Nondominated sorting is a key operation used in multiobjective evolutionary algorithms (MOEA). Worst case time complexity of this algorithm is ${O(MN^{2})}$ , where ${N}$ is the number of solutions and ${M}$ is the number of objectives. For stochastic algorithms like MOEAs, it is important to devise an algorithm which has better average case performance. In this paper, we propose a new algorithm that makes use of faster scalar sorting algorithm to perform nondominated sorting. It finds partial orders of each solution from all objectives and use these orders to skip unnecessary solution comparisons. We also propose a specific order of objectives that reduces objective comparisons. The proposed method introduces a weighted binary search over the fronts when the rank of a solution is determined. It further reduces total computational effort by a large factor when there is large number of fronts. We prove that the worst case complexity can be reduced to ${\Theta }({MNC}_{{max}}\mathrm {log}_{{2}} {(F+1)})$ , where the number of fronts is ${F}$ and the maximum number of solutions per front is ${C}_{\mathrm {max}}$ ; however, in general cases, our worst case complexity is still ${O(MN^{2})}$ . Our best case time complexity is ${O}({MN}\mathrm {log} {N})$ . We also achieve the best case complexity ${O}({MN}\mathrm {log} {N+N^{2}})$ , when all solutions are in a single front. This method is compared with other state-of-the-art algorithms—efficient nondomination level update, deductive sort, corner sort, efficient nondominated sort and divide-and-conquer sort—in four different datasets. Experimental results show that our method, namely, bounded best order sort, is computationally more efficient than all other competing algorithms.

中文翻译：

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大量前沿的一种有效的非支配排序算法

非支配排序是多目标进化算法（MOEA）中使用的关键操作。最坏的情况是该算法的时间复杂度是 $ {O（MN ^ {2}）} $ ， 在哪里 $ {N} $ 是解决方案的数量， $ {M} $ 是目标数。对于诸如MOEA的随机算法，重要的是设计一种具有更好平均情况性能的算法。在本文中，我们提出了一种新算法，该算法利用更快的标量排序算法来执行非支配排序。它从所有目标中找到每个解决方案的部分顺序，并使用这些顺序跳过不必要的解决方案比较。我们还提出了减少目标比较的特定目标顺序。当确定解决方案的等级时，所提出的方法在前面引入了加权二进制搜索。当存在大量前沿时，它还可以在很大程度上减少总的计算工作量。我们证明最坏情况下的复杂度可以降低到 $ {\ Theta}（{MNC} _ {{max}} \ mathrm {log} _ {{2}} {（F + 1）}）$ ，其中前线的数量是 $ {F} $ 每个前沿的最大解决方案数是 $ {C} _ {\ mathrm {max}} $ ; 但是，在一般情况下，我们最坏情况下的复杂度仍然 $ {O（MN ^ {2}）} $ 。我们最好的情况是时间复杂度是 $ {O}（{MN} \ mathrm {log} {N}）$ 。我们还实现了最佳的案例复杂性 $ {O}（{MN} \ mathrm {log} {N + N ^ {2}}）$ ，当所有解决方案都在同一方面时。在四个不同的数据集中，将该方法与其他最新算法（有效的非支配级别更新，演绎排序，角点排序，有效的非支配排序和分治法排序）进行了比较。实验结果表明，我们的方法（即有界最佳顺序排序）比所有其他竞争算法在计算效率上更高。