Lecture 7
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Data Structures Notes for Lecture 7 Non Linear Data Structures By Samaher Hussein Ali 2007-2008
Graphics اﻟﻤﺨﻄﻂ هﻮ ﻋﺒﺎرة ﻋﻦ ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻟﻌﻨﺎﺻﺮ ﺗﻤﺜﻞ ﺑﻨﻘﺎط )رؤوس(ﺗﺴﻤﻰ ) ( verticesوهﺬﻩ اﻟﻌﻨﺎﺻﺮ ﺗﺮﺑﻄﻬﺎ ﻋﻼﻗﺎت ﺗﻤﺜﻞ ﺑﺨﻄﻮط ﺗﺴﻤﻰ اﻟﺤﻮاف)
. ( edges
ﻣﺜﺎل-:
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)G=(V,E }V(G)={1,2,3,4,5,6 })E(G)={(1,2).(1,3),(2,3),(2,4),(3,5),(3,6 اﻟﻤﺨﻄﻄﺎت ﻧﻮﻋﻴﻦ-: * ﻣﺨﻄﻂ ﻏﻴﺮ ﻣﺘﺠﻪ Undirected Graph هﻮ اﻟﻤﺨﻄﻂ اﻟﺬي ﺗﻜﻮن اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻋﻨﺎﺻﺮﻩ ﻏﻴﺮ ﻣﺮﺗﺒﺔ أي ان اﻻﺗﺠﺎﻩ ﻏﻴﺮ ﻣﻬﻢ ﻓﻲ ﺗﻠﻚ اﻟﻌﻼﻗﺔ أي إن اﻟﺤﺎﻓﺔ ) (1,2هﻲ ﻧﻔﺴﻬﺎ )(2,1آﻤﺎ ﻓﻲ اﻟﻤﺜﺎل أﻋﻼﻩ. Directed Graph * ﻣﺨﻄﻂ ﻣﺘﺠﻪ هﻮ اﻟﻤﺨﻄﻂ اﻟﺬي ﺗﻜﻮن اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻋﻨﺎﺻﺮﻩ ﻣﺮﺗﺒﺔ أي ان اﻻﺗﺠﺎﻩ ﻣﻬﻢ ﻓﻲ ﺗﻠﻚ اﻟﻌﻼﻗﺔ أي إن اﻟﺤﺎﻓﺔ ) (1,2هﻲ ﻟﻴﺲ ﻧﻔﺴﻬﺎ ).(2,1 ﻣﺜﺎل-:
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*اﻟﻤﺴﺎر Path هﻮ ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻟﻤﺴﺘﻘﻴﻤﺎت )اﻟﺨﻄﻮط( اﻟﺘﻲ ﺗﺮﺑﻂ ﺑﻴﻦ أي ﻧﻘﻄﺘﻴﻦ ﻓﻲ اﻟﻤﺨﻄﻂ أو هﻮ اﻟﻤﺴﺎرﻣﻦ ﻧﻘﻄﺔ اﻻﻧﻄﻼق اﻟﻰ ﻧﻘﻄﺔ اﻟﻬﺪف . آﺬﻟﻚ ﺗﻘﺴﻢ اﻟﻤﺨﻄﻄﺎت إﻟﻰ-: *ﻣﺨﻄﻄﺎت ﻣﺘﺼﻠﺔ
