Simplified, stable parallel merging

TL;DR

This paper presents a simplified and stable parallel merge algorithm that reduces complexity and constant factors, making it efficient for shared-memory parallel systems and suitable for stable parallel merge sort implementations.

Contribution

It demonstrates that the additional merge step is unnecessary, simplifying the algorithm and ensuring stability with minimal synchronization.

Findings

Runs in O(n/p + log n) operations with p processors

Requires only a single synchronization step

Can be implemented on EREW PRAM and shared-memory systems

Abstract

This note makes an observation that significantly simplifies a number of previous parallel, two-way merge algorithms based on binary search and sequential merge in parallel. First, it is shown that the additional merge step of distinguished elements as found in previous algorithms is not necessary, thus simplifying the implementation and reducing constant factors. Second, by fixating the requirements to the binary search, the merge algorithm becomes stable, provided that the sequential merge subroutine is stable. The stable, parallel merge algorithm can easily be used to implement a stable, parallel merge sort. For ordered sequences with $n$ and $m$ elements, $m\leq n$, the simplified merge algorithm runs in $O(n/p+\log n)$ operations using $p$ processing elements. It can be implemented on an EREW PRAM, but since it requires only a single synchronization step, it is also a candidate for implementation on other parallel, shared-memory computers.