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TL;DR

This paper analyzes algorithms for extending vertex functions to discrete Morse functions on simplicial complexes, offering a more efficient description suitable for high-dimensional data.

Contribution

It provides a new, faster description of an existing algorithm for discrete Morse function extension using the directed Hasse diagram.

Findings

Faster runtime for high-dimensional data

No increase in space complexity

Improved understanding of the algorithm's structure

Abstract

Given a simplicial complex $K$ and an injective function $f$ from the vertices of $K$ to $\mathbb{R}$, we consider algorithms that extend $f$ to a discrete Morse function on $K$. We show that an algorithm of King, Knudson and Mramor can be described on the directed Hasse diagram of $K$. Our description has a faster runtime for high dimensional data with no increase in space.