Hierarchical Shape Construction and Complexity for Slidable Polyominoes under Uniform External Forces

Authors

Jose Balanza-Martinez, The University of Texas Rio Grande Valley
David Caballero, The University of Texas Rio Grande Valley
Angel A. Cantu, The University of Texas Rio Grande Valley
Mauricio Flores, The University of Texas Rio Grande Valley
Timothy Gomez, The University of Texas Rio Grande Valley
Austin Luchsinger, The University of Texas Rio Grande Valley
Rene Reyes, The University of Texas Rio Grande Valley
Robert Schweller, The University of Texas Rio Grande Valley
Tim Wylie, The University of Texas Rio Grande Valley

Document Type

Conference Proceeding

Publication Date

2020

Abstract

Advances in technology have given us the ability to create and manipulate robots for numerous applications at the molecular scale. At this size, fabrication tool limitations motivate the use of simple robots. The individual control of these simple objects can be infeasible. We investigate a model of robot motion planning, based on global external signals, known as the tilt model. Given a board and initial placement of polyominoes, the board may be tilted in any of the 4 cardinal directions, causing all slidable polyominoes to move maximally in the specified direction until blocked.

We propose a new hierarchy of shapes and design a single configuration that is strongly universal for any w × h bounded shape within this hierarchy (it can be reconfigured to construct any w × h bounded shape in the hierarchy). This class of shapes constitutes the most general set of buildable shapes in the literature, with most previous work consisting of just the first-level of our hierarchy. We accompany this result with a O(n⁴ log n)-time algorithm for deciding if a given hole-free shape is a member of the hierarchy. For our second result, we resolve a long-standing open problem within the field: We show that deciding if a given position may be covered by a tile for a given initial board configuration is PSPACEcomplete, even when all movable pieces are 1 × 1 tiles with no glues. We achieve this result by a reduction from Non-deterministic Constraint Logic for a one-player unbounded game.

Comments

© 2020 by SIAM. Original published version available at https://doi.org/10.1137/1.9781611975994.160

Recommended Citation

Balanza-Martinez, J., Gomez, T., Caballero, D., Luchsinger, A., Cantu, A. A., Reyes, R., Flores, M., Schweller, R., & Wylie, T. (2019). Hierarchical Shape Construction and Complexity for Slidable Polyominoes under Uniform External Forces. Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, 2625–2641. https://doi.org/10.1137/1.9781611975994.160

Publication Title

Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms

DOI

10.1137/1.9781611975994.160