Fast M\"obius and Zeta Transforms

TL;DR

This paper introduces efficient algorithms for M"obius and zeta transforms on finite posets, significantly reducing computation time from quadratic to linear in the size of the poset by leveraging DAG representations and chain decompositions.

Contribution

The paper presents novel $O(nk)$ time and space algorithms for M"obius and zeta transforms on posets, improving over traditional $O(n^2)$ methods, using DAG-based chain decomposition.

Findings

Algorithms handle posets with millions of nodes in seconds.

Parallelized implementations further improve performance.

Benchmarks demonstrate efficiency on sparse DAGs.

Abstract

M\"obius inversion of functions on partially ordered sets (posets) $\mathcal{P}$ is a classical tool in combinatorics. For finite posets it consists of two, mutually inverse, linear transformations called zeta and M\"obius transform, respectively. In this paper we provide novel fast algorithms for both that require $O(nk)$ time and space, where $n = |\mathcal{P}|$ and $k$ is the width (length of longest antichain) of $\mathcal{P}$, compared to $O(n^2)$ for a direct computation. Our approach assumes that $\mathcal{P}$ is given as directed acyclic graph (DAG) $(\mathcal{E}, \mathcal{P})$. The algorithms are then constructed using a chain decomposition for a one time cost of $O(|\mathcal{E}| + |\mathcal{E}_\text{red}| k)$, where $\mathcal{E}_\text{red}$ is the number of edges in the DAG's transitive reduction. We show benchmarks with implementations of all algorithms including parallelized versions. The results show that our algorithms enable M\"obius inversion on posets with millions of nodes in seconds if the defining DAGs are sufficiently sparse.