Connected Graphics
وهﻮ اﻟﻤﺨﻄﻂ اﻟﺬي ﺗﻮﺟﺪ ﻓﻴﻪ ﻣﺴﺎرات ﺑﻴﻦ أي ﻧﻘﻄﺘﻴﻦ ﻣﻦ ﻧﻘﺎﻃﻪ )وﻳﻤﻜﻦ ان ﻳﻜﻮن ﻣﺘﺠﻪ او ﻏﻴﺮ ﻣﺘﺠﻪ( آﻤﺎ ﻓﻲ اﻟﻤﺜﺎﻟﻴﻦ اﻋﻼﻩ. * ﻣﺨﻄﻄﺎت ﻏﻴﺮ اﻟﻤﺘﺼﻠﺔ Unconnected Graphics وهﻮ اﻟﻤﺨﻄﻂ اﻟﺬي ﺗﻮﺟﺪ ﻓﻴﻪ ﻧﻘﺎﻃﻪ ﻏﻴﺮ ﻣﺮﺗﺒﻄﺔ ﺑﻤﺴﺎرات. ﻣﺜﺎل-:
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ﻣﺨﻄﻂ ﻏﻴﺮ ﻣﺘﺼﻞ
Graph Representation ﺗﻤﺜﻴﻞ اﻟﻤﺨﻄﻄﺎت -:ان اﺧﺘﻴﺎر ﻃﺮﻳﻘﺔ ﺗﻤﺜﻴﻞ اﻟﻤﺨﻄﻂ ﺗﻌﺘﻤﺪ ﻋﻠﻰ ﻧﻮع اﻟﺘﻄﺒﻴﻖ اﻟﻤﻄﻠﻮب اﻧﺠﺎزﻩ وﻃﺒﻴﻌﺔ وﻇﺎﺋﻔﻪ وهﻨﺎﻟﻚ ﻃﺮﻳﻘﺘﻴﻦ ﻟﺘﻤﺜﻴﻞ اﻟﻤﺨﻄﻄﺎت هﻲ-: *اﺳﺘﺨﺪام ﻣﺼﻔﻮﻓﺔ اﻟﺘﺠﺎور Adjancency Matrix هﻨﺎ ﻳﺘﻢ ﺗﻤﺜﻴﻞ اﻟﻤﺨﻄﻂ ﺑﻤﺼﻔﻮﻓﺔ ﻣﺮﺑﻌﺔ درﺟﺘﻬﺎ ﻣﺴﺎوﻳﺔ ﻟﻌﺪد ﻋﻨﺎﺻﺮ اﻟﻤﺨﻄﻂ ﻓﺎذا آﺎن اﻟﻤﺨﻄﻂ ﻳﺤﺘﻮي ﻋﻠﻰ ﺛﻼث ﻋﻨﺎﺻﺮ ﻓﺎﻧﻪ ﻳﻤﺜﻞ ﺑﻤﺼﻔﻮﻓﺔ ) (3*3اﻣﺎ اذا آﺎن ﻳﺤﺘﻮي ﻋﻠﻰ ارﺑﻌﺔ ﻋﻨﺎﺻﺮ ﻓﺎﻧﻪ ﻳﻤﺜﻞ ﺑﻤﺼﻔﻮﻓﺔ ) (4*4وهﻜﺬا. ﻣﺜﺎل -:ﻳﻮﺿﺢ آﻴﻒ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﻣﺨﻄﻂ ﻣﺘﺠﻪ ﺑﻤﺼﻔﻮﻓﺔ اﻟﺘﺠﺎور S1
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ﻳﻤﺜﻞ هﺬا اﻟﻤﺨﻄﻂ ﺑﻤﺼﻔﻮﻓﺔ ﻣﺮﺑﻌﺔ ﻣﻦ اﻟﺪرﺟﺔ ) (4وﻋﻨﺎﺻﺮهﺎ ) ( Sijﺣﻴﺚ iﻳﻤﺜﻞ ﻧﻘﻄﺔ اﻟﺒﺪاﻳﺔ و
jﻳﻤﺜﻞ ﻧﻘﻄﺔ اﻟﻨﻬﺎﻳﺔ
وﻓﻲ ﺣﺎﻟﺔ وﺟﻮد ﺣﺎﻓﺔ)ﻣﺴﺎر ( ﺑﻴﻦ ﻧﻘﻄﺘﻴﻦ ﻳﺘﻢ وﺿﻊ)واﺣﺪ( ﻟﻴﻤﺜﻞ اﻟﻤﻮﻗﻊ وﺑﻌﻜﺴﻪ ﻳﺘﻢ وﺿﻊ )ﺻﻔﺮ( 4 1 0
ﻧﻘﺎط اﻟﻨﻬﺎﻳﺔ 3 0 0
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1ﻧﻘﺎط اﻟﺒﺪاﻳﺔ
