An Efficient Nondominated Sorting Algorithm for Large Number of Fronts

Nondominated sorting is a key operation used in multiobjective evolutionary algorithms (MOEA). Worst case time complexity of this algorithm is <inline-formula> <tex-math notation="LaTeX">{O(MN^{2})} </tex-math></inline-formula>, where <inline-formula> <tex-...

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Vydané v:IEEE transactions on cybernetics Ročník 49; číslo 3; s. 859 - 869
Hlavní autori: Roy, Proteek Chandan, Deb, Kalyanmoy, Islam, Md. Monirul
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: United States IEEE 01.03.2019
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Predmet:
Algorithms
Best case complexity
bounded best order sort (BBOS)
Classification
Complexity
Cybernetics
Evolutionary algorithms
layers of maxima
many-objective optimization
Multiple objective analysis
nondominated sorting
Objectives
Optimization
Pareto ranking
Sociology
Sorting
Sorting algorithms
State of the art
Statistics
Time complexity
worst case complexity
ISSN:2168-2267, 2168-2275, 2168-2275
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Shrnutí:Nondominated sorting is a key operation used in multiobjective evolutionary algorithms (MOEA). Worst case time complexity of this algorithm is <inline-formula> <tex-math notation="LaTeX">{O(MN^{2})} </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">{N} </tex-math></inline-formula> is the number of solutions and <inline-formula> <tex-math notation="LaTeX">{M} </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">{\Theta }({MNC}_{{max}}\mathrm {log}_{{2}} {(F+1)}) </tex-math></inline-formula>, where the number of fronts is <inline-formula> <tex-math notation="LaTeX">{F} </tex-math></inline-formula> and the maximum number of solutions per front is <inline-formula> <tex-math notation="LaTeX">{C}_{\mathrm {max}} </tex-math></inline-formula>; however, in general cases, our worst case complexity is still <inline-formula> <tex-math notation="LaTeX">{O(MN^{2})} </tex-math></inline-formula>. Our best case time complexity is <inline-formula> <tex-math notation="LaTeX">{O}({MN}\mathrm {log} {N}) </tex-math></inline-formula>. We also achieve the best case complexity <inline-formula> <tex-math notation="LaTeX">{O}({MN}\mathrm {log} {N+N^{2}}) </tex-math></inline-formula>, 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.
Bibliografia:ObjectType-Article-1
SourceType-Scholarly Journals-1
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ISSN:2168-2267
2168-2275
2168-2275
DOI:10.1109/TCYB.2017.2789158