Ordered Dags: HypercubeSort

TL;DR

This paper introduces a general method for converting DAGs with a single source into priority queue data structures, applying it to hypercube DAGs to develop a sorting algorithm with b5(n log^2 n) complexity and exploring properties relating path length and vertex degree.

Contribution

It generalizes heap insertion to arbitrary DAGs and presents a novel sorting algorithm based on hypercube DAGs, also establishing new relationships between DAG properties.

Findings

Developed a general DAG-to-priority-queue conversion method.

Created a hypercube-based sorting algorithm with b5(n log^2 n) complexity.

Derived a relationship between longest path length and maximum degree in a DAG.

Abstract

We generalise the insertion into a binary heap to any directed acyclic graph (DAG) with one source vertex. This lets us formulate a general method for converting any such DAG into a data structure with priority queue interface. We apply our method to a hypercube DAG to obtain a sorting algorithm of complexity $\mathcal{O}(n\log^2 (n))$. As another curious application, we derive a relationship between length of longest path and maximum degree of a vertex in a DAG.