Make Evolutionary Multiobjective Algorithms Scale Better with Advanced Data Structures: Van Emde Boas Tree for Non-dominated Sorting

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

“…Hitting Set Conjecture (HSC). For any constant ϵ > 0, and m where ω(log n) ≤ m < (log n) O (1) , there is no O(mn 2−ϵ )-time algorithm for the Hitting Set Problem.…”

“…We list two here. First, there may exist an algorithm with worst case running time O mn 2 ⇑ log O (1) n . Second, algorithms may perform better on restricted input instances.…”

“…The non-dominated sorting problem is completely solved when m = 2 or 3 with a worst-case time complexity of Θ(n log n) [7,9]. For a fixed m > 2, O(n log m−1 n)-time algorithms are known using divide-and-conquer (D&C), often referred to as Jensen's sort [5], and the best known running time for this approach is O(n log m−2 n log log n)-time algorithm [1]. For general m, the first O(mn 2 )-time algorithm is due to Deb et al [2].…”

“…We show that the O(mn 2 ) running time is nearly optimal by proving an almost matching conditional lower bound: for any ϵ > 0, and ω(log n) ≤ m ≤ (log n) O (1) , there is no O(mn 2−ϵ )time algorithm for NDS or NDS1 unless a popular conjecture in fine-grained complexity theory is false. To complete our results, we present an algorithm for NDS with an expected running time O(mn + n 2 ⇑m + n log 2 n) on uniform random inputs.…”

Filter Sort Is $$\varOmega (N^3)$$ in the Worst Case

“…The divide-and-conquer flavour in question is suggested long time ago in [15], and it was applied for the first time to non-dominated sorting in [12]. Subsequent development involved modifications to reliably work on every input [4] and various practical runtime improvements typically involving hybridization with other algorithms [18,19], and word-RAM data structures [3].…”