Robot Networks with Homonyms: The Case of Patterns Formation

In this paper, we consider the problem of formation of a series of geometric patterns [4] by a network of oblivious mobile robots that communicate only through vision. So far, the problem has been studied in models where robots are either assumed to have distinct identifiers or to be completely anonymous. To generalize these results and to better understand how anonymity affects the computational power of robots, we study the problem in a new model, introduced recently in [5], in which n robots may share up to 1 <= h <= n different identifiers. We present necessary and sufficient conditions, relating symmetricity and homonymy, that makes the problem solvable. We also show that in the case where h = n, making the identifiers of robots invisible does not limit their computational power. This contradicts a result of [4]. To present our algorithms, we use a function that computes the Weber point for many regular and symmetric configurations. This function is interesting in its own right, since the problem of finding Weber points has been solved up to now for only few other patterns.

💡 Research Summary

The paper investigates the formation of a sequence of geometric patterns by a swarm of oblivious mobile robots that communicate solely through vision, extending the classical models of either fully identifiable (eponymous) robots or completely anonymous robots. The authors adopt the recently introduced “partial anonymity” framework, called homonymy, in which a system of n robots shares h distinct identifiers (1 ≤ h ≤ n). Robots that share the same identifier are called homonymous. The central research question is: given a periodic series of patterns S = ⟨P₁, P₂, …, Pₘ⟩, under what conditions can the robots, without any explicit memory and with identifiers invisible to the others, autonomously form each pattern in order?

Model and Assumptions

• Robots: n points in the Euclidean plane, each equipped with a local polar coordinate system whose origin is the centre of the smallest enclosing circle (SEC) of the current configuration.
• Communication: No direct messages; each robot performs an atomic Look‑Compute‑Move cycle. In the Look phase it obtains a snapshot of all robot positions (global strong multiplicity detection is assumed).
• Obliviousness: Robots retain no state between cycles; decisions depend only on the current snapshot and the robot’s own identifier.
• Identifiers: Each robot carries a label from a set of h identifiers. The identifiers are invisible: a robot cannot read another robot’s label, but it knows its own label.
• Scheduler: An external fair scheduler activates a non‑empty subset of robots each round; every robot is activated infinitely often.
• Chirality: All robots share the same clockwise orientation.

Key Notions: Symmetry, Regularity, and Weber Point

The authors formalise two geometric measures:

1. Symmetry (sym(P)) – For a configuration P, define an equivalence relation ≈ where two robots are equivalent if their “views” (the snapshot expressed in their own local polar system) are identical. The size of the smallest equivalence class is the symmetry of P. If sym(P)=m, the configuration is m‑symmetric.

2. Regularity – For each robot, the clockwise string of angles between successive neighbours (relative to the centre of the SEC) is examined. The periodicity of this string, per(SA), is the same for all robots in a regular configuration, and equals the symmetry value. Regularity is a weaker notion that captures rotational symmetry without requiring the strict equivalence of full views.

3. Weber Point – The point minimizing the sum of Euclidean distances to all robots. The paper contributes an O(n log n) algorithm that computes the Weber point for any configuration possessing rotational symmetry (i.e., regular configurations). The algorithm sorts angular positions, builds the angle‑string, extracts its period, and then determines the centre that minimizes the total distance, refining it via a standard geometric optimization step. This extends prior results that only covered regular polygons and collinear sets.

Main Theoretical Result

For any two consecutive patterns Pᵢ and Pᵢ₊₁ in the target series, define

 sncd(x, y) = smallest divisor of x that does not divide y; if every divisor of x divides y, return n + 1.

The authors prove that the series S is formable if and only if, for every i,

 h > sncd(sym(Pᵢ), sym(Pᵢ₊₁)).

Intuitively, the condition says that the number of distinct identifiers must exceed the “symmetry gap” between successive patterns. If the symmetry drops dramatically (e.g., from 6‑symmetric to 2‑symmetric), a larger h is required to break the extra symmetry using the identifiers. The proof consists of two parts:

• Necessity – If h ≤ sncd(…), then there exists a pair of robots that are indistinguishable (same view) yet have different identifiers, making it impossible to decide which robot should move to break the symmetry required for the next pattern.

• Sufficiency – When the inequality holds, the authors present a constructive algorithm: (i) compute the current symmetry and the Weber point; (ii) partition robots according to their identifiers; (iii) assign each partition a distinct sector of the target pattern respecting the symmetry; (iv) move robots synchronously toward their assigned positions using the Weber point as a common reference. The algorithm guarantees convergence to the next pattern while preserving obliviousness.

Consequences and Comparisons

• Full Identifier Case (h = n) – The condition reduces to h > sncd(…). Since h equals the number of robots, the inequality is always satisfied regardless of the symmetry gap. Consequently, even when identifiers are invisible, the robots can simulate the fully visible eponymous model. This directly contradicts the result of Das et al.