This paper addresses the sensing-throughput tradeoff problem by using cluster-based cooperative spectrum sensing (CSS) schemes in two-layer hierarchical cognitive radio networks (CRNs) with soft data fusion. The problem is formulated as a combinatorial optimization problem involving both discrete and continuous variables. To simplify the solution, a reasonable weight fusion rule (WFR) is first optimized. Thus, the problem devolves into a constrained discrete optimization problem. In order to efficiently and effectively resolve this problem, a lexicographical approach is presented that solving two optimal subproblems consecutively. Moreover, for the first optimal subproblem, a closed-form solution is deduced, and an optimal clustering scheme (CS) is also presented for the second optimal subproblem. Numerical results show that the proposed approach achieves a satisfying performance and low complexity.

A non-regular tree T with a prescribed branching sequence (s1,s2,...,sn) is a rooted and ordered tree such that its internal nodes are numbered from 1 to n in preorder and every internal node i in T has si children. Recently, Wu et al. (2010) introduced a concise representation called RD-sequences to represent all non-regular trees and proposed a loopless algorithm for generating all non-regular trees in a Gray-code order. In this paper, based on such a Gray-code order, we present efficient ranking and unranking algorithms of non-regular trees with n internal nodes. Moreover, we show that the ranking algorithm and the unranking algorithm can be run in O(n2) time and O(n2+nSn-1) time, respectively, provided a preprocessing takes O(n2Sn-1) time and space in advance, where .

In this paper, we introduce a concise representation, called right-distance sequences (or RD-sequences for short), to describe all t-ary trees with n internal nodes. A result reveals that there exists a close relationship between the representation and the well-formed sequences suggested by Zaks [Lexicographic generation of ordered trees, Theoretical Computer Science 10 (1980) 63-82]. Using a coding tree and its concomitant tables, a systematical way can help us to investigate the structural representation of t-ary trees. Consequently, we develop efficient algorithms for determining the rank of a given t-ary tree in lexicographic order (i.e., a ranking algorithm), and for converting a positive integer to its corresponding RD-sequence (i.e., an unranking algorithm). Both the ranking and unranking algorithms can be run in O(tn) time and without computing all the entries of the coefficient table.

In literature, many methods have been presented for enumerating binary trees (full binary trees) and regular k-ary trees, while no one for enumerating arbitrary trees or arbitrary k-ary trees. It is proposed in 1997 using a context-free grammar GBT (GFBT) to code binary trees (full binary trees) for enumerating them. In this paper, we use another grammar GT (GTk) to code arbitrary trees (arbitrary k-ary trees) for enumerating them. The properties of words of Ln(GT) (Ln(GTk)) are discussed in depth, including necessary and sufficient conditions for a word, prefix and suffix of Ln(GT) (Ln(GTk)), and efficient algorithms are given and analyzed for the enumeration of words of Ln(GT) (Ln(GTk)).