We study the message complexity of the Election Problem
in Hypercube networks, when the processors have a ``Sense of
Direction'', i.e., the capability to distinguish between adjacent
communication links according to some globally consistent scheme.
We present two models of Sense of Direction, which differ regarding
the way the labeling of the links in the network is done: either by
matching based on dimensions, or by distance along a Hamiltonian cycle.
In the dimension model, we give an optimal linear algorithm which uses
the natural dimensional labeling of the communication links. We prove
that, in the distance-based case, the graph symmetry of the hypercube
is broken and, thus, the leader Election does not require a global
maximum-finding algorithm: O(1) messages suffice to select the
leader, whereas the Theta(N) messages are required only for the
final broadcasting. Finally, we study the communication cost of
changing one orientation labeling to the other and prove that O(N)
messages suffice.