Finding the optimum
11.1 A pessimistic government could follow the following logic: if we are not careful, we may end up with having to build that extremely expensive connection through the mountain; so let us decide right away that building this connection is not an option, and mark it as “impossible”. Similarly, let us find the second most expensive line and mark it “impossible”, etc. Well, we cannot go on like this forever: we have to look at the graph formed by those edges that are still possible, and this “possibility graph” must stay connected. In other words, if deleting the most expensive edge that is still possible from the possibility graph would destroy the connectivity of this graph, then like it or not, we have to build this line. So we build this line (the pessimistic government ends up building the most expensive line among those that are still possible). Then they go on to find the most expensive line among those that are still possible and not yet built, mark it impossible if this does not disconnect the possibility graph etc.
Prove that the pessimistic government will have the same total cost as the optimistic.
11.2 In a more real-life government, optimists and pessimists win in unpredictable order. This means that sometimes they build the cheapest line that does not create a cycle with those lines already constructed; sometimes they mark the most expensive lines “impossible” until they get to a line that cannot be marked impossible without disconnecting the network, and then they build it. Prove that they still end up with the same cost.
11.3 If the seat of the government is town , then quite likely the first line constructed will be the cheapest line out of (to some town , say), then the cheapest line that connects either or to a new town etc. In general, there will be a connected graph of telephone lines constructed on a subset of the towns including the capital, and the next line will be the cheapest among all lines that connect to a node outside . Prove that the lucky government still obtains a cheapest possible tree.
11.4 Formulate how the pessimistic government will construct a cycle through all towns. Show by an example that they don’t always get the cheapest solution.
11.5 Is the tour in figure 29 shortest possible?
11.6 Prove that if all costs are proportional to distances, then a shortest tour cannot intersect itself.
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