Generating uniform linear extensions using few random bits (2506.14725v1)

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.