OneMax in Black-Box Models with Several Restrictions
Résumé
As in classical runtime analysis the OneMax problem is the most prominent test problem also in black-box complexity theory. It is known that the unrestricted, the memory-restricted, and the ranking-based black-box complexities of this problem are all of order n/log n, where n denotes the length of the bit strings. The combined memory-restricted ranking-based black-box complexity of OneMax, however, was not known. We show in this work that it is Θ(n) for the smallest possible size bound, that is, for (1+1) black-box algorithms. We extend this result by showing that even if elitist selection is enforced, there exists a linear time algorithm optimizing OneMax with failure probability o(1). This is quite surprising given that all previously regarded algorithms with o(n log n) runtime on OneMax, in particular the quite natural (1+(λ,λ))~GA, heavily exploit information encoded in search points of fitness much smaller than the current best-so-far solution.
Also for other settings of μ and λ we show that the (μ+λ) elitist memory-restricted ranking-based black-box complexity of OneMax is as small as (an advanced version of) the information-theoretic lower bound.
Our result enlivens the quest for natural evolutionary algorithms optimizing OneMax in o(n log n) iterations.