Parallel multiple selection by regular sampling

In this paper we present a deterministic parallel algorithm solving the multiple selection problem in congested clique model. In this problem for given set of elements S and a set of ranks $K = \{k_1 , k_2 , ..., k_r \}$ we are asking for the $k_i$-th smallest element of $S$ for $1 \leq i \leq r$. The presented algorithm is deterministic, time optimal , and needs $O(\log^*_{r+1} (n))$ communication rounds, where $n$ is the size of the input set, and $r$ is the size of the rank set. This algorithm may be of theoretical interest, as for $r = 1$ (classic selection problem) it gives an improvement in the asymptotic synchronization cost over previous $O(\log \log p)$ communication rounds solution, where $p$ is size of clique.