Recent trends in high-performance computing and deep learning have led to the proliferation of studies on large-scale deep neural network training. However, the frequent communication requirements among computation nodes drastically slows the overall training speeds, which causes bottlenecks in distributed training, particularly in clusters with limited network bandwidths. To mitigate the drawbacks of distributed communications, researchers have proposed various optimization strategies. In this paper, we provide a comprehensive survey of communication strategies from both an algorithm viewpoint and a computer network perspective. Algorithm optimizations focus on reducing the communication volumes used in distributed training, while network optimizations focus on accelerating the communications between distributed devices. At the algorithm level, we describe how to reduce the number of communication rounds and transmitted bits per round. In addition, we elucidate how to overlap computation and communication. At the network level, we discuss the effects caused by network infrastructures, including logical communication schemes and network protocols. Finally, we extrapolate the potential future challenges and new research directions to accelerate communications for distributed deep neural network training.