We give a detailed analysis of the optimization time of the [Formula: see text]-Evolutionary Algorithm under two simple fitness functions (OneMax and LeadingOnes). The problem has been approached in the evolutionary algorithm literature in various ways and with different degrees of rigor. Our asymptotic approximations for the mean and the variance represent the strongest of their kind. The approach we develop is based on an asymptotic resolution of the underlying recurrences and can also be extended to characterize the corresponding limiting distributions. While most of our approximations can be derived by simple heuristic calculations based on the idea of matched asymptotics, the rigorous justifications are challenging and require a delicate error analysis.

Several simple, classical, little-known algorithms in the statistics and computer science literature for generating random permutations by coin tossing are examined, analyzed, and implemented. These algorithms are either asymptotically optimal or close to being so in terms of the expected number of times the random bits are generated. In addition to asymptotic approximations to the expected complexity, we also clarify the corresponding variances, as well as the asymptotic distributions. A brief comparative discussion with numerical computations in a multicore system is also given.

Divide-and-conquer recurrences of the form f ( n ) = f (⌊ n/2⌋ ) + f ( ⌈ n/2⌉ ) + g ( n ) ( n ⩾ 2), with g ( n ) and f (1) given, appear very frequently in the analysis of computer algorithms and related areas. While most previous methods and results focus on simpler crude approximation to the solution, we show that the solution always satisfies the simple identity f ( n ) = n P (log 2 n ) − Q ( n ) under an optimum (iff) condition on g ( n ). This form is not only an identity but also an asymptotic expansion because Q ( n ) is of a smaller order than linearity. Explicit forms for the continuous periodic function P are provided. We show how our results can be easily applied to many dozens of concrete examples collected from the literature and how they can be extended in various directions. Our method of proof is surprisingly simple and elementary but leads to the strongest types of results for all examples to which our theory applies.

Skylines emerged as a useful notion in database queries for selecting representative groups in multivariate data samples for further decision making, multiobjective optimization, or data processing, and the $k$-dominant skylines were naturally introduced to resolve the abundance of skylines when the dimensionality grows or when the coordinates are negatively correlated. We prove in this paper that the expected number of $k$-dominant skylines is asymptotically zero for large samples when $1\leq k\leq d-1$ under two reasonable (continuous) probability assumptions of the input points, $d$ being the (finite) dimensionality, in contrast to the asymptotic unboundedness when $k=d$. In addition to such an asymptotic zero-infinity property, we also establish a sharp threshold phenomenon for the expected $(d-1)$-dominant skylines when the dimensionality is allowed to grow with $n$, the sample size. Several related issues, such as the dominant cycle structures, the numerical aspects, and the practical implications, are also briefly studied.