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arxiv: 2604.16112 · v1 · submitted 2026-04-17 · 💻 cs.DC

Logarithmic-Time Geodesically Convex Decomposition in Programmable Matter

Henning Hillebrandt , Andreas Padalkin , Christian Scheideler , Daniel Warner , Julian Werthmann This is my paper

Pith reviewed 2026-05-10 07:36 UTC · model grok-4.3

classification 💻 cs.DC
keywords amoebot modelprogrammable mattergeodesic convexitydecomposition algorithmreconfigurable circuitslogarithmic timedistributed roboticsshape formation
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The pith

Arbitrary amoebot structures decompose into O(number of holes) geodesically convex regions in O(log n) rounds via reconfigurable circuits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a general method for breaking any connected amoebot structure on the triangular grid into a linear number of simple convex pieces. Prior decomposition techniques required hole-free or otherwise restricted shapes, but this approach handles arbitrary holes by exploiting circuits for fast global signaling. A reader would care because decomposition is a basic building block for shape formation, navigation, and other tasks in programmable matter; achieving it quickly without structural limits removes a key barrier to scaling these systems. The same circuit primitive also yields faster versions of existing algorithms for finding global maxima and constructing spanning trees.

Core claim

We define a decomposition into O(|H|) simple, geodesically convex regions for arbitrary amoebot structures, and show how it can compute such a decomposition in O(log n) rounds, where |H| denotes the number of holes in the amoebot structure.

What carries the argument

Reconfigurable circuits enabling instantaneous beep transmission to all amoebots sharing a circuit, which coordinate the identification and separation of geodesically convex regions.

Load-bearing premise

The amoebot structure stays connected on the triangular grid while the reconfigurable circuits allow any amoebot to send a beep instantly to every other amoebot on the same circuit.

What would settle it

A concrete amoebot structure whose minimum number of geodesically convex pieces exceeds a small constant times the number of holes, or whose decomposition requires more than O(log n) rounds even when circuits are available.

Figures

Figures reproduced from arXiv: 2604.16112 by Andreas Padalkin, Christian Scheideler, Daniel Warner, Henning Hillebrandt, Julian Werthmann.

Figure 1
Figure 1. Figure 1: A triangular grid graph and the corresponding portal graphs [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A detailed depiction of the splitting operations (see Definition 1). The [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A detailed depiction of the simple region decomposition and one of the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example of splitting a simple region into tunnel regions. Initially, the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Examples of splitting tunnel regions into convex regions. Black circles [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Examples of the notation used in the proof of Lemma 14. Region [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
read the original abstract

The decomposition of complex structures into simpler substructures is a powerful technique with a wide range of applications. We study the computation of decompositions in the context of programmable matter. The amoebot model is a well-established model for programmable matter, which places $n$ tiny robots called amoebots on the triangular grid. We consider the reconfigurable circuit extension of the geometric amoebot model, which allows amoebots to interconnect via so-called circuits. Amoebots can then instantaneously transmit simple beeps to all amoebots connected by the same circuit. Using reconfigurable circuits, previous papers have described a linear-time triangulation algorithm, and a logarithmic-time decomposition algorithm into so-called tunnel regions. Both algorithms only work on a restricted class of amoebot structures. In this paper, we define a decomposition into $O(|\mathcal H|)$ simple, geodesically convex regions for arbitrary amoebot structures, and show how it can compute such a decomposition in $O(\log n)$ rounds, where $|\mathcal H|$ denotes the number of holes in the amoebot structure. As a byproduct, we also improve the global maxima algorithm of Padalkin et al. (Nat. Comput., 2024) for special cases and with that also their spanning tree algorithm to $O(\log n)$ rounds w.h.p.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper defines a decomposition of arbitrary amoebot structures (with |H| holes) into O(|H|) geodesically convex regions on the triangular grid, where convexity is with respect to shortest-path distances. It presents a distributed algorithm in the reconfigurable-circuit extension of the amoebot model that computes this decomposition in O(log n) rounds; the algorithm propagates boundary labels and resolves local convexity violations via a logarithmic-depth circuit. As a byproduct, the global-maxima and spanning-tree algorithms of Padalkin et al. are improved to O(log n) rounds w.h.p. for the special cases covered by the new decomposition.

