9 Finding the Optimum
9.1 Finding the Best Tree
A country withntowns wants to construct a new telephone network to
connect all towns. Of course, they don’t have to build a separate line be-
tween every pair of towns; but they do need to build a connected network;
in our terms, this means that the graph of direct connections must form a
connected graph. Let’s assume that they don’t want to build a direct line
between towns that can be reached otherwise (there may be good reasons
for doing so, as we shall see later, but at the moment let’s assume their only
goal is to get a connected network). Thus they want to build a minimal
connected graph with these nodes, i.e., a tree.
We know that no matter which tree they choose to build, they have to
constructn−1 lines. Does this mean that it does not matter which tree
they build? No, because lines are not equally easy to build. Some lines
between towns may cost much more than some other lines, depending on
how far the towns are, whether there are mountains or lakes between them,
etc. So the task is to find a spanning tree whose total cost (the sum of costs
of its edges) is minimal.
How do we know what these costs are? Well, this is not something mathe-
matics can tell you; it is the job of the engineers and economists to estimate
the cost of each possible line in advance. So we just assume that these costs
are given.
At this point, the task seems trivial (very easy) again: Just compute the
cost of each tree on these nodes, and select the tree with smallest cost.