An Almost-Optimal Upper Bound on the Push Number of the Torus Puzzle

TL;DR

This paper presents an algorithm that significantly improves the upper bound on the push number for the Torus Puzzle, narrowing the gap between known bounds from a factor of (\u00f8f8f8f8) to (\u00f8f8f8f8).

Contribution

The authors develop an algorithm with an almost-optimal upper bound of O(mn d7 a0log f8f8f8f8) rotations, improving previous bounds for the push number of the Torus Puzzle.

Findings

Achieved an O(mn d7 a0log f8f8f8f8) upper bound on the push number.

Reduced the gap between upper and lower bounds from d7f8 to d7a0log f8f8f8f8.

Provided an algorithm for a more restricted model that implies bounds for the original puzzle.

Abstract

We study the Torus Puzzle, a solitaire game in which the elements of an input $m \times n$ matrix need to be rearranged into a target configuration via a sequence of unit rotations (i.e., circular shifts) of rows and/or columns. Amano et al. proposed a more permissive variant of the above puzzle, where each row and column rotation can shift the involved elements by any amount of positions. The number of rotations needed to solve the original and the permissive variants of the puzzle are respectively known as the \emph{push number} and the \emph{drag number}, where the latter is always smaller than or equal to the former and admits an existential lower bound of $\Omega(mn)$. While this lower bound is matched by an $O(mn)$ upper bound, the push number is not so well understood. Indeed, to the best of our knowledge, only an $O(mn \cdot \max\{ m, n \})$ upper bound is currently known. In this paper, we provide an algorithm that solves the Torus Puzzle using $O(mn \cdot \log \max \{m, n\})$ unit rotations in a model that is more restricted than that of the original puzzle. This implies a corresponding upper bound on the push number and reduces the gap between the known upper and lower bounds from $\Theta(\max\{m,n\})$ to $\Theta(\log \max\{m, n\})$.