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Underwater acoustic sensor networks (UWASNs) are composed of underwater sensors that use sound to transmit information collected in the ocean. Since the sound speed is lower than radio wave, UWASNs suffer from much lower throughput and higher delay compared with terrestrial wireless sensor networks (TWSNs). Current methods manage to alleviate the bottleneck by replacing mutual handshakes with reservation mechanisms that consume lower overhead. However, their throughput improvement and delay reduction are very limited (e.g., the throughput is only 30% of the theoretical maximum for TLohi in 8-node networks), and most of their analysis and simulations are based on single-hop communication. In this work, we tackle the above challenges by proposing a priority scheduling approach for multi-hop topologies. First, we find that the scheduling problem of UWASNs is very different from that of TWSNs, and analyze the shortest schedule for the whole UWASN. Then, we design an efficient priority scheduling protocol at the MAC layer. Our approach performs parallel transmissions and prioritizes the scheduling by allocating longer time to heavier-traffic nodes. We have conducted extensive evaluations which show that the proposed protocol not only improves the throughput and delay performance greatly, but also benefits the fairness.