ﺧﺼﺎﺋﺺ اﻟﻤﺼﻔﻮﻓﺔ اﻟﺘﻲ ﺗﻤﺜﻞ اﻟﻤﺨﻄﻂ اﻟﻤﺘﺠﻪ-: .1ﻣﺠﻤﻮع اﻟﻘﻴﻢ ﻓﻲ اﻟﻌﻤﻮد ﺗﻤﺜﻞ ﻋﺪد اﻟﺨﻄﻮط اﻟﺪاﺧﻠﺔ ﻟﺘﻠﻚ اﻟﻨﻘﻄﺔ .2ﻣﺠﻤﻮع اﻟﻘﻴﻢ ﻓﻲ اﻟﺼﻒ ﺗﻤﺜﻞ ﻋﺪد اﻟﺨﻄﻮط اﻟﺨﺎرﺟﺔ ﻣﻦ داﺧﻠﺔ ﺗﻠﻚ اﻟﻨﻘﻄﺔ .3اﻟﻤﺼﻔﻮﻓﺔ اﻟﻤﺴﺘﺨﺪﻣﺔ ﻟﺨﺰن اﻟﻤﺨﻄﻂ اﻟﻤﺘﺠﻪ ﺗﻤﺘﺎز ﺑﺎﻧﻬﺎ ﻏﻴﺮ ﻣﺘﻨﺎﻇﺮة ﻣﺜﺎل -:ﻳﻮﺿﺢ آﻴﻒ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﻣﺨﻄﻂ ﻏﻴﺮ ﻣﺘﺠﻪ ﺑﻤﺼﻔﻮﻓﺔ اﻟﺘﺠﺎور T1
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ﻳﻤﺜﻞ هﺬا اﻟﻤﺨﻄﻂ ﺑﻤﺼﻔﻮﻓﺔ ﻣﺮﺑﻌﺔ ﻣﻦ اﻟﺪرﺟﺔ ) (5وﻋﻨﺎﺻﺮهﺎ ) ( Tijﺣﻴﺚ iﻳﻤﺜﻞ ﻧﻘﻄﺔ اﻟﺒﺪاﻳﺔ و
jﻳﻤﺜﻞ ﻧﻘﻄﺔ اﻟﻨﻬﺎﻳﺔ
وﻓﻲ ﺣﺎﻟﺔ وﺟﻮد ﺣﺎﻓﺔ)ﻣﺴﺎر ( ﺑﻴﻦ ﻧﻘﻄﺘﻴﻦ ﻳﺘﻢ وﺿﻊ)واﺣﺪ( ﻟﻴﻤﺜﻞ اﻟﻤﻮﻗﻊ وﺑﻌﻜﺴﻪ ﻳﺘﻢ وﺿﻊ )ﺻﻔﺮ( 5
ﻧﻘﺎط اﻟﻨﻬﺎﻳﺔ 3 1 1
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1ﻧﻘﺎط اﻟﺒﺪاﻳﺔ 2
ﺧﺼﺎﺋﺺ اﻟﻤﺼﻔﻮﻓﺔ اﻟﺘﻲ ﺗﻤﺜﻞ اﻟﻤﺨﻄﻂ ﻏﻴﺮ اﻟﻤﺘﺠﻪ-: .1ﻣﺠﻤﻮع اﻟﻘﻴﻢ ﻓﻲ اﻟﻌﻤﻮد ﺗﻤﺜﻞ ﻋﺪد اﻟﺨﻄﻮط اﻟﺪاﺧﻠﺔ ﻟﺘﻠﻚ اﻟﻨﻘﻄﺔ .2ﻣﺠﻤﻮع اﻟﻘﻴﻢ ﻓﻲ اﻟﺼﻒ ﺗﻤﺜﻞ ﻋﺪد اﻟﺨﻄﻮط اﻟﺨﺎرﺟﺔ ﻣﻦ داﺧﻠﺔ ﺗﻠﻚ اﻟﻨﻘﻄﺔ .3اﻟﻤﺼﻔﻮﻓﺔ اﻟﻤﺴﺘﺨﺪﻣﺔ ﻟﺨﺰن اﻟﻤﺨﻄﻂ ﻏﻴﺮ اﻟﻤﺘﺠﻪ ﺗﻤﺘﺎز ﺑﺎﻧﻬﺎ ﻣﺘﻨﺎﻇﺮة ﺣﻮل اﻟﻤﺤﻮر )اﻟﻤﺜﻠﺚ اﻻﻋﻠﻰ ﻳﻨﺎﻇﺮ اﻟﻤﺜﻠﺚ اﻻﺳﻔﻞ( وﺑﺬﻟﻚ ﻳﻤﻜﻦ اﺧﺘﺼﺎر ﻧﺼﻒ اﻟﻤﺴﺎﺣﺔ اﻟﺨﺰﻧﻴﺔ ﻣﻦ ﺧﻼل ﺗﻤﺜﻴﻞ اﺣﺪ اﻟﻤﺜﻠﺜﻴﻦ ﻓﻘﻂ. *اﺳﺘﺨﺪام ﻗﻮاﺋﻢ اﻟﺘﺠﺎور
Adjancency Lists
. Linked ﺗﺴﺘﺨﺪم اﻟﻘﻮاﺋﻢ اﻟﻤﻮﺻﻮﻟﺔ List
ﻟﺘﻤﺜﻴﻞ اﻟﻤﺨﻄﻂ ﺣﻴﺚ ان آﻞ ﻋﻘﺪة ﻣﻦ ﻋﻘﺪ اﻟﻤﺨﻄﻂ ﺗﻤﺜﻞ ﺑﻘﺎﺋﻤﺔ ﻣﺘﺼﻠﺔ ﺗﺤﺘﻮي اﺳﻤﺎء اﻟﻌﻘﺪ
