Divide-and-conquer based non-dominated sorting with Reduced Comparisons

https://doi.org/10.1016/j.swevo.2019.100580Get rights and content

Abstract

Non-dominated sorting has attracted a lot of attention of the research community due to its use in solving multi- and many-objective optimization problems. In recent years, several approaches for non-dominated sorting have been proposed. In this paper, we have developed a non-dominated sorting framework, namely DCNSRC (Divide-and-Conquer based Non-dominated Sorting with Reduced Comparisons). Based on this framework, two approaches have been proposed by varying the search technique. These approaches perform a lower number of dominance comparisons than various other approaches. The duplicate solutions are also handled efficiently. These approaches save various comparisons while comparing the two solutions. The proposed approaches are validated using some theoretical analyses. The number of dominance comparisons performed by the proposed framework are theoretically analyzed in three different scenarios, both in the worst and the best cases. Experimental results on synthetic datasets and the benchmark problems show the superiority of the proposed approach over state-of-the-art algorithms.

Introduction

Non-dominated sorting is one of the key steps in Pareto-based multi-objective evolutionary algorithms (MOEAs). Non-dominated sorting sorts the solutions into different non-dominated fronts based on their dominance relationships. In addition of being a key step in MOEAs, the application of the optimization algorithm based on non-dominated sorting can be found in many other fields such as reactive power planning [1], generation expansion planning [2,3], feature selection for facial expression recognition [4], decision making in water distribution networks [5], vehicle routing problems [6], etc. In MOEAs, the set of solutions are often referred to as the population. Let P=s1,s2,,sN be a population of N solutions where each solution si, 1 ≤ i ≤ N is an n-dimensional vector si=si1,si2,,sinT. The objective function of si is F(si)=f1(si),f2(si),,fM(si)T where fm(si) is the objective value of solution si for mth (1 ≤ m ≤ M) objective. Let us assume that all the objectives are of minimization type. In non-dominated sorting, the solutions are sorted based on the dominance relationships among them, which is defined as follows. A solution si is said to dominate another solution sj, denoted by si ≺ sj if the two following conditions are satisfied:

  • 1

    fm(si)fm(sj),m1,2,,M

  • 2

    fm(si)<fm(sj),m1,2,,M

Two solutions si and sj are said to be non-dominated, if neither dominates the other, i.e., neither si ⊀ sj nor sj ⊀ si. In non-dominated sorting, population P of size N is divided into K(1 ≤ K ≤ N) fronts F=F1,F2,,FK arranged in decreasing order of their dominance. The division of these solutions in different fronts is such that all the following conditions are satisfied:

  • 1

    k=1KFk=P

  • 2

    ∀si, sj ∈ Fk: si ⊀ sj and sj ⊀ si, 1 ≤ k ≤ K

  • 3

    ∀s ∈ Fk, ∃s′ ∈ Fk−1: s′ ≺ s, 2 ≤ k ≤ K

There have been different approaches proposed in the past for non-dominated sorting. Srinivas et al. [7] proposed the first approach for non-dominated sorting. In this approach, a solution can be compared with other solutions a maximum of N − 1 times. Thus, its worst case time complexity is O(MN3) with space complexity of O(N). The worst case occurs when all the solutions are in different fronts. The best case time complexity of this approach is O(MN2) when all the solutions are non-dominated with each other. The fast non-dominated sorting [8] approach has improved the worst case time complexity to O(MN2), but with an increase in the space complexity to O(N2). In this approach, a solution is compared with other solutions only once. A recursive approach was developed by Jensen et al. [9] with a time complexity of O(NlogM1N). For two objectives, the time complexity of this approach is O(NlogN). This approach is not suitable when two solutions share the same value for an objective. This limitation of Jensen's approach is removed by Fortin et al. [10]. The average case time complexity of Fortin's approach is O(NlogM1N). However, the worst case time complexity of Fortin's approach is O(MN2). Tang et al. [11] proposed a fast method for constructing the non-dominated set based on arena's principle. The best case time complexity of this approach is O(MNN) [12].

