Tile Reconfiguration by a Finite Automaton

TL;DR

The paper addresses reconfiguring a passive tile configuration into a target shape using a single finite-state automaton agent on a triangular lattice, with the agent able to distinguish target nodes from non-target nodes. For simply connected targets, it achieves a worst-case optimal reconfiguration bound of $O(mn)$ by boundary traversal, supply-compaction, and column-based delivery while maintaining connectivity. For hole-containing targets, it develops pebble/counter-based strategies and a depot-based approach to enable exploration and placement, attaining $O(n^4)$ for bottleneck-free holes and $O(mn^2)$ or $O(mn^3)$ with counters or two pebbles in more general settings. The results advance understanding of shape reconfiguration under severe computational constraints and establish foundational benchmarks for reconfiguring with target recognition and environment modification.

Abstract

Shape formation is one of the most thoroughly studied problems in programmable matter and swarm robotics. However, in many models, the class of shapes that can be formed is highly restricted due to the particles' limited memory. In the hybrid model, an active agent with the computational power of a deterministic finite automaton can form shapes by lifting and placing passive tiles on the triangular lattice. We study the shape reconfiguration problem where the agent additionally has the ability to distinguish so-called target nodes from non-target nodes and needs to form a target shape from the initial tile configuration. We present a worst-case optimal $O(mn)$ algorithm for simply connected target shapes, where $m$ is the initial number of unoccupied target nodes and $n$ is the total number of tiles. Furthermore, we show how an agent can reconfigure a large class of target shapes with holes in $O(n^4)$ steps.

Figures (14)

• Figure 1: An agent on tiles and global compass directions on the triangular lattice.
• Figure 2: An example instance of the \ref{['phase:srProblem']}. The light blue line encircles the target shape $\mathcal{T}$. The blue tiles are target tiles. The yellow tiles are supply tiles and need to be moved to untiled target nodes (demand nodes). \ref{['fig:example_instance_initial']}: The positions of the tiles in the initial shape $\mathcal{I}$. \ref{['fig:example_instance_finished']}: The final shape after all supply tiles have been moved to the target shape $\mathcal{T}$. In this example, the target shape $\mathcal{T}$ is simply connected while the initial shape $\mathcal{I}$ is not.
• Figure 3: The initial set of tiled target nodes $\mathcal{I} \cap \mathcal{T}$ is not connected.
• Figure 4: Boundaries \ref{['fig:boundaries_target']}$B(\mathcal{T})$, \ref{['fig:boundaries_target_tile']}$B(T \cap \mathcal{T})$, and \ref{['fig:boundaries_supply_component']}$B(T \setminus \mathcal{T})$. The agent $r$ is on node $p$.
• Figure 5: An agent traversing the tile boundary $B(T)$. \ref{['fig:lhr_outside_a']}--\ref{['fig:lhr_outside_b']}: LHR. \ref{['fig:rhr_outside_a']}--\ref{['fig:rhr_outside_b']}: RHR.

Theorems & Definitions (16)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Lemma 9
• Lemma 10