An Optimal Algorithm for Sorting Pattern-Avoiding Sequences

We present a deterministic comparison-based algorithm that sorts sequences avoiding a fixed permutation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi$</tex> in linear time, even if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi$</tex> is a priori unkown. Moreover, the dependence of the multiplicative constant on the pattern <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pi$</tex> 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.