2-stack pushall sortable permutations

TL;DR

This paper introduces 2-stack pushall permutations, a subclass of 2-stack sortable permutations, and provides an efficient algorithm to determine their sortability, advancing the understanding of the general 2-stack sorting problem.

Contribution

The paper defines 2-stack pushall permutations, explores their relation to 2-stack sortable permutations, and presents an optimal O(n^2) algorithm for their recognition.

Findings

Introduced 2-stack pushall permutations as a subclass of 2-stack sortable permutations.

Developed an optimal O(n^2) algorithm to decide 2-stack pushall sortability.

Described all possible sortings for 2-stack pushall permutations.

Abstract

In the 60's, Knuth introduced stack-sorting and serial compositions of stacks. In particular, one significant question arise out of the work of Knuth: how to decide efficiently if a given permutation is sortable with 2 stacks in series? Whether this problem is polynomial or NP-complete is still unanswered yet. In this article we introduce 2-stack pushall permutations which form a subclass of 2-stack sortable permutations and show that these two classes are closely related. Moreover, we give an optimal O(n^2) algorithm to decide if a given permutation of size n is 2-stack pushall sortable and describe all its sortings. This result is a step to the solve the general 2-stack sorting problem in polynomial time.