Optimal antimatroid sorting

TL;DR

This paper generalizes optimal sorting algorithms to broader classes of restricted total orders defined by antimatroids, extending previous work on topological orderings and providing optimal solutions for various graph and formula-based constraints.

Contribution

It introduces a simple generalization of topological heapsort applicable to antimatroids, enabling optimal sorting under diverse restrictions beyond partial orders.

Findings

Optimal algorithms for sorting with antimatroid constraints

Extension of topological heapsort to broader classes of restrictions

Applications to monotone formulas, chordal graphs, and rooted graph searches

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.