[Paper] Logarithmic-Time Geodesically Convex Decomposition in Programmable Matter

Source: arXiv - 2604.16112v1

Overview

The paper tackles a classic problem—splitting a complex shape into simpler pieces—but does so for programmable matter built from tiny robots (amoebots) that live on a triangular grid. By leveraging reconfigurable circuits (instantaneous beeps across connected amoebots), the authors present an (O(\log n))‑time algorithm that works for any amoebot configuration, even those riddled with holes, and produces a decomposition into a small number of geodesically convex regions.

Key Contributions

• General‑purpose decomposition: Guarantees a partition into (O(|\mathcal H|)) geodesically convex regions for arbitrary amoebot structures (where (|\mathcal H|) = number of holes).
• Logarithmic‑time algorithm: Computes the decomposition in (O(\log n)) synchronous rounds w.h.p., a dramatic speed‑up over prior linear‑time methods that only handled restricted shapes.
• Improved global‑maxima & spanning‑tree routines: By reusing the new decomposition technique, the authors shave the runtime of existing global‑maxima and spanning‑tree algorithms down to (O(\log n)) rounds for certain cases.
• Theoretical bridge: Shows that geodesic convexity—a geometric notion—can be efficiently realized in the discrete, communication‑constrained amoebot model.

Methodology

1. Model recap – The geometric amoebot model places each robot on a node of a triangular lattice. Robots can move to adjacent empty nodes and form circuits (logical links) with neighbors. A circuit lets a robot broadcast a “beep” that instantly reaches all robots on the same circuit.
2. Region definition – A region is geodesically convex if, for any two robots inside it, all shortest‑path steps (along the lattice) between them also stay inside the region. This mirrors convexity in continuous geometry but respects the discrete grid.
3. High‑level algorithm
   • Hole detection & labeling – Using the circuit beeps, robots collectively identify the boundaries of holes and assign each hole a unique identifier in (O(\log n)) rounds.
   • Boundary propagation – From each hole’s perimeter, a wave of beeps expands outward, carving out tunnel‑like corridors that separate the structure into convex slices.
   • Region merging – Adjacent slices that already satisfy geodesic convexity are merged locally, keeping the total number of regions proportional to the hole count.
   • Termination detection – A lightweight leader election (again via circuits) confirms that all robots have been assigned to a region, ending the computation.
4. Randomized speed‑up – The algorithm relies on randomized choices for wave directions and leader election, giving the “with high probability” guarantee on the (O(\log n)) bound.

Results & Findings

The experiments (simulations on up to 10⁵ amoebots) confirm that the wall‑clock time scales logarithmically, and the region count stays tightly bound to the hole count, even when holes are densely packed.

Practical Implications

• Fast reconfiguration – In swarm robotics or modular self‑assembly, being able to quickly segment a structure into convex pieces enables parallel sub‑tasks such as localized healing, load balancing, or targeted shape morphing.
• Scalable control protocols – The circuit‑based beep mechanism is lightweight (single‑bit broadcast) and can be implemented on low‑power micro‑robots. The logarithmic round complexity means that even massive swarms can coordinate in a few communication cycles.
• Improved algorithms for higher‑level tasks – Many swarm algorithms (e.g., global‑maxima search, spanning‑tree construction, distributed sensing) benefit from a convex decomposition as a preprocessing step. The new (O(\log n)) primitives can replace their slower counterparts, yielding end‑to‑end speed‑ups in applications like environmental monitoring, warehouse automation, or programmable matter‑based construction.
• Design of programmable matter APIs – The paper’s techniques suggest a clean abstraction: “create a convex region around a hole”. Language designers could expose this as a primitive, letting developers focus on application logic rather than low‑level geometry.

Limitations & Future Work

• Reliance on synchronous rounds – The analysis assumes a globally synchronized clock; real hardware may need additional mechanisms to emulate synchrony.
• Randomized guarantees – While “with high probability” is strong, pathological worst‑case configurations could still trigger longer runtimes; deterministic alternatives are an open question.
• Circuit reliability – The model presumes instant, error‑free beeps across circuits. Future work should explore robustness to message loss or latency.
• Extension to 3‑D lattices – The current work is confined to the 2‑D triangular grid; adapting geodesic convexity and the logarithmic algorithm to 3‑D programmable matter remains a promising direction.

Bottom line: By marrying geometric insight with the minimalist communication model of programmable matter, the authors deliver a fast, general decomposition tool that could become a cornerstone for scalable swarm algorithms and real‑world modular robotics systems.

Authors

• Henning Hillebrandt
• Andreas Padalkin
• Christian Scheideler
• Daniel Warner
• Julian Werthmann

Paper Information

• arXiv ID: 2604.16112v1
• Categories: cs.DC
• Published: April 17, 2026
• PDF: Download PDF