اﻟﺘﻲ ﺗﺘﺼﻞ ﺑﻬﺎ .
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ﻳﻮﺿﺢ آﻴﻒ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﻣﺨﻄﻂ ﻏﻴﺮ ﻣﺘﺠﻪ ﺑﻘﻮاﺋﻢ اﻟﺘﺠﺎور-:ﻣﺜﺎل 1
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2* ﻋﺪد اﻟﻌﻘﺪ اﻟﺘﻲ ﻳﻤﻜﻦ ان أﻣﺜﻠﻬﺎ ﻟﺬﻟﻚ اﻟﻤﺨﻄﻂ= ﻋﺪد اﻟﺤﻮاف (No of nodes=No of edges*2) -:( ﻟﺬﻟﻚ ﻳﻤﻜﻦ ان ﻳﻤﺜﻞ ﺑﻘﻮاﺋﻢ اﻟﻤﺘﺠﺎورة آﻤﺎ ﻳﺄﺗﻲ1,2,3,4)هﺬا اﻟﻤﺨﻄﻂ ﻳﺤﺘﻮي ﻋﻠﻰ ارﺑﻌﺔ رؤس هﻲ Vertex1
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Vertex2 Vertex3 Vertex4
-:اﻟﺘﻌﺮﻳﻒ اﻟﺒﺮﻣﺠﻲ Public Type ptr vertex as integer {or any other type} Link as variant End type Public Type pointer vertex as integer {or any other type} Link as ptr End type Public heads(1 to n) as pointer ﻳﻮﺿﺢ آﻴﻒ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﻣﺨﻄﻂ ﻣﺘﺠﻪ ﺑﻘﻮاﺋﻢ اﻟﺘﺠﺎور-:ﻣﺜﺎل 2
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ﻋﺪد اﻟﻌﻘﺪ اﻟﺘﻲ ﻳﻤﻜﻦ ان أﻣﺜﻠﻬﺎ ﻟﺬﻟﻚ اﻟﻤﺨﻄﻂ= ﻋﺪد اﻟﺤﻮاف )(No of nodes=No of edges هﺬا اﻟﻤﺨﻄﻂ ﻳﺤﺘﻮي ﻋﻠﻰ ارﺑﻌﺔ رؤس هﻲ) (1,2,3,4ﻟﺬﻟﻚ ﻳﻤﻜﻦ ان ﻳﻤﺜﻞ ﺑﻘﻮاﺋﻢ اﻟﻤﺘﺠﺎورة آﻤﺎ ﻳﺄﺗﻲ-: null 3
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Vertex1 Vertex 2
