## 1 Introduction  
The collaborative relay technique is an effective technique to achieve higher spectrum efficiency (SE) due to its ability to save the node's power consumption and realize the spatial diversity [1, 2]. The authors in [3, 4] propose a unidirectional decode-and-forward relay transmission scheme because it can increase the transmission rate and reduce the bit error rate. However, unidirectional relay transmission uses two time slots to complete one direction signal transmission, which reduces the time slot utilization and causes the SE loss [5]. For this problem, the authors in [6, 7] propose a bidirectional relay transmission scheme that utilizes two time slots for bidirectional signals exchange. However, refs. [6, 7]are based on single relay transmission. Under single relay transmission, once the relay node fails, the entire relay link will be subject to outage. Subsequently, the authors in [8; 9] propose a bidirectional multi-relay (BMR) transmission strategy, which improves the network SE while increasing the time slot utilization. In the BMR transmission, the channel quality on either side of each relay node differs, which may result in communication outages or insufficient channel capacity. Therefore, it is crucial to select the best relay node(s) to achieve reliable transmission. In [10], the authors propose a relay selection algorithm based on the signal-to-noise ratio (SNR) threshold. However, since this algorithm only relies on SNR for relay selection, it may allow interfering nodes to act as relays, thus affecting the quality of signal transmission. In [11], the authors use an opportunity relay selection algorithm to transmit signals. Although this algorithm provides better performance in some cases, it increases the complexity and management cost of the network. And it is also difficult to cope with dynamic network environments. For these problems, the authors in [12] propose an optimal relay selection algorithm based on SNR and channel gain, thus improving the transmission rate. However, ref. [12] only considers the single channel gain to be better, while neglecting the case that the global channel gain should be better. Meanwhile, refs. [10; 11; 12] only use the relay link to transmit signals. It means that they neglect the potential of direct link (DL) being better. The DL means the link directly between the transmit and receive nodes [13]. Besides the collaborative relay technique, the cognitive wireless network (CWN) allows secondary users (SUs) to share spectrum resources with primary users (PUs) without affecting the normal communication of the PUs, which can also improve spectrum utilization [14; 15]. Considering the advantages of CWN and collaborative relay technique, combining CWN with the collaborative relay technique to construct a cognitive relay network (CRN) has continued to receive extensive attention from researchers. In [16], the authors study an adaptive cooperative transmission strategy in cognitive bidirectional relay network to improve the performance of the SUs. At the same time, the optimal relay selection approach for this bidirectional relay network to minimize outage probability is also provided. In [17], the authors propose a hop-by-hop relay selection strategy for multi-hop CWN, which provides a lower outage probability and a higher network throughput. Refs. [16; 17] effectively improve the network SE to a certain extent, but they neglect the energy efficiency (EE) issue of the CRN. To deal with the issue of energy inefficiency in CRN, simultaneous wireless information and power transfer (SWIPT) techniques are investigated in [18; 19; 20]. As mentioned in [21; 22], SWIPT enables the simultaneous transmission of signals and energy through radio frequency signals. Thus, it can effectively extend the lifetime of energy-constrained wireless networks. Generally, there are two protocols with SWIPT techniques, namely, time switching and power splitting (PS) protocols.