Generating uniform linear extensions using few random bits

TL;DR

This paper introduces a new algorithm that efficiently generates uniform linear extensions of a partial order using significantly fewer random bits and computational steps than previous methods.

Contribution

The paper presents a novel method that reduces the randomness and computational complexity required to generate uniform linear extensions of partial orders.

Findings

Uses fewer random bits than previous algorithms.

Reduces computational complexity from O(n^3 log n) to approximately 2.75 n^3 log n operations.

Achieves uniform sampling with high efficiency.

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.