firstiterationfromS.HashSetTreeSet47Y,V,T X474748Collection
ListSet
ArrayListLinkedListSortedSet
HashSetTreeSet484849494950A
Stack!50AStack!5051X
V
YZTS515152Assumetheedgesareinsertedintothegraphasfollows:
S,T//SameasT,S
S,Z
T,Y
T,Z
V,X
V,Z
X,Y
X,ZPerformadepth-firstiterationfromS.52Assumetheedgesareinserte5253
Stack Vertexreturnedbynext()
S
Z,T
SX,V,T ZY,V,T X
V,T Y
TV TTopBottom53Stack Vertexreturnedby53545454555555565656orderisunimportant:T某collection的objects的兩兩關係X6NetworkClass開發三道工序:StartingatA,theneighborsofAareBandC,butnotD.ArrayListLinkedListSortedSetXGraphtheory圖論
FromWikipedia,thefreeencyclopedia2DirectedGraphs5
GraphAlgorithmsT,Z SYZS,ZPerformadepth-firstiterationfromS.CollectionX,Y,T,Z,Xisacycle.Trees,andBCDAssumetheedgesareinsertedintothegraphasfollows:YZ57orderisunimportant:57575858585959596060606161616262626363636464646565656666666767676868686969692DirectedGraphsV,T YV,XStack Vertexreturnedbynext()72Inmathematicsandcomputerscience,graphtheoryisthestudyofgraphs,mathematicalstructuresusedtomodelpair-wiserelationsbetweenobjectsfromacertaincollection.TArrayListLinkedListSortedSetArrayListLinkedListSortedSet某collection的objects的兩兩關係5
GraphAlgorithms5
GraphAlgorithmsthedirectionofthearrow.Assumetheedgesareinsertedintothegraphasfollows:6ANetworkClassAssumetheedgesareinsertedintothegraphasfollows:xvV,ZXVT702DirectedGraphs707071717172727273737374747475757576767677KeyIdea
C72A10B
priorityqueue(pq):<C,7><B,10>;remove
<C,7>checkC’sneighbors,find<B,9>;andaddittopq;markCthepredecessorofB
Now,pq:<B,9><B,10>77KeyIdea77787878797979808080818181828282838383848484858585868686878787888888898989Forabreadth-firstiterationofadirectedgraph,Performadepth-firstiterationfromS.Graphtheory圖論
FromWikipedia,thefreeencyclopediapriorityqueue(pq):<C,7><B,10>;remove<C,7>YZT,Z5GraphAlgorithmsThisundirectedgraphisacyclic.6NetworkClass開發三道工序:ArrayListLinkedListSortedSetNetworkClass許多實際問題可用graph來表示。YZCollectionYZYZIsY,Z,T,S,Z,X,Yacycle?Yes!Performabreadth-firstiterationfromS.ArrayListLinkedListSortedSet90Forabreadth-firstiteration909115.6ANetworkClass9115.6ANetworkClass9192929293939394NetworkClass94NetworkClass94959595969696979797989898999999100100100101101101102102102103103103104104104105105105106NetworkClass106NetworkClass106107107107108108108109109109110Acollectioninwhichorderisunimportant:Alinkedlist!
110Acollectioninwhich110111111111112112112113113113114114114115115115116116116117117117118118118119NetworkClass119NetworkClass119120120120121121121122122122123123123124124124125125125126126126127127127128128128129129129130130130131131131132132132133133133134134134135135135136136136137137137138138138139139139140140140141141141142142142143143143144144144145Chapter15
Graphs,Trees,andNetworks1Chapter15
Graphs,145146Chap.1515.1UndirectedGraphs15.2DirectedGraphs15.3Trees15.4Networks15.5GraphAlgorithms
MinimumSpanningTreeShortestPath2Chap.1515.1UndirectedGraph146147
Chap.15(Cont.)15.6NetworkClass開發三道工序:3Chap.15(Cont.)147148Graphtheory圖論
FromWikipedia,thefreeencyclopediaInmathematicsandcomputerscience,graphtheoryisthestudyofgraphs,mathematicalstructuresusedtomodelpair-wiserelationsbetweenobjectsfromacertaincollection.
某collection的objects的兩兩關係4Graphtheory圖論
FromWikipedi148149ApplicationsofGraphs
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。5ApplicationsofGraphs14915015.1UndirectedGraphs615.1UndirectedGraphs150151 Example:
X
YZTAns:
5Vertices
6EdgesS7 Example:151152Collection
ListSet
ArrayListLinkedListSortedSet
HashSetTreeSet815215391531541015415511155S6NetworkClass開發三道工序:Alinkedlist!X某collection的objects的兩兩關係Performadepth-firstiterationfromS.Stack Vertexreturnedbynext()Z,T SHashSetTreeSetListSetxv5
GraphAlgorithms6ANetworkClass6ANetworkClassSqueue Vertexreturnedbynext()Z,T SV,T YYZ開發algorithm來處理graph，因此主要應用是在computerscience。156X
YZTX,Y,T,Z,Shaslength4.ShortestpathfromXtoS=?
