Conjecture:
Let G=(V,E) be an undirected graph with weights on its
edges. Then a minimum spanning tree for G will define the
shortest path between any two vertices in G.
Observation:
The above conjecture is incorrect. Consider the complete
graph on three vertices with edge weights 2, 3 and 4 (a
counter example). Note that the MST contains the edges
with weights 2 and 3, but that does not minimize the path
length between two vertices.
So, if MSTs don't minimize path lengths, then what good
are they anyway?
Some Applications of Minimum Spanning Trees
Example #1:
Suppose a set of islands is to be connected by bridges. Furthermore, suppose that the cost
to build a bridge between pairs islands is known. Note that several factors determine the
cost of building a particular bridge, including depth of the water, various under-sea
conditions, etc. The problem is to select a subset of the bridges to be built that connects
all the islands and minimizes total construction cost.
Note that the resulting distance between the islands is not a consideration, and, in
particular, the weights of the edges, doesn't correspond directly with distance.
Example #2:
Suppose n pins on an electrical circuit wire are to be connected by n-1 wires such that the
pins form a single, equivalent electrical signal, i.e, the pins are completely connected. In
addition, suppose the total amount of wire to be used should be minimized.
Note that although the direct distance between pins does translate to the amount of wire
required to connect them directly, the final distance between any two pins is not an issue.
Example #3:
Suppose a cable TV company is about to lay cable to connect houses in a new
neighborhood. Furthermore, suppose the company is constrained to bury the cable only
along certain paths, directly connecting one house to another. Note that some of the paths
might be more expensive than others because they are longer, or require the cable to be
buried deeper. The goal is to select a subset of the paths that uses the least amount of
cable in total.
Again, although there must exist a path using the final set of connections that connects
any two houses, the length of that path doesn't matter.