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

2026-01-13

AI Summary

This work addresses the problem of optimizing the upper bound on the push number—the number of unit row and column rotations required—in the toroidal jigsaw puzzle. To overcome the looseness of the existing upper bound of $O(mn \cdot \max\{m, n\})$, the paper proposes a novel algorithm based on a refined scheduling strategy. Operating under a more restrictive model, the algorithm achieves puzzle completion in only $O(mn \cdot \log \max\{m, n\})$ operations. This result significantly narrows the gap between the known upper and lower bounds on the push number from $\Theta(\max\{m, n\})$ to $\Theta(\log \max\{m, n\})$, bringing the theoretical analysis substantially closer to optimality and offering a key advance in understanding the computational complexity of toroidal jigsaw puzzles.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: The sharing economy

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 puzzle in the original and in 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 $\Theta(\log \max\{m, n\})$.

Problem

• Torus Puzzle
• push number
• unit rotations
• upper bound
• matrix rearrangement

Innovation

• Torus Puzzle
• push number
• unit rotation
• upper bound
• combinatorial puzzle