Optimal Partitioning Which Maximizes the Sum of the Weighted Averages

We consider an optimal partitioning problem that occurs in the assignment of computer jobs to a multiple cache and in other combinatorial optimization problems: For a given set of n elements, where each element i has a given frequency p_i and a specific weight w_i, we would like to divide the elements into m mutually exclusive groups such that the sum over all the groups of the average group weight is maximal. We characterize the optimal solution and present an algorithm which is polynomial in n for obtaining the optimal partitioning. Optimal partitioning policies for two groups has an especially simple characterization: There exist two numbers α and β, with minw_i < α < β ≤ maxw_i, such that all the elements with weight w_i satisfying α ≤ w_i ≤ β belong to one group and all other elements belong to the other group. A modification of this policy gives the optimal partitioning for an arbitrary number of groups.