Parallel multiple selection by regular sampling

TL;DR

This paper introduces a deterministic parallel algorithm for the multiple selection problem in the congested clique model, achieving optimal time and minimal communication rounds, with improvements over previous methods especially for the classic case.

Contribution

The paper presents a new deterministic parallel algorithm for multiple selection that reduces communication rounds and improves synchronization costs in the congested clique model.

Findings

Optimal time complexity achieved

Communication rounds reduced to $O(\log^*_{r+1} (n))$

Improves synchronization cost for the classic selection problem

Abstract

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.