The problem of "Shortest Connectivity", which is discussed here, has a
long and convoluted history. Many scientists from many fields as well as
laymen have stepped on its stage. Usually, the problem is known as
Steiner's Problem and it can be described more precisely in the
following way: Given a finite set of points in a metric space, search
for a network that connects these points with the shortest possible
length. This shortest network must be a tree and is called a Steiner
Minimal Tree (SMT). It may contain vertices different from the points
which are to be connected. Such points are called Steiner points.
Steiner's Problem seems disarmingly simple, but it is rich with
possibilities and difficulties, even in the simplest case, the Euclidean
plane. This is one of the reasons that an enormous volume of literature
has been published, starting in 1 the seventeenth century and continuing
until today. The difficulty is that we look for the shortest network
overall. Minimum span- ning networks have been well-studied and solved
eompletely in the case where only the given points must be connected.
The novelty of Steiner's Problem is that new points, the Steiner points,
may be introduced so that an intercon- necting network of all these
points will be shorter. This also shows that it is impossible to solve
the problem with combinatorial and geometric methods alone.