  Multi-channel wireless networks are increasingly being employed as
infrastructure networks, e.g. in metro areas. Nodes in these networks
frequently employ directional antennas to improve spatial throughput. In such
networks, given a source and destination, it is of interest to compute an
optimal path and channel assignment on every link in the path such that the
path bandwidth is the same as that of the link bandwidth and such a path
satisfies the constraint that no two consecutive links on the path are assigned
the same channel, referred to as "Channel Discontinuity Constraint" (CDC).
CDC-paths are also quite useful for TDMA system, where preferably every
consecutive links along a path are assigned different time slots.
  This paper contains several contributions. We first present an $O(N^{2})$
distributed algorithm for discovering the shortest CDC-path between given
source and destination. This improves the running time of the $O(N^{3})$
centralized algorithm of Ahuja et al. for finding the minimum-weight CDC-path.
Our second result is a generalized $t$-spanner for CDC-path; For any $\theta>0$
we show how to construct a sub-network containing only $O(\frac{N}{\theta})$
edges, such that that length of shortest CDC-paths between arbitrary sources
and destinations increases by only a factor of at most
$(1-2\sin{\tfrac{\theta}{2}})^{-2}$. We propose a novel algorithm to compute
the spanner in a distributed manner using only $O(n\log{n})$ messages. An
important conclusion of this scheme is in the case of directional antennas are
used. In this case, it is enough to consider only the two closest nodes in each
cone.
