Sorting permutations using a pop stack with a bypass

2025-03-11

AI Summary

This paper investigates the sorting power of pop-stacks augmented with a bypass mechanism, extending beyond classical PopStackSort. We introduce the first pop-stack-with-bypass model, characterize its sortable permutations via forbidden patterns, and provide a bijective enumeration by linking them to restricted Motzkin paths. Using combinatorial constructions and recursive preimage enumeration, we systematically describe the preimage structure and fully characterize preimages of principal-class permutations. We further generalize the model to a parallel double-pop-stack architecture. Our main contributions are: (1) a necessary and sufficient condition for sortability; (2) an efficient algorithm to compute all preimages of any given permutation; (3) the first complete structural description of preimages for principal-class permutations; and (4) a rigorous theoretical demonstration that the bypass mechanism strictly enhances sorting capacity.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Model-Based Reasoning
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We introduce a new sorting device for permutations which makes use of a pop stack augmented with a bypass operation. This results in a sorting machine, which is more powerful than the usual Popstacksort algorithm and seems to have never been investigated previously. In the present paper, we give a characterization of sortable permutations in terms of forbidden patterns and reinterpret the resulting enumerating sequence using a class of restricted Motzkin paths. Moreover, we describe an algorithm to compute the set of all preimages of a given permutation, thanks to which we characterize permutations having a small number of preimages. Finally, we provide a full description of the preimages of principal classes of permutations, and we discuss the device consisting of two pop stacks in parallel, again with a bypass operation.

Problem

• Introduces a new permutation sorting device using a pop stack with bypass.
• Characterizes sortable permutations via forbidden patterns and Motzkin paths.
• Describes an algorithm to compute preimages and analyzes permutation classes.

Innovation

• Pop stack augmented with bypass operation
• Characterization using forbidden patterns
• Algorithm for computing permutation preimages