In this paper, the competitive mechanism is introduced to the production and propagation processes of the autowave of neural network. The autowave-competition neural network (ACNN) is proposed to successfully resolve the problem of single-source shortest paths (SSSP). The algorithm for shortest paths based on ACNN is presented. Compared with other neural network based approaches, the new algorithm has the following advantages: less number of neurons needed, simple structure of neurons and networks, readily available software and hardware. When ACNN is employed to resolve the shortest path problem, the computational complexity is only related to the hop number of the shortest path, but independent of the complexity of path graph, the number of the existing paths in the graph and the precision of the length of edges. Simulations show that the proposed algorithm is plausible and effictive.