TransportationNetwork

# TransportationNetwork - Transportation Network Lecture 10...

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Transportation Network Lecture 10 Discrete Mathematical Structures

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Transport Networks 6 5 4 3 2 1 2 2 3 3 3 4 4 5 The unique node with in-degree 0 The source The unique node with out-degree 0 The sink Capacity of edge, C i,j It is assume that all edges are in one direction.
Conservation of flow: Here: 3=1+0+2 Flows 6 5 4 3 2 1 (2, 1 ) (2, 0 ) (3, 2 ) (3, 2 ) (3, 3 ) (4, 2 ) (4, 3 ) (5, 3 ) ( C i,j , F i,j ), For any edge, F i,j C i,j Value of the flow is 5

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Maximum Flows 1 4 3 2 (6, 4 ) (2, 2 ) (4, 2 ) (4, 0 ) (6, 6 ) Value of flow: 6 1 4 3 2 (6, 4 ) (2, 2 ) (4, 4 ) (4, 2 ) (6, 6 ) Value of flow: 8 Basic Problems: (1) Largest value of flow? (2) A flow with the largest value? 1 4 3 2 (6, 6 ) (2, 0 ) (4, 4 ) (4, 4 ) (6, 6 ) Value of flow: 10
How to get the maximum flow? 6 5 4 3 2 1 2 4 5 4 7 5 6

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How to get the maximum flow? 6 5 4 3 2 1 4 5 3 (7,5 ) (5,5) (6,5) 2
Path in symmetric closure of N 1 4 5 Properly oriented 6,2 5,0 6 7,0 Value of flow increased by 4 1 4 5 6,6 5,4 6 7,4

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How to get the maximum flow? 6 5 4 3 2 1 2,2 4,2 5 4 7 5 6,2
How to get the maximum flow? 6 5 4 3 2 1 2,2 4,2 5 4 7,4 5,4 6,6

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How to get the maximum flow? 6 5 4 3 2 1 2,2 4,4 5,2 4,2 7,4 5,4 6,6
Path in symmetric closure of N 1 3 4 Improperly oriented 4,2 2,2 2 5,2 5 6 7,4 5,4 Value of flow increased by 1 1 3 4 4,3 2,1 2 5,3 5 6 7,5 5,5

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How to get the maximum flow? 6 5 4 3 2 1 2, 1 4,4 5,3 4,3 7, 5 5, 5 6,6
So, just think so: How to find the max flow: Find all paths of source and sink in symmetric closure of N Increase the flow of each path, till: one of F ij = Cij (i,j) in N one of F ij = 0 (j,i) in N Just sum the flows incident to sink Question: how to increase it?

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Maximum Flows 1 4 3 2 (6, 6 ) (2, 0 ) (4, 2 ) (4, 2 ) (6, 6 ) Value of flow: 8 1 4 3 2 (6, 6 ) (2, 0 ) (4, 4 ) (4, 4 ) (6, 6 ) Value of flow: 10 1 4 3 (4, 2 ) (4, 2 ) 2
Maximum Flows 1 4 3 2 (6, 4 ) (2, 2 ) (4, 4 ) (4, 2 ) (6, 6 ) Value of flow: 8 1 4 3 2 (6, 6 ) (2, 0 ) (4, 4 ) (4, 4 ) (6, 6 ) Value of flow: 10 1 4 3 2 (2, 2 ) (4, 2 ) 2 This edge is not in N , but in N’ s symmetric closure

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Excess of Capacity 1 4 3 2 (2, 2 ) (4, 2 ) 2 This edge is not in N , but in N’ s symmetric closure π :1 2 3 4 is not a path in N , but in G , the symmetric closure. (1,2) is in N , this edge has a excess capacity 2 (=6-4) (2,3) is not in N , this edge has a excess capacity 2
General Senario C i,j is the capacity of edge ( i , j ) F i,j is the flow on edge ( i , j ) edges in N edges in s( N ), but not in N 6 5 4 3 2 1 e 1,4 e 3,6 e 2,5 e 2,3 e 2,4 e 4,5 e 5,6 e 4,2 e 6,3 e

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TransportationNetwork - Transportation Network Lecture 10...

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