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Graphics اﻟﻤﺨﻄﻂ هﻮ ﻋﺒﺎرة ﻋﻦ ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻟﻌﻨﺎﺻﺮ ﺗﻤﺜﻞ ﺑﻨﻘﺎط )رؤوس(ﺗﺴﻤﻰ ) ( verticesوهﺬﻩ اﻟﻌﻨﺎﺻﺮ ﺗﺮﺑﻄﻬﺎ ﻋﻼﻗﺎت ﺗﻤﺜﻞ ﺑﺨﻄﻮط ﺗﺴﻤﻰ اﻟﺤﻮاف)
. ( edges
ﻣﺜﺎل-:
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)G=(V,E }V(G)={1,2,3,4,5,6 })E(G)={(1,2).(1,3),(2,3),(2,4),(3,5),(3,6 اﻟﻤﺨﻄﻄﺎت ﻧﻮﻋﻴﻦ-: * ﻣﺨﻄﻂ ﻏﻴﺮ ﻣﺘﺠﻪ Undirected Graph هﻮ اﻟﻤﺨﻄﻂ اﻟﺬي ﺗﻜﻮن اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻋﻨﺎﺻﺮﻩ ﻏﻴﺮ ﻣﺮﺗﺒﺔ أي ان اﻻﺗﺠﺎﻩ ﻏﻴﺮ ﻣﻬﻢ ﻓﻲ ﺗﻠﻚ اﻟﻌﻼﻗﺔ أي إن اﻟﺤﺎﻓﺔ ) (1,2هﻲ ﻧﻔﺴﻬﺎ )(2,1آﻤﺎ ﻓﻲ اﻟﻤﺜﺎل أﻋﻼﻩ. Directed Graph * ﻣﺨﻄﻂ ﻣﺘﺠﻪ هﻮ اﻟﻤﺨﻄﻂ اﻟﺬي ﺗﻜﻮن اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻋﻨﺎﺻﺮﻩ ﻣﺮﺗﺒﺔ أي ان اﻻﺗﺠﺎﻩ ﻣﻬﻢ ﻓﻲ ﺗﻠﻚ اﻟﻌﻼﻗﺔ أي إن اﻟﺤﺎﻓﺔ ) (1,2هﻲ ﻟﻴﺲ ﻧﻔﺴﻬﺎ ).(2,1 ﻣﺜﺎل-:
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*اﻟﻤﺴﺎر Path هﻮ ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻟﻤﺴﺘﻘﻴﻤﺎت )اﻟﺨﻄﻮط( اﻟﺘﻲ ﺗﺮﺑﻂ ﺑﻴﻦ أي ﻧﻘﻄﺘﻴﻦ ﻓﻲ اﻟﻤﺨﻄﻂ أو هﻮ اﻟﻤﺴﺎرﻣﻦ ﻧﻘﻄﺔ اﻻﻧﻄﻼق اﻟﻰ ﻧﻘﻄﺔ اﻟﻬﺪف . آﺬﻟﻚ ﺗﻘﺴﻢ اﻟﻤﺨﻄﻄﺎت إﻟﻰ-: *ﻣﺨﻄﻄﺎت ﻣﺘﺼﻠﺔ
Connected Graphics
وهﻮ اﻟﻤﺨﻄﻂ اﻟﺬي ﺗﻮﺟﺪ ﻓﻴﻪ ﻣﺴﺎرات ﺑﻴﻦ أي ﻧﻘﻄﺘﻴﻦ ﻣﻦ ﻧﻘﺎﻃﻪ )وﻳﻤﻜﻦ ان ﻳﻜﻮن ﻣﺘﺠﻪ او ﻏﻴﺮ ﻣﺘﺠﻪ( آﻤﺎ ﻓﻲ اﻟﻤﺜﺎﻟﻴﻦ اﻋﻼﻩ. * ﻣﺨﻄﻄﺎت ﻏﻴﺮ اﻟﻤﺘﺼﻠﺔ Unconnected Graphics وهﻮ اﻟﻤﺨﻄﻂ اﻟﺬي ﺗﻮﺟﺪ ﻓﻴﻪ ﻧﻘﺎﻃﻪ ﻏﻴﺮ ﻣﺮﺗﺒﻄﺔ ﺑﻤﺴﺎرات. ﻣﺜﺎل-:
