Optimal antimatroid sorting

Abstract

The classical comparison-based sorting problem asks us to find the underlying total order of a given set of elements, where we can only access the elements via comparisons. In this paper, we study a restricted version, where, as a hint, a set $T$ of possible total orders is given, usually in some compressed form. Recently, an algorithm called topological heapsort with optimal running time was found for the case where $T$ is the set of topological orderings of a given directed acyclic graph, or, equivalently, $T$ is the set of linear extensions of a given partial order [Haeupler et al. 2024]. We show that a simple generalization of topological heapsort is applicable to a much broader class of restricted sorting problems, where $T$ corresponds to a given antimatroid. As a consequence, we obtain optimal algorithms for the following restricted sorting problems, where the allowed total orders are restricted by: a given set of monotone precedence formulas; the perfect elimination orders of a given chordal graph; or the possible vertex search orders of a given connected rooted graph.