An Optimal Algorithm for Sorting Pattern-Avoiding Sequences

TL;DR

This paper introduces a linear-time deterministic comparison-based algorithm for sorting sequences that avoid a fixed permutation pattern, matching the theoretical lower bound and applicable to permutations of bounded twin-width.

Contribution

It provides the first optimal linear-time algorithm for sorting pattern-avoiding sequences, leveraging the Marcus-Tardos theorem for efficient multi-way merging.

Findings

Achieves linear-time sorting for pattern-avoiding sequences

Matches the information-theoretic lower bound on complexity

Extends to permutations of bounded twin-width

Abstract

We present a deterministic comparison-based algorithm that sorts sequences avoiding a fixed permutation $\pi$ in linear time, even if $\pi$ is a priori unkown. Moreover, the dependence of the multiplicative constant on the pattern $\pi$ matches the information-theoretic lower bound. A crucial ingredient is an algorithm for performing efficient multi-way merge based on the Marcus-Tardos theorem. As a direct corollary, we obtain a linear-time algorithm for sorting permutations of bounded twin-width.