An efficient approach based on a divide-and-conquer strategy was proposed by Fang et al. [13]. This approach uses the dominance tree to reduce the number of comparisons. The worst case time complexity of this algorithm is O(MN2) when all the solutions are non-dominated with respect to each other. The lower bound time complexity of this algorithm is O(MNlogN). This approach considers one solution as dominated by another if both are identical. Two sorting algorithms (climbing sort and deductive sort) were proposed by McClymont et al. [14]. These algorithms use the dominance relation between the solutions to reduce the number of dominance comparisons. These algorithms have a worst case time complexity of O(MN2). The space complexity of deductive sort is O(N), and its best case time complexity is O(MNN). Zhang et al. [12] proposed an efficient approach for non-dominated sorting, called efficient non-dominated sort (ENS). Two variants of ENS – one using sequential search (ENS-SS) and another using binary search (ENS-BS) were proposed. The worst case time complexity is O(MN2) for these two variants. The best case time complexity of ENS-SS is O(MNN), whereas the best case time complexity of ENS-BS is O(MNlogN). Buzdalov et al. [15] proved that the time complexity of non-dominated sorting is O(NlogM1N). Mishra et al. [16,17] proposed an approach based on a divide-and-conquer strategy. This algorithm has a best case time complexity of O(MNlogN) and a worst case time complexity of O(MN2).

In some recently proposed approaches [[18], [19], [20]] when a solution is inserted in a front, it is not always compared with all the other solutions in that front. Based on this idea, an approach known as Best Order Sort (BOS) has been proposed, which is very efficient in terms of the number of dominance comparisons that it performs [18]. However, BOS is not suitable for duplicate solutions. Recently, BOS has been generalized to handle duplicate solutions by removing the comparison set concept.2 In this paper, we call this modified BOS algorithm as BOS∗. A tree based efficient non-dominated sorting approach known as T-ENS [19] is proposed with a worst case time complexity of O(MN2). T-ENS has a better best case time complexity, which is O(MNlogN/logM). Recently, an efficient non-dominated sorting approach that uses a non-dominated tree (ENS-NDT) [20] was developed with a worst case time complexity of O(MN2). The best case time complexity of ENS-NDT is the same as ENS-BS, i.e., O(MNlogN). Zhou et al. [21] have developed a dominance degree based approach for non-dominated sorting. The authors proposed the dominance degree matrix for the solutions in the population and also proposed an efficient approach to obtain this matrix. Based on the dominance degree matrix, an efficient approach called dominance degree approach for non-dominated sorting (DDA-NS) is proposed. In this approach, the comparisons between the objective function values of the solutions are only performed in the process of constructing the dominance degree matrix. Once the dominance degree matrix is obtained, the additional comparisons are required to assign solutions to different fronts. The additional comparisons are in the form of integer value comparisons between the elements of the dominance degree matrix. So, in this approach, two types of comparisons (objective value comparisons and integer value comparisons) occur. The average number of objective value comparisons required is O(MNlogN), and the number of integer value comparisons required is O(N2). The number of integer additions is O(MN2). Few other approaches like [[22], [23], [24], [25]] have been recently proposed for non-dominated sorting.

In some of the MOEAs, the population is decomposed into sub-populations and then non-dominated sorting is performed in each sub-population as done in Ref. [26]. The authors have used the fast non-dominated sorting approach proposed by Deb et al. [8] in Ref. [26]. However, other efficient sorting approaches can also be used to make non-dominated sorting more efficient.

