We investigate here the problem
of establishing communication in an ad-hoc
mobile network, that is, we assume the extreme case
of a total absence of any fixed network infrastructure
(for example a case of rapid deployment of a set of
mobile hosts in an unknown terrain). We propose, in
such a case, that a small
subset of the deployed hosts (which we call the support)
should be used to manage network operations.
However, the vast majority
of the hosts are moving arbitrarily according
to application needs.
We then provide a simple, correct and efficient protocol
for communication establishment
that avoids message flooding.
Our protocol manages to establish communication between
any pair of mobile hosts in small, a-priori
guaranteed time bounds even in the worst case of arbitrary motions of the hosts that do not belong to
the support (provided that they do not deliberately try
to avoid the support).
These time bounds, interestingly, do not depend,
on the number of mobile hosts that do not
belong in the support. They depend only on the size of the area
Our protocol can be implemented
in very efficient ways by exploiting knowledge of the space of motions
or by adding more power to the hosts of the support.
Our results exploit and further develop some
fundamental properties of random walks in finite graphs.