## 1 Introduction  
In wireless underwater sensor networks (WUSNs), reliability and void hole avoidance in packets' advancement have paramount significance. Reliability in packets' advancement ensures that packets are not corrupted by the undesirable channel characteristics, which in effect, eases the information extraction at the end target. Furthermore, it reduces the complexity in electronics of the transmitter and receiver, which would otherwise be required for unreliable data routing trajectories. Moreover, reliable information advancement to the sea surface saves the amount of power in a node's battery. That is, reliable information delivery reduces the probability of retransmission of the lost or corrupted packets along the channel. Therefore, the power used in retransmission of the packets is saved for the transmission of other packets. This, in effect, achieves efficient utilization of battery power, which has a limited extent in WUSNs [1]. A void-hole is a condition in which a node cannot find another node towards the ultimate destination for data advancement, is completely isolated from other nodes of the network, and cannot communicate with them [2]. This scenario interrupts data delivery to the end destination, and consequently, network performance is compromised. The presence of a void-hole is more severe when it exists close to the water surface. This is because it interrupts the process of information advancement from high depth nodes (nodes located at a higher depth than the void-hole). When a significant number of void-holes exists near the water surface, this results in complete collapse of the network, as higher depth nodes located under the void-holes are not able to transfer their packets to the sea surface. Network protocols that address reliability in information advancement and void-hole avoidance find applications in undersea disaster control [3], naval warfare [4], and sea monitoring [5]. They are also of significant importance in terrestrial networks [6]. In WUSNs, the existing protocols suffer from degraded performance in sparse conditions when the number of nodes reduces in the network, as these protocols are usually designed by considering a significant number of sensor nodes in the network. In particular, reliability, computational complexity and void-hole avoidance are some of the key challenges that arise during sparse network conditions. For instance, protocols that focus on reliability or void-hole mitigation exist in the literature [7; 8; 9]. However, these protocols intrinsically have either high computational complexity or are dependent on the geographical coordinates of nodes. Such demerits limit the performance and operation of these protocols. Computational complexity makes the operation of these protocols a challenging task and inherently inserts extra delay in information transfer from the sea bottom to surface. The dependency of nodes on the geographical information of nodes is also challenging, as the currents in the sea keep changing the positions of nodes. As a result, error probability in position estimation of nodes increases, which in turn, compromises the throughput of the network. Specifically, in [7], geographical information of nodes is required, and the surface sink has to send beacon signals periodically. Moreover, the fitness function for selecting nodes requires a number of parameters. All these requirements complicate the process of information routing. Similarly, the work in [8] requires distance and hop calculation of nodes using constant beacon signals from the water surface, which inserts extra delay into the communications and increases the computational time. Continuing in a similar fashion, the work in [9] requires construction of routing pipes based on the geographical information of nodes, which is computationally complex and such pipes are difficult to construct since nodes move with ocean tides. Moreover, these protocols are dependent on the condition when every node has enough neighbor nodes for information advancement to the water surface.