There are some approaches [[27], [28], [29], [30], [31], [32], [33]] which have been proposed for incremental non-dominated sorting. This incremental non-dominated sorting is generally used in steady-state evolutionary algorithms [34,35] where a solution needs to be inserted into the existing set of non-dominated fronts once a new solution is generated. Drozdik et al. [27] have developed an M-front based approach with worst case time complexity of O(MN2) and best case time complexity of O(MN). The average case time complexity of this approach is O(M2N21M1). An approach was proposed by Buzdalov et al. [28], which is only suitable for two objectives. A dominance matrix based approach was developed by Mishra et al. [31], which restricts the multiple comparisons between the same pairs of solutions in different generations of a steady-state evolutionary algorithm. By generalizing their previous approach for two objectives, Yakupov et al. [33] proposed an approach with time complexity O(NlogM2N). The incorporation of a new solution in the population does not change the entire non-domination level structure of the solutions. By exploiting this property, Li et al. [30] proposed an efficient non-domination level update approach to insert a new solution in the existing set of non-dominated fronts. The worst case time complexity of this approach is O(MN2), however, the maximum number of dominance comparisons is 1/4N2. The best case time complexity of this approach is O(M). This best case time complexity is a great improvement over the best case time complexity of the non-dominated sorting approaches. Recently, Mishra et al. [32] also proposed a generalized approach for the non-domination level update problem with constant space complexity. A dominance binary tree based approach was also discussed. The worst and the best case time complexities of this approach are the same as the time complexity of the approach proposed by Li et al. [30].

Even though there have been several approaches proposed for non-dominated sorting, there is still a chance of improvement in terms of reducing the number of dominance comparisons. In general, when two solutions are compared to obtain their dominance relationships, all the objective values can be considered. Also, very few existing approaches handle duplicate solutions efficiently. So, we also focused on the non-dominated sorting problem to handle these issues in order to make the approach more efficient.

We have focused on developing a framework based on a divide-and-conquer strategy. Divide-and-conquer based approaches are easy to parallelize, so our approach also has this property. However, in this paper, we have not focused on the parallelism. There have been some approaches proposed based on a divide-and-conquer strategy such as [13,16,17]. However, in these approaches, while comparing two solutions, all the objective values between the solutions are compared. Also, before inserting a solution into a particular front, the solution is compared with all the solutions of that front. Duplicate solutions are also not handled efficiently in these approaches. In the current paper, a divide-and-conquer based strategy is developed, which can handle the drawbacks of the existing divide-and-conquer strategies. The key difference between other divide-and-conquer based approaches like [13,16,17] and ours is that when two solutions are compared to determine their dominance relationship, not all the objective values are considered. So, two solutions are compared efficiently. Apart from this difference from other divide-and-conquer based approaches, our proposed approach is also capable of handling duplicate solutions in the population efficiently. Also, before inserting a solution to a particular front, the solution is not compared with all the solutions of that front. However, in many approaches, a solution is compared with all the solutions of a front before being inserted into that front.

In this paper, we develop a framework which we call DCNSRC (Divide-and-Conquer based Non-dominated Sorting with Reduced Comparisons), which is proposed to reduce the number of dominance comparisons. This framework especially takes care of handling duplicate solutions efficiently. Also, in the proposed framework, when two solutions are compared to determine their dominance relationship, not all the objective values are considered. So, the solutions are also compared efficiently. Two variants of the developed framework are presented varying the search technique as in Refs. [12,16]. We have theoretically analyzed the number of dominance comparisons performed by the proposed framework in different scenarios and observed that the number of dominance comparisons is less as compared to various other approaches. When all the solutions are non-dominated, then in the best case, the number of dominance comparisons can be zero. This can happen when M ≥ N. In brief, the contributions in this paper are as follows:

  • A framework for non-dominated sorting has been developed based on a divide-and-conquer strategy. We call this framework DCNSRC (Divide-and-Conquer based Non-dominated Sorting with Reduced Comparisons).

  • The developed framework handles duplicate solutions efficiently.

  • The number of dominance comparisons between the solutions is reduced.

  • When two solutions are compared to determine their dominance relationship, not all the objective values are considered. So, the solutions are compared efficiently.

  • Two approaches have been proposed based on the developed framework by varying the search technique as in Ref. [12].

  • The number of dominance comparisons performed by DCNSRC in three different scenarios in the worst and the best cases are also theoretically analyzed.

