Black-Box Complexity: Breaking the $O(n\log n)$ Barrier of LeadingOnes

Abstract

We show that the unrestricted black-box complexity of the $n$-dimensional XOR- and permutation-invariant LeadingOnes function class is $O(n\log (n)/\log \log n)$. This shows that the recent natural looking $O(n\log n)$ bound is not tight. The black-box optimization algorithm leading to this bound can be implemented in a way that only 3-ary unbiased variation operators are used. Hence our bound is also valid for the unbiased black-box complexity recently introduced by Lehre and Witt (GECCO 2010). The bound also remains valid if we impose the additional restriction that the black-box algorithm does not have access to the objective values but only to their relative order (ranking-based black-box complexity).

12 pages, to appear in the Proc. of Artificial Evolution 2011, LNCS 7401, Springer, 2012. For the unrestricted black-box complexity of LeadingOnes there is now a tight $\hat{I}˜(n\log \log n)$ bound, cf. http://eccc.hpi-web.de/report/2012/087/