Federated learning (FL) achieves great popularity in broad areas as a powerful interface to offer intelligent services to customers while maintaining data privacy. Nevertheless, FL faces communication and computation bottlenecks due to limited bandwidth and resource constraints of edge devices. To comprehensively address the bottlenecks, the technique of dropout is introduced, where resource-constrained edge devices are allowed to collaboratively train a subset of the global model parameters. However, dropout impedes the learning efficiency of FL under unbalanced local data distributions. As a result, FL requires more rounds to achieve appropriate accuracy, consuming more communication and computation resources. In this paper, we present FLrce, an efficient FL framework with a relationship-based client selection and early-stopping strategy. FLrce accelerates the FL process by selecting clients with more significant effects, enabling the global model to converge to a high accuracy in fewer rounds. FLrce also leverages an early stopping mechanism to terminate FL in advance to save communication and computation resources. Experiment results show that FLrce increases the communication and computation efficiency by 6% to 73.9% and 20% to 79.5%, respectively, while maintaining competitive accuracy.