The rest of the paper is organized as follows. Section 2 discusses the proposed non-dominated sorting framework. An important component in the proposed framework is the merge procedure which is discussed in detail in Section 3. The complexity analysis of the framework is performed in Section 4. The experimental analysis is discussed in Section 5. Section 6 concludes the paper and provides some possible paths for future research.

Section snippets

Proposed framework

A non-dominated sorting framework, namely DCNSRC (Divide-and-Conquer based Non-dominated Sorting with Reduced Comparisons) is proposed. The proposed framework consists of two-phases. The steps of the DCNSRC framework are summarized in Algorithm 1. The framework uses various symbols which are summarized in Table 1 along with their meaning.

Merge procedure

In this section we discuss the merge procedure in detail. This procedure merges two sets of fronts, i.e., inserts all the solutions of the second set of fronts into the first set of fronts at their respective positions. The normal operation of merge sort cannot be directly applied here because of the dominance relationship. The dominance relationship does not follow the transitivity property (when two solutions are non-dominated with a particular solution, then it does not imply that both

Complexity analysis

In this section, the complexity analysis of the proposed framework is performed. When the solutions are sorted based on each objective individually, then heap sort is used which requires O(1) extra space. The sorted solutions based on each objective need to be stored in a separate list. Thus, the overall space complexity of the first phase is O(MN). The initialization of N matrices of different sizes takes O(MNlogN) space (from Section 2.2.2).

For each solution sP, a variable sameAss is

Experimental evaluation

We have compared the performance of DCNSRC-SS and DCNSRC-BS with respect to the fast non-dominated sorting approach (FNDS) [8], deductive sort (DS) [14], ENS-SS [12], ENS-BS [12], BOS [18] and DDA-NS [21]. In the DDA-NS approach, two types of comparisons (objective value comparisons and integer value comparisons) occur. However, for the other approaches (FNDS, DS, ENS-SS, ENS-BS and BOS), objective value comparisons are required so, we have computed the objective value comparisons for these

Conclusions and future work

In this paper, a framework for non-dominated sorting, namely DCNSRC, has been proposed. Based on this framework, two approaches have been proposed. The main advantage of the proposed framework is that with an increase in the number of objectives, there is a decrease in the number of dominance comparisons. Duplicate solutions are also handled efficiently in this approach. This framework also does not consider all the objective values while comparing two solutions. The worst case time complexity

Declaration of competing interest

The authors confirm that there are no known conflicts of interest.

Acknowledgements

Dr. Sriparna Saha would like to acknowledge the support of Early Career Research Award of Science and Engineering Research Board (SERB) of Department of Science and Technology India to carry out this research. The last author gratefully acknowledges support from CONACYT grant no. 2016-01-1920 (Investigación en Fronteras de la Ciencia 2016) and from a project from the 2018 SEP-Cinvestav Fund (application no. 4).

References (37)

  • M.T. Jensen

    Reducing the run-time complexity of multiobjective EAs: the NSGA-II and other algorithms

    IEEE Trans. Evol. Comput.

    (2003)
  • F.-A. Fortin et al.

    Generalizing the improved run-time complexity algorithm for non-dominated sorting

  • S. Tang et al.

    A fast method of constructing the non-dominated set: arena's principle

  • X. Zhang et al.

    An efficient approach to nondominated sorting for evolutionary multiobjective optimization

    IEEE Trans. Evol. Comput.

    (2015)
  • H. Fang et al.

    An efficient non-dominated sorting method for evolutionary algorithms

    Evol. Comput.

    (2008)
  • K. McClymont et al.

    Deductive sort and climbing sort: new methods for non-dominated sorting

    Evol. Comput.

    (2012)
  • M. Buzdalov et al.

    A provably asymptotically fast version of the generalized Jensen algorithm for non-dominated sorting

  • S. Mishra et al.

    Divide and conquer based non-dominated sorting for parallel environment

  • Cited by (1)

    1

    On sabbatical leave from Departamento de Computación CINVESTAV-IPN, Mexico City, Mexico.

    View full text