Generating uniform linear extensions using few random bits

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.