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 $(1-2sin{tfrac{theta}{2}})^{-2}$. We propose a novel algorithm to compute the spanner in a distributed manner using only $O(nlog{n})$ 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.
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