W
2
−
5
6
3
X
−
5
−
5
4
Y
4
6
5
−
−
Z
−
3
4
−
−
The university want to minimise the money spent on installing shelters.
a Which footpaths should have shelters installed for it to be possible to travel between any of the buildings when it is raining?
PL E
b If an additional £5000 can be spent on the shelters, which footpath should it be spent on? Explain your choice. 8
Give an example of a network on A, B and C in which AB has the minimum weight but there is a minimum spanning tree that does not use AB .
9
Following a heavy snowstorm a snowplough is used to clear the snow from some of the
roads between five houses. In a weighted network the houses are represented as nodes and the roads are represented as arcs. The weight of each arc is the time taken (in minutes) to clear the snow from that road.
A
B
C
D
E
−
15
33
28
−
15
−
12
−
40
33
12
−
45
−
SA
C
B
M
A
D
28
−
45
−
68
E
−
40
−
68
−
a Find the minimum time needed to clear a route to each house, given that the time needed to drive the snowplough along any road that has already been cleared of snow can be ignored. b Find the minimum time needed to clear a route to each house, given that instead, the time needed to drive the snowplough along a road that has already been cleared of snow is exactly half the time needed to clear the snow from that road.
10 Why is it not possible to have directed arcs in a minimum spanning tree?
Original material © Cambridge University Press 2018