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ﻣﺨﻄﻂ ﻏﻴﺮ ﻣﺘﺼﻞ
Graph Representation ﺗﻤﺜﻴﻞ اﻟﻤﺨﻄﻄﺎت -:ان اﺧﺘﻴﺎر ﻃﺮﻳﻘﺔ ﺗﻤﺜﻴﻞ اﻟﻤﺨﻄﻂ ﺗﻌﺘﻤﺪ ﻋﻠﻰ ﻧﻮع اﻟﺘﻄﺒﻴﻖ اﻟﻤﻄﻠﻮب اﻧﺠﺎزﻩ وﻃﺒﻴﻌﺔ وﻇﺎﺋﻔﻪ وهﻨﺎﻟﻚ ﻃﺮﻳﻘﺘﻴﻦ ﻟﺘﻤﺜﻴﻞ اﻟﻤﺨﻄﻄﺎت هﻲ-: *اﺳﺘﺨﺪام ﻣﺼﻔﻮﻓﺔ اﻟﺘﺠﺎور Adjancency Matrix هﻨﺎ ﻳﺘﻢ ﺗﻤﺜﻴﻞ اﻟﻤﺨﻄﻂ ﺑﻤﺼﻔﻮﻓﺔ ﻣﺮﺑﻌﺔ درﺟﺘﻬﺎ ﻣﺴﺎوﻳﺔ ﻟﻌﺪد ﻋﻨﺎﺻﺮ اﻟﻤﺨﻄﻂ ﻓﺎذا آﺎن اﻟﻤﺨﻄﻂ ﻳﺤﺘﻮي ﻋﻠﻰ ﺛﻼث ﻋﻨﺎﺻﺮ ﻓﺎﻧﻪ ﻳﻤﺜﻞ ﺑﻤﺼﻔﻮﻓﺔ ) (3*3اﻣﺎ اذا آﺎن ﻳﺤﺘﻮي ﻋﻠﻰ ارﺑﻌﺔ ﻋﻨﺎﺻﺮ ﻓﺎﻧﻪ ﻳﻤﺜﻞ ﺑﻤﺼﻔﻮﻓﺔ ) (4*4وهﻜﺬا. ﻣﺜﺎل -:ﻳﻮﺿﺢ آﻴﻒ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﻣﺨﻄﻂ ﻣﺘﺠﻪ ﺑﻤﺼﻔﻮﻓﺔ اﻟﺘﺠﺎور S1
S3
S4
S2
ﻳﻤﺜﻞ هﺬا اﻟﻤﺨﻄﻂ ﺑﻤﺼﻔﻮﻓﺔ ﻣﺮﺑﻌﺔ ﻣﻦ اﻟﺪرﺟﺔ ) (4وﻋﻨﺎﺻﺮهﺎ ) ( Sijﺣﻴﺚ iﻳﻤﺜﻞ ﻧﻘﻄﺔ اﻟﺒﺪاﻳﺔ و
jﻳﻤﺜﻞ ﻧﻘﻄﺔ اﻟﻨﻬﺎﻳﺔ
وﻓﻲ ﺣﺎﻟﺔ وﺟﻮد ﺣﺎﻓﺔ)ﻣﺴﺎر ( ﺑﻴﻦ ﻧﻘﻄﺘﻴﻦ ﻳﺘﻢ وﺿﻊ)واﺣﺪ( ﻟﻴﻤﺜﻞ اﻟﻤﻮﻗﻊ وﺑﻌﻜﺴﻪ ﻳﺘﻢ وﺿﻊ )ﺻﻔﺮ( 4 1 0
ﻧﻘﺎط اﻟﻨﻬﺎﻳﺔ 3 0 0
1 0
1 0 0
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1ﻧﻘﺎط اﻟﺒﺪاﻳﺔ
ﺧﺼﺎﺋﺺ اﻟﻤﺼﻔﻮﻓﺔ اﻟﺘﻲ ﺗﻤﺜﻞ اﻟﻤﺨﻄﻂ اﻟﻤﺘﺠﻪ-: .1ﻣﺠﻤﻮع اﻟﻘﻴﻢ ﻓﻲ اﻟﻌﻤﻮد ﺗﻤﺜﻞ ﻋﺪد اﻟﺨﻄﻮط اﻟﺪاﺧﻠﺔ ﻟﺘﻠﻚ اﻟﻨﻘﻄﺔ .2ﻣﺠﻤﻮع اﻟﻘﻴﻢ ﻓﻲ اﻟﺼﻒ ﺗﻤﺜﻞ ﻋﺪد اﻟﺨﻄﻮط اﻟﺨﺎرﺟﺔ ﻣﻦ داﺧﻠﺔ ﺗﻠﻚ اﻟﻨﻘﻄﺔ .3اﻟﻤﺼﻔﻮﻓﺔ اﻟﻤﺴﺘﺨﺪﻣﺔ ﻟﺨﺰن اﻟﻤﺨﻄﻂ اﻟﻤﺘﺠﻪ ﺗﻤﺘﺎز ﺑﺎﻧﻬﺎ ﻏﻴﺮ ﻣﺘﻨﺎﻇﺮة ﻣﺜﺎل -:ﻳﻮﺿﺢ آﻴﻒ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﻣﺨﻄﻂ ﻏﻴﺮ ﻣﺘﺠﻪ ﺑﻤﺼﻔﻮﻓﺔ اﻟﺘﺠﺎور T1
