Abstract
Control channel is used to transmit protocol or signal information between wireless network nodes and is a key component of wireless network. Compared with data information, protocol or signal information is usually much less, so the spectrum bandwidth requirement of control channel is also much less than that of data channel. In order to optimize the usage of the limited spectrum resources, this paper focuses on the issue of control channel selection. We propose a greedy algorithm which minimizes the total spectrum bandwidth of the set of control channels. Theoretical analysis proves that the proposed algorithm can achieve the optimal set of control whose sum of the spectrum bandwidth is the minimum. Simulation results also show that the proposed algorithm consumes less spectrum resources than other algorithms in the same wireless network environment.
1. Introduction
Recently, with the rapid development of the wireless communication technology, wireless network has been the most important infrastructure in the daily life. In order to organize each node in wireless networks to work together, control channel is used to transmit protocol or signal information between wireless network nodes and is a key component of wireless network. Control channel is usually divided into two categories: inband control channel and outofband control channel. The inband control channel means that protocol or signal information is transmitted in the same channel with data information. On the contrary, outofband control channel is different from the channel used to transmit data information. Channel assignment [1–4] is a very important issue for reducing interference and improving throughput in wireless networks, and the generation and selection of control channel are the base for solving this issue. In this paper, we study the issue of optimal selection of outofband control channel.
In wireless networks, each node needs to select one channel from its available channels as the control channel to transmit protocol or signal information with its neighbor nodes. To transmit protocol or signal information, each control channel needs to occupy spectrum bandwidth. The problem is how to select the set of control channels for the whole wireless network which minimizes the total spectrum bandwidth of the control channels. If all nodes in wireless networks have the same available channels, the channel with the minimum spectrum bandwidth will be selected as the control channel. However, in real wireless networks, the available channel set of each node may be different. For example, in cognitive wireless networks, due to the geographical location difference and activity of primary user, the available channel set of each node is quite different, and the spectrum bandwidth of each available channel is also different. In this paper, we set the spectrum bandwidth as channel weight and propose a greedy algorithm to solve the problem of optimal selection of control channel under the constraint of different nodes with different available channel sets. The proposed algorithm is composed of two parts. In the first part of the algorithm, we delete the wireless network node whose accessible channel set completely includes that of some other nodes. Then, in the second part of the algorithm, we iteratively choose the channel which has the minimum spectrum bandwidth in an accessible channel set containing the maximum spectrum bandwidth in the current iteration.
The problem of the optimal control channel selection and related research methods which are also akin to selforganization and evolutionary game theory have been paid attention widely [5–20]. Zhong has proposed an optimal algorithm for this problem [21]. But it assumes that all of the channels have the same spectrum bandwidth. However, in reality, different channels maybe have different spectrum bandwidths. To address this issue, we design a greedy algorithm to minimize the total spectrum bandwidth of the control channels. Theoretical analysis shows that our proposed algorithm can achieve the optimal control channels, where sum of the spectrum bandwidth is the minimum with the time complexity of .
2. Preliminaries
(1) To solve the selection of the control channel, we make the following assumptions. (a) Each of the wireless network nodes has a set of the available channels. In this paper, the spectrum bandwidth occupied by the channel is monotonous. We assume that, as long as the index of the channel is bigger, the spectrum bandwidth it occupies is bigger. (b) Wireless network nodes can transmit protocol or signal information with one another through their common control channels. Note that our objective is to find the optimal set of control channels with the sum of spectrum bandwidth as small as possible. Let be the set of the wireless network nodes; that is, , and let be the set of available channels; that is, (here ). For each node , the channels that can be accessed are through ; that is, . A function for the spectrum bandwidth of each channel has the monotonously increasing property. (2) The objective is to find a subset of control channels with the minimum sum of spectrum bandwidths ; that is, , , such that holds and is minimum.
3. Algorithm Overview
We describe the algorithm in this section. For an arbitrary instance, the wireless network nodes and the channels set with the associated function are all given. The pseudocode of the algorithm is presented as Algorithm 1.

4. Analysis of the Algorithm
For showing that the output of the algorithm is optimal, we first give the following lemma.
