In the first part of this paper, we develop a t-spanner for CDC-paths using spatial properties; a sub-network containing O(n/胃) links, for any 胃 > 0, such that CDC-paths increase in cost by at most a factor t = (1 鈭?#xA0;2 sin(胃/2))鈭?. We propose a novel distributed algorithm to compute the spanner using an expected number of O(n log n) fixed-size messages.
In the second part, we present a distributed algorithm to find minimum-cost CDC-paths between two nodes using O(n2) fixed-size messages, by developing an extension of Edmonds鈥?algorithm for minimum-cost perfect matching. In a centralized implementation, our algorithm runs in O(n2) time improving the previous best algorithm which requires O(n3) running time. Moreover, this running time improves to O(n/胃) when used in conjunction with the spanner developed.