Distributed Freeze Tag: a Sustainable Solution to Discover and Wake-up a Robot Swarm

TL;DR

This work addresses the distributed Freeze Tag Problem (dFTP) in the plane when initial sleeping robots are unknown and visibility is limited to distance 1. It introduces three algorithms, Separator, Grid, and Wave, that achieve near-optimal makespans under various energy budgets by combining exploration, geometric separators, and distributed $\ell$-sampling to efficiently discover and wake all robots. The main results establish tight upper and matching lower bounds: an unconstrained-energy makespan of $O(\rho + \ell^2 \log(\rho/\ell))$; a Grid-based method with energy $\Theta(\ell^2)$ yielding $O(\xi_\ell \cdot \ell)$; and a Wave method with energy $\Theta(\ell^2 \log \ell)$ achieving $O(\xi_\ell + \ell^2 \log(\xi_\ell/\ell))$. Lower bounds are provided to demonstrate optimality across regimes, and the work discusses practical aspects and open questions, such as approaching optimal energy in all cases and removing rendezvous dependencies. Overall, the paper advances sustainable, distributed wake-up strategies for robot swarms under partial information and energy constraints, with implications for scalable multi-robot exploration and coordination.

Abstract

The Freeze Tag Problem consists in waking up a swarm of robots starting with one initially awake robot. Whereas there is a wide literature of the centralized setting, where the location of the robots is known in advance, we focus in the distributed version where the location of the robots $¶$ are unknown, and where awake robots only detect other robots up to distance~$1$. Assuming that moving at distance $δ$ takes a time $δ$, we show that waking up of the whole swarm takes $O(ρ+\ell^2\log( ρ/\ell))$, where $ρ$ stands for the largest distance from the initial robot to any point of $¶$, and the $\ell$ is the connectivity threshold of $¶$. Moreover, the result is complemented by a matching lower bound in both parameters $ρ$ and $\ell$. We also provide other distributed algorithms, complemented with lower bounds, whenever each robot has a bounded amount of energy.

Figures (5)

• Figure 1: First phases of $\mathcal{A}\xspace_{\mathit{Separator}}$; $\bullet$ sleeping robots; $\circ$ awake robots; sleeping seeds; awake seeds.
• Figure 2: Recruitment and Reorganization phases of $\mathcal{A}\xspace_{\mathit{Separator}}$; recruited robots; explored area is grayed.
• Figure 3: Description of $\mathcal{A}\xspace_{\mathit{Separator}}$
• Figure 4: Depiction of the exploration process.
• Figure 5: Proof of the Lower Bound

Theorems & Definitions (53)

• Definition 1: dFTP
• Lemma 1
• Lemma 2: Distributed Makespan
• Corollary 1: Explore and Wake up
• Lemma 3
• Corollary 2
• Lemma 4: $\ell$-sampling cardinality
• Lemma 5
• Theorem 1
• Theorem 2: Lower Bound without energy constraint