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Hierarchical non-dominated sort: analysis and improvement

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Abstract

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 \({\mathcal {O}}(MN^2)\).

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Two datasets—Cloud dataset and Fixed front dataset are used in this paper which are commonly used for non-dominated sorting problem. The process to generate the data is mentioned in the relevant section.

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  1. Department of Computer Science and Engineering, Indian Institute of Information Technology Guwahati, Guwahati, 781015, Assam, India

    Ved Prakash & Sumit Mishra

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  1. Ved Prakash
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Appendix 1: Supporting materials

Appendix 1: Supporting materials

$$\begin{aligned} T_{\texttt {presort}}&= M \left[ \sum _{k=1}^{K} \left( N - \left( n_1 + n_2 + \ldots + n_{k-1} \right) \right) \log \left( N - \left( n_1 + n_2 + \ldots + n_{k-1} \right) \right) \right] \nonumber \\&\le M \left[ \sum _{k=1}^{K} N \log N \right] \nonumber \\&= M \left[ \sum _{k=1}^{N} N \log N \right] \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text { as maximum value of } K \text { is } N \nonumber \\&= M \left( N^2\log N \right) = {\mathcal {O}}(MN^2\log N) \end{aligned}$$
(33)
$$\begin{aligned} A_w&= \sum _{k=1}^{K} \Psi _kn_k = \sum _{k=1}^{K} \left[ N - \left( n_1 + n_2 + \cdots + n_{k-1} \right) \right] n_k \nonumber \\&= \sum _{k=1}^{K} Nn_k - \sum _{k=1}^{K} \left( n_1 + n_2 + \cdots + n_{k-1} \right) n_k \nonumber \\&= N \sum _{k=1}^{K} n_k - \left[ \left( n_1 \right) n_2 + \left( n_1+n_2 \right) n_3 + \cdots \right. \nonumber \\&\quad +\left. \left( n_1 + n_2 + \cdots + n_{k-1} \right) n_k \right] \nonumber \\&= N^2 - \left[ \left( n_1n_2 + n_1n_3 + \cdots + n_1n_k \right) \right. \nonumber \\&\quad +\left. \left( n_2n_3 + n_2n_4 + \cdots + n_2n_k \right) + \cdots + \left( n_{k-1}n_k \right) \right] \nonumber \\&= N^2 + \frac{1}{2}\left( n_1^2 + n_2^2 +\ldots + n_K^2 \right) - \frac{1}{2}\left( n_1^2 + n_2^2 + \cdots + n_K^2 \right) \nonumber \\&\quad - \left[ \left( n_1n_2 + n_1n_3 + \cdots + n_1n_k \right) + \left( n_2n_3 + n_2n_4 + \cdots + n_2n_k \right) + \cdots \right. \nonumber \\&\quad + \left. \left( n_{k-1}n_k \right) \right] \nonumber \\&= N^2 + \frac{1}{2}\left( n_1^2 + n_2^2+ \cdots + n_K^2 \right) - \frac{1}{2}\left( n_1 + n_2 + \cdots + n_K \right) ^2 \nonumber \\&= N^2 + \frac{1}{2} \sum _{k=1}^{K} n_k^2 - \frac{1}{2}N^2 = \frac{1}{2}N^2 + \frac{1}{2} \sum _{k=1}^{K} n_k^2 \end{aligned}$$
(34)
$$\begin{aligned} T_{\texttt {presort}}&= M \left[ \sum _{k=1}^{N} \left( N-k+1 \right) \log \left( N-k+1 \right) \right] \nonumber \\&\le M \left[ \sum _{k=1}^{N} \left( N-k+1 \right) \log N \right] \nonumber \\&= M \log N \left[ \sum _{k=1}^{N} \left( N-k+1 \right) \right] \nonumber \\&= \frac{1}{2}MN\left( N+1 \right) \log N = {\mathcal {O}}(MN^2 \log N) \end{aligned}$$
(35)
$$\begin{aligned} T_{\texttt {presort}}&= M \left[ \sum _{k=1}^{K} \left( N - \left( n_1 + n_2 + \cdots + n_{k-1} \right) \right) \log \left( N - \left( n_1 + n_2 + \cdots + n_{k-1} \right) \right) \right] \nonumber \\&\le M \left[ \sum _{k=1}^{K} N \log N \right] \nonumber \\&= M \left( KN\log N \right) = {\mathcal {O}}(MKN\log N) \end{aligned}$$
(36)
$$\begin{aligned} A_b&= \sum _{k=1}^{K} \left( \Psi _k-1 \right) = \sum _{k=1}^{K} \left[ N - \left( n_1 + n_2 + \cdots + n_{k-1} \right) - 1 \right] \nonumber \\&= \sum _{k=1}^{K} \left( N-1 \right) - \sum _{k=1}^{K} \left( n_1 + n_2 + \cdots + n_{k-1} \right) \nonumber \\&= \left( N-1 \right) K - \left[ \left( K-1 \right) n_1 + \left( K-2 \right) n_2 + \cdots + n_{K-1} \right] \nonumber \\&= \left( N-1 \right) K - \left[ \left( n_1+n_2+\cdots +n_{K-1} \right) K \right. \nonumber \\&\quad -\left. \left( n_1+2n_2+\cdots +(K-1)n_{K-1} \right) \right] \nonumber \\&= \left( N-1 \right) K - \left[ \left( N-n_K\right) K - \sum _{k=1}^{K-1}kn_k \right] \nonumber \\&= \left( n_K-1 \right) K + \sum _{k=1}^{K-1}kn_k \end{aligned}$$
(37)
$$\begin{aligned} B_b&= \frac{1}{2} \sum _{k=1}^{K} \left( n_k-1 \right) \left( n_k-2 \right) = \frac{1}{2} \sum _{k=1}^{K} \left( n_k^2 -3n_k \right) + \frac{1}{2} \sum _{k=1}^{K} 2 \nonumber \\&= K +\frac{1}{2} \sum _{k=1}^{K} \left( n_k^2 -3n_k \right) \end{aligned}$$
(38)

See Tables 3 and 4.

Table 3 Points in three fronts such that all the points in a front is only dominated by the last point (last point based on the lexicographical sort) in its preceding front
Full size table
Table 4 Points in three fronts such that all the points in a front is only dominated by the first point (first point based on the lexicographical sort) in its preceding front
Full size table

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Prakash, V., Mishra, S. Hierarchical non-dominated sort: analysis and improvement. Genet Program Evolvable Mach 25, 14 (2024). https://doi.org/10.1007/s10710-024-09487-1

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