The Ripple-Spreading Algorithm for the $k$-Color Shortest Path Problem.

The k-color shortest path problem (k-CSPP) aims to find the shortest path that traverses at most k colors on edge-colored graphs, which plays a pivotal role in the field of network reliability. This paper proposes a nature-inspired ripple-spreading algorithm (RSA) to solve such problems. The RSA is devised by mimicking ripple propagation patterns on the perturbed water surface. Based on the properties of the k-CSPP, several strategies are proposed to stop exploring non-optimal or infeasible subpaths at an early stage without losing optimality. In addition, the RSA can solve the k-CSPP with multiple destination nodes within a single run with minor modifications to the termination judgment of the algorithm. An illustrative example of how the RSA solves the k-CSPP is presented, using ripple spreading and triggering phenomena to make its processes more comprehensible. Moreover, extensive experiments against existing algorithms exhibit the outstanding efficiency and robustness of the RSA, especially on large-scale networks. The experimental data imply that the RSA speeds up the problem-solving efficiency at least ten times. The RSA can be expected to make a difference in the real-world problems modeled as the k-CSPP.