Efficient parallel multiselection on hypercubes

We study efficient parallel solutions to the problem of selecting r elements at specified ranks from a set of n arbitrary elements, known as multiselection, on a hypercube with p processors, p,r⩽n. We propose two parallel algorithms based on different approaches, where one requires processors to operate in the SIMD mode, and the other in the MIMD mode. Our SIMD algorithm runs in time O((log n log log n) min{r, log n}) when p=Θ(n), and O(n^ε min{r, (1-ε) log n}) when p=n^ε for any 0<ε<1, where the latter is cost optimal when r⩾p. Our MIMD algorithm runs in O(log n log log n log r) time when p=Θ(n), and in O(n^ε log r) time when p=n^ε for any 0<ε<1, which is cost optimal for any r. Both algorithms are more efficient than the possible straightforward solutions and that of direct simulation of the optimal EREW algorithm