Stratified sampling is a technique that consists in separating the elements of a population into nonoverlapping groups, called strata. This paper describes a new algorithm to solve the one-dimensional case, which reduces the stratification problem to just determining strata boundaries. Assuming that the number L of strata and the total sample size n are predetermined, we obtain the strata boundaries by taking into consideration an objective function associated with the variance. In order to solve this problem, we have implemented an algorithm based on the iterative local search metaheuristic. Computational results obtained from a real data set are presented and discussed.

The Euclidean Steiner tree problem in R(n) (ESTP) is that of finding the shortest interconnecting network spanning p given nodes in the Euclidean R(n), with the possible use of extra nodes. Combinatorial explosion precludes the use of exact methods for large high-dimensional ESTP instances, but very few heuristic approaches have so far been proposed for them. Here we introduce a microcanonical optimization algorithm that works over a topology-describing data structure associated to the ESTP solutions, and which is proven able to find close-to-minimum Steiner trees in reasonable computational time, even for configurations of up to p=50 points in n=50 dimensions. Moreover, its performance is shown to increase with n, which makes it especially suited for high-dimensional clustering problems such as those of phylogenetic inference, an instance of which is considered here.

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We propose algorithmic frameworks based on the iterated local search (ILS) metaheuristic to obtain goodquality solutions for the Euclidean Steiner tree problem (ESTP) in n dimensions. This problem consists in finding a tree with minimal total length that spans p points given in an n-dimensional Euclidean space and, eventually, also some additional points whose insertion contributes to reduce the total length of the tree. These ILS approaches make use of both the tree enumeration structure, called topology-describing vector, and the exact minimization step of a well-known branch-and-bound method for the ESTP. Computational results are provided.

We report a new optimal resolution for the statistical stratification problem under proportional sampling allocation among strata. Consider a finite population of N units, a random sample of n units selected from this population and a number L of strata. Thus, we have to define which units belong to each stratum so as to minimize the variance of a total estimator for one desired variable of interest in each stratum, and consequently reduce the overall variance for such quantity. In order to solve this problem, an exact algorithm based on the concept of minimal path in a graph is proposed and assessed. Computational results using real data from IBGE (Brazilian Central Statistical Office) are provided.