The generalized sorting problem is to find the first k largest elements among n input elements and to report them in a sorted order. In this paper, we propose a fast generalized sorting algorithm under the single hop wireless networks model with collision detection (WNCD). The algorithm is based on the maximum finding algorithm and the sorting algorithm. The key point of our algorithm is to use successful broadcasts to build broadcasting layers logically and then to distribute the data elements into those logic layers properly. Thus, the number of broadcast conflicts is reduced. We prove that the average time complexity required for our generalized sorting algorithm is Θ(k + log(n - k)). When k = 1, our generalized sorting algorithm does the work of finding maximum, and when k = n, it does the work of sorting. Thus, the analysis of our algorithm builds a connection between the two extremely special cases which are maximum finding and sorting.

Given a set of n stations, the initialization problem is to assign each station a unique identification number, from 1 to n. In the single-hop wireless Networks with collision detection, Nakano and Olariu proposed an algorithm to build a partition tree and solve the problem. In this paper, we shall classify the partition tree into four parts. By reviewing the classification, we find that three ideas can improve the algorithm. We show that it needs 2.88n time slots for solving the problem containing n stations. After applying our three ideas, the number of time slots will be improved to 2.46n.