arXiv subjects

Gabriel Le Bouder

Publications and source records attributed to Gabriel Le Bouder.

3 recordsLinked to original sources

arXiv2026

Freeze-Tag with Return

In the standard Freeze-Tag Problem (FTP), an initially awake robot (the source) is in charge of waking up a swarm of sleeping robots by moving towards them, given that all the awake robots can participate in the awakening process. The goal is to minimize the makespan to wake up all robots assuming they move at unit speed. In this paper we introduce the Freeze-Tag-with-Return Problem (FTRP) variant, where the robots must eventually return to their initial positions. In the Euclidean plane with $n$ sleeping robots lying on the unit disk centered at the initial position of the source, we show a non-trivial relationship between FTP and FTRP by proving that the difference between the optimal makespan of both problems never exceeds $1.959$, and is at least $1.732$ in the worst-case. We also present several upper and lower bounds on the optimal makespan. In particular, we show that if the sleeping robots are in convex positions, then the optimal makespan is at most $2 + 2\sqrt{2}$, which is achieved by some instances. From an algorithmic point-of-view, we present single-exponential algorithms for general distance functions. In metric spaces, these algorithms are asymptotically optimal under the ETH, which we show via an NP-hardness reduction on unweighted graphs.

cs.DS↗

arXiv2025

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

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.

cs.DS↗

arXiv2023

Optimal Space Lower Bound for Deterministic Self-Stabilizing Leader Election Algorithms

Given a boolean predicate $Π$ on labeled networks (e.g., proper coloring, leader election, etc.), a self-stabilizing algorithm for $Π$ is a distributed algorithm that can start from any initial configuration of the network (i.e., every node has an arbitrary value assigned to each of its variables), and eventually converge to a configuration satisfying $Π$. It is known that leader election does not have a deterministic self-stabilizing algorithm using a constant-size register at each node, i.e., for some networks, some of their nodes must have registers whose sizes grow with the size $n$ of the networks. On the other hand, it is also known that leader election can be solved by a deterministic self-stabilizing algorithm using registers of $O(\log \log n)$ bits per node in any $n$-node bounded-degree network. We show that this latter space complexity is optimal. Specifically, we prove that every deterministic self-stabilizing algorithm solving leader election must use $Ω(\log \log n)$-bit per node registers in some $n$-node networks. In addition, we show that our lower bounds go beyond leader election, and apply to all problems that cannot be solved by anonymous algorithms.

cs.DC↗