Hierarchical non-dominated sort: analysis and improvement

Pareto dominance-based multiobjective evolutionary algorithms use non-dominated sorting to rank their solutions. In the last few decades, various approaches have been proposed for non-dominated sorting. However, the running time analysis of some of the approaches has some issues and they are imprecise. In this paper, we focus on one such algorithm namely hierarchical non-dominated sort (HNDS), where the running time is imprecise and obtain the generic equations that show the number of dominance comparisons in the worst and the best case. Based on the equation for the worst case, we obtain the worst-case running time as well as the scenario where the worst case occurs. Based on the equation for the best case, we identify a scenario where HNDS performs less number of dominance comparisons than that presented in the original paper, making the best-case analysis of the original paper unrigorous. In the end, we present an improved version of HNDS which guarantees the claimed worst-case time complexity by the authors of HNDS which is \({\mathcal {O}}(MN^2)\).