1. Network Problems A D O B T E C 70 40 10 55 60 80 50 20
You need to take a trip by car to another town that you have never visited before. Therefore, you are studying a map to determine the shortest route to your destination. Depending on which route you choose, there are five other towns (call them A, B, C, D, E) that you might pass through on the way. The map shows the mileage along each road that directly connects two towns without any intervening towns. These numbers are summarized in the following table, where a dash indicates that there is no road directly connecting these two towns without going through any other towns.
Formulate this problem as a shortest-path problem by drawing a network where nodes represent towns, links represent roads, and numbers indicate the length of each link in miles. O A B C D E T 40 60 50 10 70 20 55 80

2. [40] A D [0] O B T E C [40] A D [0] [50] O B T E C [50] 70 40 10 55 6020 55 80
[0]
[40] O A B C D E T 40 60 50 10 70 20 55 80
[0]
[50]
[50]

3. [40] A D [0] [50] O B T E C [90] [50] [40] A [100] D [0] [50] O B T E60 50 10 70 20 55 80
[0]
[50]
[90]
[50]
[40] O A B C D E T 40 60 50 10 70 20 55 80
[100]
[0]
[50]
[90]
[50]

4. [40] A [100] D [0] [160] [50] O B T E C [90] [50] [40] A [100] D [0]70 20 55 80
[100]
[0]
[160]
[50]
[90]
[50]
[40] A
[100] 40
[0] 10 D
[160]
[50] 60 O B T 10 40 50 E C
[90]
[50]

5. [4] [10] [0] [16] [5] [9] [5] [4] [7] [10] [0] [8] [17] [11] [6] [3]
Use the algorithm described in Sec to find the shortest path through each of the following networks, where the numbers represent actual distances between the corresponding nodes.
[4]
[10]
[0]
[16]
[5]
[9]
[5]
[4]
[7]
[10]
[0]
[8]
[17]
[11]
[6]
[3]
[9]
[14]

6. x x [4] [-4] (flow, cap remaining) (4,0) (4,2) (4,0)
For networks (a) and (b), use the augmenting path algorithm described in Sec. 9.5 to find the flow pattern giving the maximum flow from the source to the sink, given that the arc capacity from node i to node j is the number nearest node i along the arc between these nodes.
(flow, cap remaining)
(4,0)
(4,2) x
(4,0)
[4] x
[-4]

7. x x x x x x x x x x x x [7] [-7] [8] [-8] [9] [-9]
(flow, cap remaining)
(4,0)
(4,2) x
(4,0)
[7] x
[-7]
(3,1)
(3,0) x
(3,6)
(flow, cap remaining)
(4,0)
(4,2) x
(4,0)
[8] x
[-8]
(4,0) x
(3,0) x
(4,5)
(1,2)
(1,3)
(flow, cap remaining)
(4,0)
(4,2) x
(4,0)
[9] x
[-9]
(4,0) x
(3,0) x
(5,4)
(1,2) x
(1,0)
(2,2)

8. Min cut = 9
[9]
[-9]