Deciding Foot-sortability and Minimal 2-bounded Non-foot-sortable Sock Orderings

TL;DR

This paper investigates the problem of determining whether sock orderings can be sorted using a stack, providing a general solution for orderings with each color appearing at most twice and introducing an efficient $O(N\log N)$ algorithm.

Contribution

It extends previous work by resolving foot-sortability for more general sock orderings and presents a fast algorithm for deciding foot-sortability in this setting.

Findings

Resolved foot-sortability for sock orderings with each color appearing at most twice.

Developed an $O(N\log N)$ algorithm for deciding foot-sortability.

Extended prior results from alignment-free 2-uniform sock orderings.

Abstract

A sock ordering is a sequence of socks with different colors. A sock ordering is foot-sortable if the sequence of socks can be sorted by a stack so that socks with the same color form a contiguous block. The problem of deciding whether a given sock ordering is foot-sortable was first considered by Defant and Kravitz, who resolved the case for alignment-free 2-uniform sock orderings. In this paper, we resolve the problem in a more general setting, where each color appears in the sock ordering at most twice. A key component of the argument is a fast algorithm that determines the foot-sortability of a sock ordering of length $N$ in time $O(N\log N)$, which is also an interesting result on its own.