Distributed Selection in $O ( \log n )$ Time with $O ( n \log \log n )$ Messages

TL;DR

This paper presents a distributed algorithm that finds the $k$th smallest element in a network of $n$ processors in logarithmic time using near-linear messages, leveraging AKS sorting network simulation.

Contribution

It introduces the first known parallel distributed $k$-selection algorithm with $O( ext{log} n)$ time and $O(n ext{log} ext{log} n)$ messages, simulating AKS sorting network comparisons.

Findings

Achieves $O( ext{log} n)$ time complexity for distributed $k$-selection.

Uses $O(n ext{log} ext{log} n)$ messages, near-linear in $n$.

Proves $ ext{log} n$ as the lower bound for data aggregation problems.

Abstract

We consider the selection problem on a completely connected network of $n$ processors with no shared memory. Each processor initially holds a given numeric item of $b$ bits allowed to send a message of $\max ( b, \lg n )$ bits to another processor at a time. On such a communication network ${\cal G}$, we show that the $k$th smallest of the $n$ inputs can be detected in $O ( \log n )$ time with $O ( n \log \log n )$ messages. The possibility of such a parallel algorithm for this distributed $k$-selection problem has been unknown despite the intensive investigation on many variations of the selection problem carried out since 1970s. The main trick of our algorithm is to simulate the comparisons and swaps performed by the AKS sorting network, the $n$-input sorting network of logarithmic depth discovered by Ajtai, Koml{\'o}s and Szemer{\'e}di in 1983. We also show the universal time lower bound $\lg n$ for many basic data aggregation problems on ${\cal G}$, confirming the asymptotic time optimality of our parallel algorithm.