T3
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T5
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ﻳﻤﺜﻞ هﺬا اﻟﻤﺨﻄﻂ ﺑﻤﺼﻔﻮﻓﺔ ﻣﺮﺑﻌﺔ ﻣﻦ اﻟﺪرﺟﺔ ) (5وﻋﻨﺎﺻﺮهﺎ ) ( Tijﺣﻴﺚ iﻳﻤﺜﻞ ﻧﻘﻄﺔ اﻟﺒﺪاﻳﺔ و
jﻳﻤﺜﻞ ﻧﻘﻄﺔ اﻟﻨﻬﺎﻳﺔ
وﻓﻲ ﺣﺎﻟﺔ وﺟﻮد ﺣﺎﻓﺔ)ﻣﺴﺎر ( ﺑﻴﻦ ﻧﻘﻄﺘﻴﻦ ﻳﺘﻢ وﺿﻊ)واﺣﺪ( ﻟﻴﻤﺜﻞ اﻟﻤﻮﻗﻊ وﺑﻌﻜﺴﻪ ﻳﺘﻢ وﺿﻊ )ﺻﻔﺮ( 5
ﻧﻘﺎط اﻟﻨﻬﺎﻳﺔ 3 1 1
2 1 0
1 0 1
4 0 0
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4 5
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1ﻧﻘﺎط اﻟﺒﺪاﻳﺔ 2
ﺧﺼﺎﺋﺺ اﻟﻤﺼﻔﻮﻓﺔ اﻟﺘﻲ ﺗﻤﺜﻞ اﻟﻤﺨﻄﻂ ﻏﻴﺮ اﻟﻤﺘﺠﻪ-: .1ﻣﺠﻤﻮع اﻟﻘﻴﻢ ﻓﻲ اﻟﻌﻤﻮد ﺗﻤﺜﻞ ﻋﺪد اﻟﺨﻄﻮط اﻟﺪاﺧﻠﺔ ﻟﺘﻠﻚ اﻟﻨﻘﻄﺔ .2ﻣﺠﻤﻮع اﻟﻘﻴﻢ ﻓﻲ اﻟﺼﻒ ﺗﻤﺜﻞ ﻋﺪد اﻟﺨﻄﻮط اﻟﺨﺎرﺟﺔ ﻣﻦ داﺧﻠﺔ ﺗﻠﻚ اﻟﻨﻘﻄﺔ .3اﻟﻤﺼﻔﻮﻓﺔ اﻟﻤﺴﺘﺨﺪﻣﺔ ﻟﺨﺰن اﻟﻤﺨﻄﻂ ﻏﻴﺮ اﻟﻤﺘﺠﻪ ﺗﻤﺘﺎز ﺑﺎﻧﻬﺎ ﻣﺘﻨﺎﻇﺮة ﺣﻮل اﻟﻤﺤﻮر )اﻟﻤﺜﻠﺚ اﻻﻋﻠﻰ ﻳﻨﺎﻇﺮ اﻟﻤﺜﻠﺚ اﻻﺳﻔﻞ( وﺑﺬﻟﻚ ﻳﻤﻜﻦ اﺧﺘﺼﺎر ﻧﺼﻒ اﻟﻤﺴﺎﺣﺔ اﻟﺨﺰﻧﻴﺔ ﻣﻦ ﺧﻼل ﺗﻤﺜﻴﻞ اﺣﺪ اﻟﻤﺜﻠﺜﻴﻦ ﻓﻘﻂ. *اﺳﺘﺨﺪام ﻗﻮاﺋﻢ اﻟﺘﺠﺎور
Adjancency Lists
. Linked ﺗﺴﺘﺨﺪم اﻟﻘﻮاﺋﻢ اﻟﻤﻮﺻﻮﻟﺔ List
ﻟﺘﻤﺜﻴﻞ اﻟﻤﺨﻄﻂ ﺣﻴﺚ ان آﻞ ﻋﻘﺪة ﻣﻦ ﻋﻘﺪ اﻟﻤﺨﻄﻂ ﺗﻤﺜﻞ ﺑﻘﺎﺋﻤﺔ ﻣﺘﺼﻠﺔ ﺗﺤﺘﻮي اﺳﻤﺎء اﻟﻌﻘﺪ
اﻟﺘﻲ ﺗﺘﺼﻞ ﺑﻬﺎ .
3
