A New Algorithm Using the Non-Dominated Tree to Improve Non-Dominated Sorting

Abstract

Non-dominated sorting is a technique often used in evolutionary algorithms to determine the quality of solutions in a population. The most common algorithm is the Fast Non-dominated Sort (FNS). This algorithm, however, has the drawback that its performance deteriorates when the population size grows. The same drawback applies also to other non-dominating sorting algorithms such as the Efficient Non-dominated Sort with Binary Strategy (ENS-BS). An algorithm suggested to overcome this drawback is the Divide-and-Conquer Non-dominated Sort (DCNS) which works well on a limited number of objectives but deteriorates when the number of objectives grows. This article presents a new, more efficient algorithm called the Efficient Non-dominated Sort with Non-Dominated Tree (ENS-NDT). ENS-NDT is an extension of the ENS-BS algorithm and uses a novel Non-Dominated Tree (NDTree) to speed up the non-dominated sorting. ENS-NDT is able to handle large population sizes and a large number of objectives more efficiently than existing algorithms for non-dominated sorting. In the article, it is shown that with ENS-NDT the runtime of multi-objective optimization algorithms such as the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) can be substantially reduced.

Generalized offline orthant search

“…For non-dominated sorting, we also used for comparison the three state-of-the-art algorithms, namely, Best Order Sort [20], the k-d tree-based algorithm called ENS-NDT [9], and the hybrid algorithm combining the divide-and-conquer idea [2] and the ENS algorithm [26] similar to how this was done in this paper. The implementations were borrowed from the GitHub repository dedicated to non-dominated sorting 3 .…”

“…The divide-and-conquer nature of the algorithm developed in [2,8,12] made it impractical to use with relatively small n and large k. A recent work [16] suggested resorting to an O(n 2 k) algorithm for smaller values of n within the main divide-and-conquer algorithm, which made the entire algorithm faster and brought it to competitiveness even at smaller n. Meanwhile, an impressive number of worst-case quadratic algorithms has been developed in order to reduce the running time of non-dominated sorting: [5,17,18,24,26] and many more. Of them, a separate attention deserves Best Order Sort [20] as well as the very recent k-d tree algorithm [9]. These algorithms, while still being prone to degeneration to the Θ(n 2 k) running time, are typically very fast in practice.…”

SETNDS: A SET-Based Non-Dominated Sorting Algorithm for Multi-Objective Optimization Problems

“…In this section, we will discuss and analyze the methods for non-dominated sorting that have been researched and utilized in MOPs. Through reviewing lots of broad research on non-dominated sorting [2,[6][7][8][9][10][11][12][13][14][15][16][17][18][19], the common methods for sorting multi-objective solutions can be classified into two classes: one front after another (OFAA) and one solution after another (OSAA). The basic idea of OFAA is to compare each solution with others to obtain non-dominated solutions, then assign them to the current rank and remove them temporarily.…”

Worst-case conditional hardness and fast algorithms with random inputs for non-dominated sorting

“…For general m, the first O(mn 2 )-time algorithm is due to Deb et al [2]. Since then there have been several algorithms achieving the same worst-case bounds, but focusing on practical running time [3,4,8,10,11,13]. Until now, the O(mn 2 )-time bound has stood for almost two decades.…”