Energy-Constrained Programmable Matter Under Unfair Adversaries

Individual modules of programmable matter participate in their system's collective behavior by expending energy to perform actions. However, not all modules may have access to the external energy source powering the system, necessitating a local and distributed strategy for supplying energy to modules. In this work, we present a general energy distribution framework for the canonical amoebot model of programmable matter that transforms energy-agnostic algorithms into energy-constrained ones with equivalent behavior and an $\mathcal{O}(n^2)$-round runtime overhead -- even under an unfair adversary -- provided the original algorithms satisfy certain conventions. We then prove that existing amoebot algorithms for leader election (ICDCN 2023) and shape formation (Distributed Computing, 2023) are compatible with this framework and show simulations of their energy-constrained counterparts, demonstrating how other unfair algorithms can be generalized to the energy-constrained setting with relatively little effort. Finally, we show that our energy distribution framework can be composed with the concurrency control framework for amoebot algorithms (Distributed Computing, 2023), allowing algorithm designers to focus on the simpler energy-agnostic, sequential setting but gain the general applicability of energy-constrained, asynchronous correctness.Comment: 31 pages, 4 figures, 1 table. Submitted to OPODIS 202

Asynchronous Deterministic Leader Election in Three-Dimensional Programmable Matter

Over three decades of scientific endeavors to realize programmable matter, a substance that can change its physical properties based on user input or responses to its environment, there have been many advances in both the engineering of modular robotic systems and the corresponding algorithmic theory of collective behavior. However, while the design of modular robots routinely addresses the challenges of realistic three-dimensional (3D) space, algorithmic theory remains largely focused on 2D abstractions such as planes and planar graphs. In this work, we present the 3D geometric space variant for the well-established amoebot model of programmable matter, using the face-centered cubic (FCC) lattice to represent space and define local spatial orientations. We then give a distributed algorithm for the classical problem of leader election that can be applied to 2D or 3D geometric amoebot systems, proving that it deterministically elects exactly one leader in $\mathcal{O}(n)$ rounds under an unfair sequential adversary, where $n$ is the number of amoebots in the system. We conclude by demonstrating how this algorithm can be transformed using the concurrency control framework for amoebot algorithms (DISC 2021) to obtain the first known amoebot algorithm, both in 2D and 3D space, to solve leader election under an unfair asynchronous adversary.Comment: 16 pages, 4 figures, 2 table