ﻳﻮﺿﺢ آﻴﻒ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﻣﺨﻄﻂ ﻏﻴﺮ ﻣﺘﺠﻪ ﺑﻘﻮاﺋﻢ اﻟﺘﺠﺎور-:ﻣﺜﺎل 1
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2* ﻋﺪد اﻟﻌﻘﺪ اﻟﺘﻲ ﻳﻤﻜﻦ ان أﻣﺜﻠﻬﺎ ﻟﺬﻟﻚ اﻟﻤﺨﻄﻂ= ﻋﺪد اﻟﺤﻮاف (No of nodes=No of edges*2) -:( ﻟﺬﻟﻚ ﻳﻤﻜﻦ ان ﻳﻤﺜﻞ ﺑﻘﻮاﺋﻢ اﻟﻤﺘﺠﺎورة آﻤﺎ ﻳﺄﺗﻲ1,2,3,4)هﺬا اﻟﻤﺨﻄﻂ ﻳﺤﺘﻮي ﻋﻠﻰ ارﺑﻌﺔ رؤس هﻲ Vertex1
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Vertex2 Vertex3 Vertex4
-:اﻟﺘﻌﺮﻳﻒ اﻟﺒﺮﻣﺠﻲ Public Type ptr vertex as integer {or any other type} Link as variant End type Public Type pointer vertex as integer {or any other type} Link as ptr End type Public heads(1 to n) as pointer ﻳﻮﺿﺢ آﻴﻒ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﻣﺨﻄﻂ ﻣﺘﺠﻪ ﺑﻘﻮاﺋﻢ اﻟﺘﺠﺎور-:ﻣﺜﺎل 2
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ﻋﺪد اﻟﻌﻘﺪ اﻟﺘﻲ ﻳﻤﻜﻦ ان أﻣﺜﻠﻬﺎ ﻟﺬﻟﻚ اﻟﻤﺨﻄﻂ= ﻋﺪد اﻟﺤﻮاف )(No of nodes=No of edges هﺬا اﻟﻤﺨﻄﻂ ﻳﺤﺘﻮي ﻋﻠﻰ ارﺑﻌﺔ رؤس هﻲ) (1,2,3,4ﻟﺬﻟﻚ ﻳﻤﻜﻦ ان ﻳﻤﺜﻞ ﺑﻘﻮاﺋﻢ اﻟﻤﺘﺠﺎورة آﻤﺎ ﻳﺄﺗﻲ-: null 3
null
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4 1
Vertex1 Vertex 2
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Vertex3
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Vertex 4