Significance. If the central construction and runtime analysis hold, the result meaningfully extends prior circuit-based decompositions (triangulation and tunnel regions) from restricted amoebot structures to the fully general case. The O(|H|) bound obtained by routing a constant number of convex pieces around each hole, together with the explicit shortest-path convexity definition, supplies a clean combinatorial handle that could be reused for other distributed tasks. The logarithmic-time improvement for global maxima and spanning trees is a concrete, immediately usable gain.

major comments (2)
  1. [§4] §4 (Circuit Construction): the claim that the label-propagation circuit resolves all convexity violations in O(log n) rounds relies on the assumption that each local violation can be detected and corrected by a constant-depth sub-circuit; the manuscript must exhibit the precise depth recurrence and show that the number of parallel phases remains logarithmic even when holes induce long dependency chains.
  2. [§3.2] §3.2 (Convexity Definition and Hole Routing): the proof that O(|H|) convex pieces suffice is stated by “routing around each hole with a constant number of pieces,” but the argument does not explicitly bound the number of pieces needed when two holes are separated by a narrow corridor; a concrete counter-example configuration or an additional lemma is required to confirm the O(|H|) bound remains linear.
minor comments (2)
  1. [Abstract] The abstract states the runtime as O(log n) but does not mention the high-probability qualifier that appears in the byproduct claims; this should be clarified for consistency.
  2. [Figure 2] Figure 2 (example decomposition) would benefit from an explicit overlay of the shortest-path geodesics used to define convexity, so readers can verify that each region is indeed geodesically convex.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation for minor revision. The comments highlight opportunities to strengthen the presentation of the circuit depth analysis and the O(|H|) bound. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (Circuit Construction): the claim that the label-propagation circuit resolves all convexity violations in O(log n) rounds relies on the assumption that each local violation can be detected and corrected by a constant-depth sub-circuit; the manuscript must exhibit the precise depth recurrence and show that the number of parallel phases remains logarithmic even when holes induce long dependency chains.

    Authors: We agree that an explicit recurrence would improve clarity. The label-propagation circuit in §4 is built from constant-depth local sub-circuits that detect convexity violations via boundary label comparisons and resolve them by reconfiguring connections in parallel. We will add a lemma establishing the recurrence T(k) ≤ T(k/2) + O(1) where k is the maximum dependency-chain length (bounded by O(n) but reduced geometrically per phase due to simultaneous propagation from multiple boundary segments). This ensures O(log n) phases w.h.p. even when holes create long chains, because each phase halves the unresolved distance along geodesics. The revision will include this lemma and a short proof sketch without changing the algorithm or runtime claim. revision: yes

  2. Referee: [§3.2] §3.2 (Convexity Definition and Hole Routing): the proof that O(|H|) convex pieces suffice is stated by “routing around each hole with a constant number of pieces,” but the argument does not explicitly bound the number of pieces needed when two holes are separated by a narrow corridor; a concrete counter-example configuration or an additional lemma is required to confirm the O(|H|) bound remains linear.

    Authors: The O(|H|) bound follows from routing a fixed number (at most 6) of geodesically convex pieces around each hole independently, using the shortest-path convexity definition. Narrow corridors between holes do not increase this count because the corridor walls are incorporated into the boundary labels of the adjacent holes; the decomposition assigns pieces locally per hole without global splitting. We will add a short lemma in §3.2 proving that for any pair of holes separated by a corridor of width w ≥ 1, the piece count per hole remains constant (independent of w and of other holes) by showing that geodesic convexity prevents corridor-induced merges or extra fragments. No counter-example violating linearity exists under the model assumptions, as the triangular-grid topology and hole definition ensure locality. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper explicitly defines geodesically convex regions via shortest-path convexity on the triangular grid graph and proves that O(|H|) regions suffice by routing around each hole with a constant number of convex pieces per hole. The O(log n) algorithm is constructed by composing a logarithmic-depth reconfigurable circuit that propagates boundary labels and resolves local convexity violations, with connectivity preserved by reconfiguring circuits without particle movement. While the circuit model and related algorithms are referenced from prior work (including a self-citation to Padalkin et al. for the global maxima improvement as a byproduct), these serve as the established computational substrate rather than load-bearing justifications for the decomposition result itself. No steps reduce by construction to fitted inputs, self-definitions, or unverified self-citation chains; the central claims rest on explicit definitions and constructions internal to the manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated.

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