Lemma 1. When the algorithm terminates, is a subset of such that, for any , there exists , s.t., .
Proof. Let the value of be after Steps . We first prove that, for any , there exists such that . Note that, for any , there is an iteration which removes from . Then there are two possible cases in this iteration from Steps . (1)If , will be removed from and is added to . Hence . (2)If , then we have since is removed from in this iteration according to the algorithm. Furthermore, we should have , otherwise we would have , since , where is the left nodes in the current iteration. However, in such a case should have been removed in Steps , which is a contradiction. To sum up, we have and .
To conclude the above two possible cases, we get that, for any , we have such that . For any , there must exist such that and from Steps . According to the above conclusion, we get that there must exist such that . This completes the proof of the lemma.
Then the main result of the paper is presented as follows.
Theorem 2. The subset of is the optimal control channels with the minimum spectrum width and the time complexity of the algorithm is .
Proof. We first claim that is a subset of with the smallest cardinality such that, for all , there exists , such that .
From the proof of Lemma 1, and the algorithm outputs a subset such that, for all , there exists such that . Considering any subset of such that, for all , there exists such that . Now sort the elements of in decreasing order: .
By induction, we show that , , where is the element added into in the th iteration of the “While” cycle in the algorithm.
From the algorithm we define , where . Provided that , then, for all , according to the algorithm. Hence, there is no such that , which is a contradiction. Then . Suppose that , where . Let the value of be at the end of the th iteration. Therefore, , where . Suppose that ; then we have . On the other hand, for all , then . According to the algorithm, we have (otherwise would have been removed from in the th iteration). Then, for all . In summary, we have that there is no such that , which is a contradiction. Thus, So for any , we have . Therefore, assuming that , due to , we have . Consequently, there is no such that , which is a contradiction. Thus . The claim holds. And since the spectrum bandwidth occupied by the channel is monotonous, we have , , and then . So the algorithm outputs the optimal control channels with the minimum spectrum width when it terminates.
At each iteration of Steps in the algorithm, for each wireless network node, there are times comparison. So there are a total of times.
Obviously, the number of iterations in Step is not more than . And it takes time to find in Step and time for to find comparison in Steps .
To conclude, the time complexity of the algorithm is . This completes the proof.
5. Evaluation
In this section, we use the numerical simulation method to evaluate the proposed algorithm compared with the proposed algorithm by Zhong [21].
The simulation topology is as shown in Figure 1. The set of available channels of each node at time is also shown in Figure 1, and the set will change over time. The left column of each table shows the index of each channel, and the right column shows the spectrum bandwidth of each channel. For easy of explanation, we set the total number of available channels to be 6, and index all channels by the ascending order of spectrum bandwidth of each channel, which means that the bigger the index of the channel, bigger the spectrum bandwidth of the channel.
In the simulation, for each run, we use the random number generator to generate the random set of available channels for each node. The simulation executes 60 times of runs in total, sums all simulation results (number of control channels, total spectrum bandwidth of control channels) of each run, and calculates the average values.
Figure 2 shows that number of control channels between our proposed algorithm and Zhong’s algorithm is same. The reason behind this result is that these two algorithms use similar method to find the minimum number of control channels. However, Figure 3 shows that there is a big difference of total spectrum bandwidth of control channels between these two algorithms. The results show that our proposed algorithm not only tries to find optimal number of control channels but also reduces the total spectrum.
6. Concluding Remarks
In this paper, we design an optimal algorithm to find the set of control channels with the minimum spectrum bandwidth for wireless networks. We also present simulation results to show that the proposed algorithm consumes less spectrum resources than Zhong’s algorithm in the same wireless network environment. Furthermore, in some other applications, such as spectrum allocation which usually sorts all channels by descending order, the function for spectrum bandwidth of each channel is monotonously decreasing. It is easy to modify the proposed algorithm and obtain similar results for the case.
Acknowledgments
The authors thank the referees for their helpful comments and suggestions. This work is supported by the National Science Foundation of China under Grant no. 61301159, Open Fund of Lab of Military Network Technology, and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (13KJB1100188).