Algorithmes auto-stabilisants pour la construction darbres couvrants et la gestion dentites autonomes

In the context of large-scale networks, the consideration of faults is an evident necessity. This document is focussing on the self-stabilizing approach which aims at conceiving algorithms “repairing themselves” in case of transient faults, that is of faults implying an arbitrary modification of the states of the processes. The document focuses on two different contexts, covering the major part of my research work these last years. The first part of the document is dedicated to the design and analysis of self-stabilizing algorithms for networks of processes. The second part of the document is dedicated to the design and analysis of self-stabilizing algorithms for autonomous entities (i.e., software agents, robots, etc.) moving in a network.

💡 Research Summary

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This habilitation thesis investigates self‑stabilizing algorithms for two broad contexts: (i) large‑scale distributed networks of processes, and (ii) autonomous entities such as mobile robots or software agents moving in a network. The central theme is the design of algorithms that can recover from arbitrary transient faults—situations where the state of any subset of nodes may be corrupted—without external intervention.

Part I – Self‑stabilizing spanning‑tree constructions
The first part extends classic self‑stabilizing tree construction (BFS, DFS, shortest‑path trees) to more demanding optimization problems. The author presents memory‑optimal algorithms for (a) Minimum‑Weight Spanning Tree (MST), (b) Minimum‑Degree Spanning Tree, and (c) Steiner Tree. A unifying framework is introduced that encodes each node’s local state in O(log n) bits, which is provably sufficient to store parent identifiers, edge weights, degree counters, or terminal‑set membership. By carefully designing local label‑update rules, the algorithms guarantee convergence to a globally optimal structure while keeping per‑node memory at the information‑theoretic lower bound.

The thesis surveys prior work (Gupta & Srimani, Higham & Lyan, Korman‑Kutten‑Masuza) and shows how their approaches can be combined or refined to improve convergence time and reduce message complexity. Trade‑offs among three key metrics—memory size, convergence time, and solution quality—are analyzed in depth. For MST, the proposed algorithm achieves O(log n) bits per node and polynomial‑time convergence, matching the best known distributed MST algorithms while being self‑stabilizing. Similar results are obtained for the minimum‑degree and Steiner variants, with explicit bounds on the approximation ratios when exact optimality is computationally infeasible.

Part II – Self‑stabilizing algorithms for autonomous entities
The second part shifts focus to mobile robots (or agents) that operate under harsh conditions: faults may affect both the robots’ internal state and the communication network. The author defines a fault model where robots can experience arbitrary state corruption and where messages may be lost or delayed. Within this model, fundamental distributed tasks are examined: naming (assigning unique identifiers), leader election, and perpetual exploration (continuous coverage of the environment).

Key contributions include:

• Impossibility proofs showing that deterministic self‑stabilizing naming and election are impossible in a fully asynchronous setting without additional assumptions.
• Deterministic algorithms that succeed under modest synchrony assumptions (e.g., round‑based execution) or with minimal extra information such as an initial color or a bounded number of faulty robots. These algorithms use only O(log n) bits of memory and converge in O(n) rounds.
• Probabilistic algorithms that achieve naming and election with high probability even in the fully asynchronous model, using random back‑off and collision detection; expected convergence time is O(n log n).
• A study of the CORD‑A (discrete global vision) model, where robots have limited sensing range but share a common coordinate system. The thesis presents both lower bounds (minimum number of robots required for perpetual exploration) and constructive algorithms that achieve exploration with the minimal number of robots, using constant‑size memory per robot.

The work emphasizes the interplay between the number of robots, the strength of the sensing model, and the fault tolerance level. It demonstrates that even with very weak assumptions—minimal memory, limited communication, and high fault rates—robots can still self‑stabilize to perform non‑trivial tasks.

Conclusions and research outlook
Overall, the thesis establishes that self‑stabilization does not inherently impose prohibitive memory overhead; optimal or near‑optimal solutions for classic network problems can be achieved with O(log n) bits per node. Moreover, the same principles can be transferred to mobile autonomous systems, where self‑stabilizing protocols enable robust naming, election, and exploration despite severe faults. Future research directions identified include multi‑objective optimization of memory, convergence speed, and solution quality; extensions to dynamic networks with continuous topology changes; and experimental validation on real robot platforms to bridge theory and practice.