Optimal antimatroid sorting

2025-07-18

AI Summary

This paper studies optimal comparison sorting under antimatroid constraints: given a family of candidate total orders compactly represented by an antimatroid, how to efficiently sort within this restricted space. We generalize topological heap sort to the antimatroid framework, designing a novel priority-queue-based algorithm that exploits structural properties of antimatroids. Our method unifies several important restricted sorting settings, including linear extensions satisfying monotone prefix formulas, perfect elimination orderings of chordal graphs, and vertex search orderings of connected graphs. We prove that the algorithm achieves the information-theoretic lower bound on comparison complexity, thereby establishing the first optimal comparison sorting result beyond partial orders—extending it to the broader class of antimatroid-constrained domains. This work significantly broadens both the applicability and theoretical foundations of constrained sorting.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Extends topological heapsort to antimatroid-based sorting problems
• Optimizes sorting for total orders restricted by antimatroids
• Solves sorting with precedence, chordal graph, or search order constraints

Innovation

• Generalized topological heapsort for antimatroids
• Optimal sorting with monotone precedence formulas
• Efficient algorithms for chordal graph orders