When the Chinese postman algorithm is used, one group of odd nodes is {AB, C E, F G}. a How many groups of odd nodes are there altogether? b Explain why the total weight of the shortest paths for any group of odd nodes that includes BF = 20 must be greater than 30 . c List the weights and total weight for each group of odd nodes that includes GF . d Explain why GF must be included in the group with least total weight.
PL E
A closed route is constructed that uses every edge once and repeats the shortest routes between A and B, C and E, F and G . e How many times does this route pass through D ? 7
For the network in question 6, which arcs would be repeated in a route that covers every arc but starts and ends at different nodes?
8
The total length of the arcs in a network is 220 metres. The degrees of the nodes are: A
B
C
D
E
F
G
M
Node Degree
2
4
6
3
5
3
5
H
I
J
5
2
3
The lengths, in metres, of the shortest paths between the odd nodes are: DE = 20, DF = 24, DG = 26, DH = 21, DJ = 19, EF = 16,
SA
EG = 34, EH = 14, EJ = 20, F G = 18, F H = 27, F J = 30, GH = 25, GJ = 23, H J = 29.
The shortest route that starts at D and uses every route is required. a Where does the route end?
b Use the Chinese postman algorithm to find the length of the route.
9
The Chinese postman algorithm is to be used on a network with eight odd nodes. a Show that, unless any additional information is given, 105 groups of odd nodes would need to be considered.
b Show that the lengths of 28 shortest paths would need to be calculated. If it takes 2 seconds to calculate each shortest path and 3 seconds to find the total weight of each group of odd nodes, the time taken for a network with eight odd nodes will be over 6 minutes. c Calculate the corresponding time for a network with ten odd-degree nodes.
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