Generating uniform linear extensions using few random bits

Mark Huber

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arXiv · 2506.14725

Generating uniform linear extensions using few random bits

Mark Huber

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Abstract

A \emph{linear extension} of a partial order $\preceq$ over items $A = \{ 1, 2, \ldots, n \}$ is a permutation $\sigma$ such that for all $i < j$ in $A$, it holds that $\neg(\sigma(j) \preceq \sigma(i))$. Consider the problem of generating uniformly from the set of linear extensions of a partial order. The best method currently known uses $O(n^3 \ln(n))$ operations and $O(n^3 \ln(n)^2)$ iid fair random bits to generate such a permutation. This paper presents a method that generates a uniform linear extension using only $2.75 n^3 \ln(n)$ operations and $ 1.83 n^3 \ln(n) $ iid fair bits on average.

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Mark Huber. 2025-06-17. Generating uniform linear extensions using few random bits. https://arxiv.org/abs/2506.14725

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