Sorting Pattern-Avoiding Permutations via 0-1 Matrices Forbidding Product Patterns

TL;DR

This paper investigates the extremal properties of pattern-avoiding 0-1 matrices related to permutation sorting, providing bounds that improve understanding of sorting complexity for pattern-avoiding sequences.

Contribution

It establishes nearly tight bounds on the density of certain pattern-avoiding matrices, linking matrix theory to permutation sorting complexity.

Findings

Bounds on extremal functions involve the inverse-Ackermann function

Sorting permutation-avoiding sequences can be done in near-linear time

Analysis connects forbidden matrix theory with dynamic optimality conjecture

Abstract

We consider the problem of comparison-sorting an $n$-permutation $S$ that avoids some $k$-permutation $\pi$. Chalermsook, Goswami, Kozma, Mehlhorn, and Saranurak prove that when $S$ is sorted by inserting the elements into the GreedyFuture binary search tree, the running time is linear in the extremal function $\mathrm{Ex}(P_\pi\otimes \text{hat},n)$. This is the maximum number of 1s in an $n\times n$ 0-1 matrix avoiding $P_\pi \otimes \text{hat}$, where $P_\pi$ is the $k\times k$ permutation matrix of $\pi$, $\otimes$ the Kronecker product, and $\text{hat} = \left(\begin{array}{ccc}&\bullet&\\\bullet&&\bullet\end{array}\right)$. The same time bound can be achieved by sorting $S$ with Kozma and Saranurak's SmoothHeap. In this paper we give nearly tight upper and lower bounds on the density of $P_\pi\otimes\text{hat}$-free matrices in terms of the inverse-Ackermann function $\alpha(n)$. \[ \mathrm{Ex}(P_\pi\otimes \text{hat},n) = \left\{\begin{array}{ll} \Omega(n\cdot 2^{\alpha(n)}), & \mbox{for most $\pi$,}\\ O(n\cdot 2^{O(k^2)+(1+o(1))\alpha(n)}), & \mbox{for all $\pi$.} \end{array}\right. \] As a consequence, sorting $\pi$-free sequences can be performed in $O(n2^{(1+o(1))\alpha(n)})$ time. For many corollaries of the dynamic optimality conjecture, the best analysis uses forbidden 0-1 matrix theory. Our analysis may be useful in analyzing other classes of access sequences on binary search trees.