Sorting an Array Using the Topological Sort of a Corresponding Comparison Graph

TL;DR

This paper introduces a graph-based stable sorting method using topological sort on comparison graphs, achieving optimal time complexity and improved space efficiency over traditional algorithms like MergeSort.

Contribution

It presents a novel graph-based sorting algorithm leveraging topological sort and Hamiltonian path relations, with a new merge process for space efficiency.

Findings

Runs in optimal O(n log n) time

More space-efficient than MergeSort

Utilizes a divide-and-conquer approach

Abstract

The quest for efficient sorting is ongoing, and we will explore a graph-based stable sorting strategy, in particular employing comparison graphs. We use the topological sort to map the comparison graph to a linear domain, and we can manipulate our graph such that the resulting topological sort is the sorted array. By taking advantage of the many relations between Hamiltonian paths and topological sorts in comparison graphs, we design a Divide-and-Conquer algorithm that runs in the optimal $O(n \log n)$ time. In the process, we construct a new merge process for graphs with relevant invariant properties for our use. Furthermore, this method is more space-efficient than the famous {\sc MergeSort} since we modify our fixed graph only.