## 1 Introduction  
The Internet of Things (IoT) seeks to connect all aspects of our world through a unified infrastructure, enabling us to control and monitor their status anytime. IoT research has made great progress in the commercial and industrial fields [1, 2, 3], enhancing the potential for its application in underwater environments. Consequently, research on the Internet of Underwater Things (IoUT) is increasingly capturing the interest of researchers [4, 5]. IoUT technology plays a crucial role in military defense, smart coasts and oceans, underwater detection, disaster prediction, and other civilian and military fields [6, 7, 8]. The most important branch of IoUT research is underwater acoustic sensor networks (UASNs). The construction of UASNs for data collection has gradually become an effective ocean exploration method [9]. In terrestrial sensor networks, electromagnetic waves are undoubtedly an efficient medium for wireless transmission. However, in underwater environments, electromagnetic waves suffer from severe attenuation, which makes long-distance propagation challenging. In addition, although light waves can propagate underwater, they are significantly affected by scattering and absorption, resulting in very short transmission distances. Therefore, acoustic waves hold distinct advantages in underwater communication as they can penetrate water layers and transmit over longer distances. However, there are also many drawbacks, such as high transmission losses, multipath effects, limited bandwidth, long delays, and Doppler propagation, which pose huge challenges in conducting relevant research [10]. In addition, underwater nodes are affected by movement caused by water currents and tides, causing the network topology to change continuously, which complicates network management and communication. The accurate locations of underwater nodes are particularly difficult to obtain since Global Positioning System (GPS) signals cannot propagate underwater. Additionally, since underwater nodes are typically battery-powered, managing energy consumption effectively to prolong the network lifetime is a key challenge. These problems make it impossible to apply terrestrial wireless sensor network technology directly to an underwater environment, so it is urgent to design an energy-efficient, scalable, and adaptive underwater acoustic routing protocol. To address these challenges, various routing protocols have been proposed for UASNs. The main objective of the first category of routing protocols is to improve the energy efficiency of sensor nodes, which in turn helps to prolong the lifespan of the entire network. In [11], the authors applied an offline centralized algorithm to transform the energy consumption balance challenge into an optimal data distribution problem. A flaw in the study is that the forwarding node is frequently selected. In [12], the authors used machine learning technology, Q-learning (QL), to solve the problem mentioned above. The selection of the forwarding node is determined by its remaining energy and the group energy of its neighboring nodes. The phenomenon of routing voids in the protocol is also avoided. In addition, ref. [13] proposed a new baseline lightweight energy-aware opportunistic routing (EnOR) protocol. EnOR employs timer-based coordination to set candidate forwarding priorities, enabling efficient time slot assignment. The way the protocol calculates the priority is just a simple algebraic operation. However, the input parameters, such as the residual energy, are often uncertain and fuzzy, which cannot be described and dealt with by a pure algebraic operation. Another branch called cluster-based routing protocols was first proposed in [14], in which whether a node is selected as a cluster head (CH) node is determined by the number of CHs required and how many times the node has been CH before. Compared to [14], ref. [15] provided a more detailed method for selecting the CH node with minimal energy consumption and data traffic load.