Ans:2
SIngeneral,howcanthesmallestpath
between2verticesbedetermined?
Wewillcoverthat.S1215615713157158X
YZTX,Y,T,Z,Xisacycle.Y,T,S,Z,T,Yisnotacycle.(repeatededges:YTandTY)SIsY,Z,T,S,Z,X,Yacycle?
Yes!14158159Collection
ListSet
ArrayListLinkedListSortedSet
HashSetTreeSetThisundirectedgraphisacyclic.15159160161601611716116215.2DirectedGraphs1815.2DirectedGraphs16216319163164201641652116516622166X,Zthedirectionofthearrow.Graphtheory圖論
FromWikipedia,thefreeencyclopediaV,Zorderisunimportant:HashSetTreeSet6NetworkClass開發三道工序:5
GraphAlgorithmsAssumetheedgesareinsertedintothegraphasfollows:TNetworkClassS,T//SameasT,SCollectionPerformadepth-firstiterationfromS.YZStack Vertexreturnedbynext()YZ167X,Z2316716815.3Trees2415.3Trees168169Collection
ListSet
ArrayListLinkedListSortedSet
HashSetTreeSet2516917015.4Networks2615.4Networks170171271711722817217329173174301741753117517615.5
GraphAlgorithms3215.5
GraphAlgorithms1761773317717834178179
xvYZT
SAssumetheverticesareenteredinalphabeticalorder.Performabreadth-firstiterationfromS.
Ans:S,T,Z,Y,V,X
3517918036180YZS,ZAcollectioninwhichForabreadth-firstiterationofadirectedgraph,orderisunimportant:Z,T SHashSetTreeSetYZListSet6NetworkClass開發三道工序:SAssumetheedgesareinsertedintothegraphasfollows:Assumetheedgesareinsertedintothegraphasfollows:orderisunimportant:ArrayListLinkedListSortedSetpriorityqueue(pq):<C,7><B,10>;remove<C,7>S,T(SameasT,S)Athealgorithmisthesame,but“neighbor”takesintoaccountS181Y1811823818218339183184X
V
YZTS40184185Assumetheedgesareinsertedintothegraphasfollows:
S,T(SameasT,S)
S,Z
T,Y
T,Z
V,X
V,Z
X,Y
X,ZPerformabreadth-firstiterationfromS.41Assumetheedgesareinserte185186
queue
Vertexreturnedbynext()ST,Z SZ,Y TY,V,X ZV,X YX VXenqueuedequeue42queue Vertexreturnedby186187431871884418818945189190Forabreadth-firstiterationofadirected
graph,thealgorithmisthesame,but“neighbor”takesintoaccountthedirectionofthearrow.A
BCDStartingatA,theneighborsofAareBandC,butnotD.DNEVER
getsenqueued,andnevergetsreturnedbynext().46Forabreadth-firstiteratio190Y,V,T XXV,XV,X YT,YHashSetTreeSetCV,ZTHashSetTreeSetZ,Y TcheckC’sneighbors,find<B,9>;andaddittopq;5VerticesT,YCollection6ANetworkClassZ,T SAcollectioninwhich6NetworkClass開發三道工序:Performadepth-firstiterationfromS.HashSetTreeSet191Y,V,T X47191192Collection
ListSet
ArrayListLinkedListSortedSet
HashSetTreeSet4819219349193194A
Stack!50AStack!194195X
V
YZTS51195196Assumetheedgesareinsertedintothegraphasfollows:
S,T//SameasT,S
S,Z
T,Y
T,Z
V,X
V,Z
X,Y
X,ZPerformadepth-firstiterationfromS.52Assumetheedgesareinserte196197
Stack Vertexreturnedbynext()
S
Z,T
SX,V,T ZY,V,T X
V,T Y
TV TTopBottom53Stack Vertexreturnedby197198541981995519920056200orderisunimportant:T某collection的objects的兩兩關係X6NetworkClass開發三道工序:StartingatA,theneighborsofAareBandC,butnotD.ArrayListLinkedListSortedSetXGraphtheory圖論
FromWikipedia,thefreeencyclopedia2DirectedGraphs5
GraphAlgorithmsT,Z SYZS,ZPerformadepth-firstiterationfromS.CollectionX,Y,T,Z,Xisacycle.Trees,andBCDAssumetheedgesareinsertedintothegraphasfollows:YZ201orderisunimportant:572012025820220359203204602042056120520662206207632072086420820965209210662102116721121268212213692132DirectedGraphsV,T YV,XStack Vertexreturnedbynext()72Inmathematicsandcomputerscience,graphtheoryisthestudyofgraphs,mathematicalstructuresusedtomodelpair-wiserelationsbetweenobjectsfromacertaincollection.TArrayListLinkedList
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