TransportationNetwork

TransportationNetwork - Transportation Network Lecture 10...

Info iconThis preview shows pages 1–18. Sign up to view the full content.

View Full Document Right Arrow Icon
Transportation Network Lecture 10 Discrete Mathematical Structures
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 4
How to get the maximum flow? 6 5 4 3 2 1 2 4 5 4 7 5 6
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
How to get the maximum flow? 6 5 4 3 2 1 4 5 3 (7,5 ) (5,5) (6,5) 2
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
How to get the maximum flow? 6 5 4 3 2 1 2,2 4,2 5 4 7 5 6,2
Background image of page 8
How to get the maximum flow? 6 5 4 3 2 1 2,2 4,2 5 4 7,4 5,4 6,6
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 10
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
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 12
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?
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 14
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
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 16
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
Background image of page 17

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 18
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 47

TransportationNetwork - Transportation Network Lecture 10...

This preview shows document pages 1 - 18. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online