Yvan Rivierre

Self-Stabilizing Small k-Dominating Sets

A self-stabilizing algorithm, after transient faults hit the system and place it in some arbitrary global state, recovers in finite time without external (e.g., human) intervention. In this paper, we propose a distributed asynchronous silent self-stabilizing algorithm for finding a minimal k-dominating set of at most n/(k+1) processes in an arbitrary identified network of size n. We propose a transformer that allows our algorithm work under an unfair daemon (the weakest scheduling assumption). The complexity of our solution is in O(n) rounds and O(D n²) steps using O(log n + k log n) bits per process where D is the diameter of the network.

Competitive Self-Stabilizing k-Clustering

In this paper, we propose a silent self-stabilizing asynchronous distributed algorithm for constructing a k-clustering of any connected network with unique IDs. Our algorithm stabilizes in O (n) rounds, using O (log n) space per process, where n is the number of processes. In the general case, our algorithm constructs O (n/k) k-clusters. If the network is a Unit Disk Graph (UDG), then our algorithm is 7.2552k+O (1) -competitive, that is, the number of k-clusters constructed by the algorithm is at most 7.2552*k + O (1) times the minimum possible number of k-clusters in any k-clustering of the same network. More generally, if the network is an Approximate Disk Graph (ADG) with approximation ratio ?, then our algorithm is 7.2552* (?^2k) +O (*?) -competitive. Our solution is based on the self-stabilizing construction of a data structure called the MIS Tree, a spanning tree of the network whose processes at even levels form a maximal independent set of the network. The MIS tree construction is the time bottleneck of our k-clustering algorithm, as it takes T (n) rounds in the worst case, while the rest of the algorithm takes O (D) rounds, where D is the diameter of the network. We would like to improve that time to be O (D), but we show that our distributed MIS tree construction is a P-complete problem.

Self-stabilizing (f,g)-Alliances with Safe Convergence

Given two functions f and g mapping nodes to non-negative integers, we give a silent self-stabilizing algorithm that computes a minimal (f, g)-alliance in an asynchronous network with unique node IDs, assuming that every node p has a degree at least g(p) and satisfies f(p) ≤ g(p). Our algorithm is safely converging in the sense that starting from any configuration, it first converges to a (not necessarily minimal) (f, g)-alliance in at most four rounds, and then continues to converge to a minimal one in at most 5n + 4 additional rounds, where n is the size of the network. Our algorithm is written in the shared memory model. It is proven assuming an unfair (distributed) daemon. Its memory requirement is O(log n) bits per process, and it takes

Competitive self-stabilizing k-clustering☆

O(\varDelta^3n)

Self-stabilizing ( f , g ) -alliances with safe convergence

steps to stabilize, where

Space-Optimal Deterministic Rendezvous

\varDelta

Algorithme autostabilisant construisant un petit ensemble k-dominant

is the degree of the network.

Self-stabilizing labeling and ranking in ordered trees

In this paper, we propose a silent self-stabilizing asynchronous distributed algorithm for constructing a kclustering of any connected network with unique IDs. Our algorithm stabilizes in O(n) rounds, using O(log n) space per process, where n is the number of processes. In the general case, our algorithm constructs O(n/k) k-clusters. If the network is a Unit Disk Graph (UDG), then our algorithm is 7.2552k+O(1)competitive, that is, the number of k-clusters constructed by the algorithm is at most 7.2552k + O(1) times the minimum possible number of k-clusters in any k-clustering of the same network. More generally, if the network is an Approximate Disk Graph (ADG) with approximation ratio λ, then our algorithm is 7.2552λ2k + O(λ)-competitive. Our solution is based on the self-stabilizing construction of a data structure called the MIS Tree, a spanning tree of the network whose processes at even levels form a maximal independent set of the network. The MIS tree construction is the time bottleneck of our k-clustering algorithm, as it takes Θ(n) rounds in the worst case, while the rest of the algorithm takes O(D) rounds, where V is the diameter of the network. We would like to improve that time to be O(D), but we show that our distributed MIS tree construction is a P-complete problem.

Algorithme de k-partitionnement auto-stabilisant et compétitif

Given two functions f and g mapping nodes to non-negative integers, we give a silent self-stabilizing algorithm that computes a minimal ( f , g ) -alliance in an asynchronous network with unique node IDs, assuming that every node p has a degree at least g ( p ) and satisfies f ( p ) ? g ( p ) . Our algorithm is safely converging in the sense that starting from any configuration, it first converges to a (not necessarily minimal) ( f , g ) -alliance in at most four rounds, and then continues to converge to a minimal one in at most 5 n + 4 additional rounds, where n is the size of the network. Our algorithm is written in the shared memory model. It is proven assuming an unfair (distributed) daemon. Its memory requirement is ? ( log n ) bits per process, and it takes O ( n ? Δ 3 ) steps to stabilize, where Δ is the degree of the network. We give an algorithm for the ( f , g ) -alliance, a generalization of several problems.Our algorithm is distributed, self-stabilizing, silent, and safe converging.The complexity analysis shows that it is better than many other related solutions.

Asymptotically Optimal Deterministic Rendezvous

In this paper, we address the deterministic rendezvous of mobile agents into any unoriented connected graph. The agents are autonomous, oblivious, move asynchronously. For this problem, we exhibit some time and space lower bounds as well as some necessary conditions. We also propose an algorithm that is space-optimal and asymptotically optimal in rounds.