Efficient weighted multiselection in parallel architectures

We study parallel solutions to the problem of weighted multiselection to select r elements on given weighted-ranks from a set S of n weighted elements, where an element is on weighted rank k if it is the smallest element such that the aggregated weight of all elements not greater than it in S is not smaller than k. We propose efficient algorithms on two of the most popular parallel architectures, hypercube and mesh. For a hypercube with p < n processors, we present a parallel algorithm running in 0(n^∈ min{r, log p}) time for p = n^1-∈, 0 < ∈ < 1, which is cost optimal when r ≥ p. Our algorithm on √p x √p- mesh runs in O(√p + _p-ⁿ log³ p) time which is the same as multiselection on mesh when r ≥ logp, and thus has the same optimality as multiselection in this case.