QR Sort: A Novel Non-Comparative Sorting Algorithm

TL;DR

QR Sort is a new non-comparative integer sorting algorithm based on the Quotient-Remainder Theorem, offering efficient, stable sorting with favorable complexity and practical performance, especially for large input ranges.

Contribution

The paper introduces QR Sort, a novel non-comparative sorting algorithm that leverages the Quotient-Remainder Theorem and provides implementation optimizations for improved efficiency.

Findings

QR Sort often outperforms established algorithms.

It achieves linear time and space complexity when input range is not too large.

The algorithm is adaptable and reliable for large value ranges.

Abstract

In this paper, we introduce and prove QR Sort, a novel non-comparative integer sorting algorithm. This algorithm uses principles derived from the Quotient-Remainder Theorem and Counting Sort subroutines to sort input sequences stably. QR Sort exhibits the general time and space complexity $\mathcal{O}(n+d+\frac{m}{d})$, where $n$ denotes the input sequence length, $d$ denotes a predetermined positive integer, and $m$ denotes the range of input sequence values plus 1. Setting $d = \sqrt{m}$ minimizes time and space to $\mathcal{O}(n + \sqrt{m})$, resulting in linear time and space $\mathcal{O}(n)$ when $m \leq \mathcal{O}(n^2)$. We provide implementation optimizations for minimizing the time and space complexity, runtime, and number of computations expended by QR Sort, showcasing its adaptability. Our results reveal that QR Sort frequently outperforms established algorithms and serves as a reliable sorting algorithm for input sequences that exhibit large $m$ relative to $n$.