## 1 Introduction  
Wireless sensor networks (WSNs) have the advantages of high flexibility in deployment and high usability in dynamic environments [1; 2]. Because of these advantages, WSNs enable developing diverse applications such as security surveillance, environmental monitoring, disaster monitoring, healthcare, target tracking, and internet of things (IoT) applications [1; 2; 3; 4]. However, to realize the long-range communication applications, a limited communication range of WSNs with low-power transmitters should be improved. Thus, multi-hop transmission techniques have been introduced into WSNs for coverage extension. The multi-hop communication allows intermediate nodes between a source node and a destination node to relay a signal of the source node, which can enable long-range communications [5; 6; 7]. In WSNs, the nodes may have insufficient space to be equipped with multiple antennas, and thus they may exploit very limited spatial diversity to enhance the signal reliability and data rate. In many studies, hence the concept of virtual antennas has been employed to generate and exploit more spatial diversity, where multiple cooperative nodes are regarded as virtual antennas of a transmitter or a receiver. Such virtual antennas have been also adopted in multi-hop WSNs so as to improve their signal reliability and data rate [8; 9; 10]. One of well-known multi-hop cooperation techniques using virtual antennas is the best path selection (BPS) in which a single path providing the best end-to-end channel quality is selected among multiple paths between a source node and a destination node for data transmission [11]. The BPS requires global channel state information (CSI) (i.e., full CSI for all hops of every path) for the best path selection. Thus, the BPS may not be employed in wireless communication networks with the limited CSI feedback which can be adopted for the restricted resources such as bandwidth and node energy. In wireless communication, one of limited resources is energy which can be mainly consumed by signal (e.g., data and control information) transmission and signal processing. The energy resource is significantly treated in a sensor node because of its small energy capability. Such an important energy issue has been also handled in wireless cellular networks to realize the green communication network, and limited CSI feedback has been considered as one of energy-efficient radio technologies [12; 13; 14; 15]. Hence, the partial path selection (PPS) utilizing local CSI, i.e., CSI for certain hops of each path, has been proposed in [16]. To enhance the limited end-to-end performance of the PPS, an efficient PPS scheme has been presented in [17], where only CSI for the hop with the smallest average channel power in each dual-hop path is used for the PPS. In addition, the end-to-end performances of the PPS have been widely studied for a decode-and-forward (DF) relaying system with an eavesdropper [18], a cognitive amplify-and-forward (AF) relaying system [19], and a non-orthogonal multiple access (NOMA)-based AF relaying system [20] in terms of the outage probability and ergodic capacity. Moreover, the PPS can be applicable to a cooperative NOMA system [21] and a relay-assisted multicast system [22] when those systems require limited CSI feedback. However, the PPS techniques in the previous works assuming the dual-hop scenario cannot be directly applied to the multi-hop scenario since many different combinations of hops for local CSI exist in the multi-hop case. Therefore, in this paper, we propose a novel PPS algorithm for a multi-hop relaying system with the limited CSI feedback, by which we can choose a pertinent subset of hops in each path to report local CSI according to the system requirement for CSI feedback overhead.