We study the message complexity of distributed algorithms in Tori
and Chordal Rings when the communication links are unlabelled, which
implies that the processors do not have ``Sense of Direction''.
We introduce the paradigm of *handrail* which allows messages to
travel with a consistent direction. We give a distributed algorithm
which confirms the conjecture that the Leader Election problem for
unlabelled Tori of N processors can be solved using Theta(N)
messages instead of O(N log N). Using the same *handrail*
paradigm, we solve the Election problem using Theta(N) messages in
unlabelled chordal rings with one chord (of length approximately
sqrt{N}). This solves a long-standing open problem of the minimal
number of unlabelled chords required to decrease to decrease the
O(N log N) message complexity.
For each topology, we give an algorithm to compute the Sense of
Direction in Theta(N) messages (improving the O(N log N)
previous results). This proves the more fundamental result
that any global distributed algorithm for these labelled topologies
can be used with a similar asymptotic complexity in the respective
unlabelled class.