Efficient Shape Reconfiguration by Hybrid Programmable Matter

Shape formation is one of the most thoroughly studied problems in most algorithmic models of programmable matter. However, few existing shape formation algorithms utilize similarities between an initial configuration and a desired target shape. In the hybrid model, an active agent with the computational capabilities 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 needs to move all tiles in an input shape to so-called target nodes, which are distinguishable from other nodes by the agent. We present a worst-case optimal $O(mn)$ algorithm for simply connected target shapes and an $O(n^4)$ algorithm for a large class of target shapes that may contain holes, where $m$ is the initial number of unoccupied target nodes and $n$ is the total number of tiles.
Comment: 23 pages, 12 figures