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

Title: Hierarchical Shape Construction and Complexity for Slidable Polyominos under Uniform External Forces
Authors: Jose Balanza-Martinez, David Caballero, Angel A. Cantu, Mauricio Flores, Timothy Gomez, Austin Luchsinger, Rene Reyes, Robert Schweller, and Tim Wylie
Conference: The 31st ACM-SIAM Symposium on Discrete Algorithms (SODA’20), 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 \emph{strongly universal} for any $w \times h$ bounded shape within this hierarchy (it can be reconfigured to construct any $w \times h$ 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 $O(n^4 \log n)$-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 PSPACE-complete, even when all movable pieces are $1 \times 1$ tiles with no glues. We achieve this result by a reduction from Non-deterministic Constraint Logic for a one-player unbounded game.

Accompanying videos related to the paper

Hierarchy Constructor: Strict Level 2 Polyomino Construction:

We show the construction of a strict level 2 polyomino using two tile types following the command sequences described in the paper. This polyomino cannot be built by the level 1 constructor, and requires the use of the level 2 constructor.

NCL Reduction:

Succesful Relocation of 1×1 Tile:

This video shows a solution to an instance of the relocation problem in ful tilt that was generated from a constraint graph. The image below shows the successive states of the the corresponding constraint graph. The target configuration is shown in the rightmost graph.

Unsuccesful Relocation of 1×1 Tile:

This video shows the an attempt at completing the relocation process when the gadgets are not in the correct state, causing a tile to get trapped.  The starting and goal configurations are the same as the previous example. The image below shows the successive states of the the corresponding constraint graph.

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Full Tilt: Universal Constructors for General Shapes with Uniform External Forces

Title: Full Tilt: Universal Constructors for General Shapes with Uniform External Forces

Authors: Jose Balanza-Martinez, David Caballero, Angel A. Cantu, Luis Angel Garcia, Austin Luchsinger, Rene Reyes, Robert Schweller, and Tim Wylie

Conference: The 30th ACM-SIAM Symposium on Discrete Algorithms (SODA’19), 2019.

Abstract:We investigate the problem of assembling general shapes and patterns in a model in which particles move based on uniform external forces until they encounter an obstacle. In this model, corresponding particles may bond when adjacent with one another. Succinctly, this model considers a 2D grid of “open” and “blocked” spaces, along with a set of slidable polyominoes placed at open locations on the board. 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. By successively applying a sequence of such tilts, along with allowing different polyominoes to stick when adjacent, tilt sequences provide a method to reconfigure an initial board configuration so as to assemble a collection of previous separate polyominoes into a larger shape.

While previous work within this model of assembly has focused on designing a specific board configuration for the assembly of a specific given shape, we propose the problem of designing \emph{universal configurations} that are capable of constructing a large class of shapes and patterns.
For these constructions, we present the notions of \emph{weak} and \emph{strong} universality which indicate the presence of “excess” polyominoes after the shape is constructed. In particular, for given integers $h,w$, we show that there exists a weakly universal configuration with $\mathcal{O}(hw)$ $1 \times 1$ slidable particles that can be reconfigured to build any $h \times w$ patterned rectangle. We then expand this result to show that there exists a weakly universal configuration that can build any $h \times w$-bounded size connected shape. Following these results, which require an admittedly relaxed assembly definition, we go on to show the existence of a strongly universal configuration (no excess particles) which can assemble any shape within a previously studied “Drop” class, while using quadratically less space than previous results.

Finally, we include a study of the complexity of deciding if a particle within a configuration may be relocated to another position, and deciding if a given configuration may be transformed into a second given configuration. In both cases, we show this problem to be PSPACE-complete, even when movable particles are restricted to $1\times 1$ and $2\times 2$ polyominoes that do not stick to one another.

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Bibtex:

Accompanying videos related to the paper

Section 3: Pattern and General Shape Builder

Construction of a Pattern:

Construction of a Drop Shape:

Construction of a Non-Drop Shape:

Section 4: Drop Shape Builder

Section 5: Relocation Gadget

Example of Correct Traversal Through our Relocation Gadget:

This video shows the correct sequence of tilts to traverse our robot through the gadget.  Note the closed exit/entrance points of our gadgets do not affect this traversal, and are only there to simplify testing

Example of Incorrect State Traversal Attempt Through our Relocation Gadget:

Here we see the robot trying to traverse our gadget while the gadget is not in the correct state to allow the robot to do so. This is only one of the possible sequences, but no sequence of tilts exists that would allow the robot to traverse through the gadget from this position.

Example of Robot Polyomino Becoming Stuck Through Correct State but Incorrect Traversal Sequence:

Here is an example of the importance of our optimal sequence to traverse through the gadget, although there are many ways to traverse, there are also many ways to enter what we call “stuck” configuration. From these configurations there is no sequence of tilts that would allow the robot to leave the gadget.

Section 6: Reconfiguration Gadget

Example of Correct Traversal Through Our Reconfiguration Gadget:

Example of Moving State Tiles to Reconfiguration Ring:

Here we remove all tiles from the inner section of our gadget and into the “reconfiguration ring”. This property of our gadget is what allows us to achieve a global configuration of our gadgets and classify this problem as a reconfiguration problem.

Example of A System of Reconfiguration Gadgets:

This shows a system of reconfiguration gadgets and the robots traversal through the system. At the end of our traversal we move all our state tiles int the reconfiguration ring and achieve a specific configuration of our entire system.

Example of Traversal Sequences that Preserve Initial State Tile Positions:

The videos here show that we can easily traverse gadgets while maintaining the a small set of positions for our state tiles. We know that from these positions we can traverse the gadget with a sequence that will allow us to keep the positions of our red tiles within this set, while not moving any tiles that started in this set into the reconfiguration ring.