Memory Lower Bounds for Deterministic Self-Stabilization L elia - PowerPoint PPT Presentation

Memory Lower Bounds for Deterministic Self-Stabilization L´ elia Blin, Laurent Feuillolet et Gabriel Le Bouder Sorbonne Universit´ e, LIP6. L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 1 / 27

Mod` ele Problems Memory bounds Conclusion Mod` ele L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 2 / 27

Mod` ele Problems Memory bounds Conclusion Syst` eme r´ eparti Syst` eme r´ eparti Mod` ele R´ eseaux asynchrones G = ( V , E ) avec identifiants. Fautes transitoires (corruptions de variables). L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 3 / 27

Mod` ele Problems Memory bounds Conclusion Mod` ele ` a Etat Mod` ele ` a Etat En une ´ etape atomique un noeud v peut • Lire ses variables et les variables de ses voisins. • Calculer. • Mettre ` a jour ses variables. L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 4 / 27

Mod` ele Problems Memory bounds Conclusion Mod` ele ` a Etat R´ eseaux non-anonyme 48 33 18 71 17 12 10 13 94 04 22 19 Identifiants Identifiants deux ` a deux distincts. ∃ c > 1 : ∀ v ∈ V , id v ∈ [1 , n c ] Les identifiants ne sont pas stock´ e dans les variables, ils ne sont pas accessibles aux voisins. L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 5 / 27

Mod` ele Problems Memory bounds Conclusion Mod` ele ` a Etat Noeud activable Un noeud est activable si au moins une des r` egles de son algorithme est ex´ ecutable. L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 6 / 27

Mod` ele Problems Memory bounds Conclusion Scheduler Ordonnanceur Synchrone ´ equitable Faiblement ´ equitable In´ equitable Definition Ordonnanceur choisit parmi les noeuds activables les noeuds qui s’activent. L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 7 / 27

Mod` ele Problems Memory bounds Conclusion Configurations Configurations Etat L’´ etat d’un noeud est l’ensemble de ses va- riables Configuration Pour un graphe G , une configuration Γ est l’ensemble des ´ etats de ses noeuds ` a un instant donn´ e. 8 8 7 4 2 4 2 2 3 3 (a) (b) L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 8 / 27

Mod` ele Problems Memory bounds Conclusion Configurations Configurations l´ egitimes D´ epend du pr´ edicat correspondant ` a la tˆ ache ` a r´ esoudre. Exemple : pr´ edicat arbre couvrant. L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 9 / 27

Mod` ele Problems Memory bounds Conclusion Configurations Ensemble de toutes les configurations L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 10 / 27

Mod` ele Problems Memory bounds Conclusion Configurations Algorithmes distribu´ ees ”classique” Conf. Conf. l´ egitime l´ egitime Initial L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 11 / 27

Mod` ele Problems Memory bounds Conclusion Configurations Auto-stabilisation propri´ et´ e de Silence L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 12 / 27

Mod` ele Problems Memory bounds Conclusion Configurations Auto-stabilisation L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 13 / 27

Mod` ele Problems Memory bounds Conclusion D´ efinition Algorithme auto-stabilisant Dijkstra, 1974 Un algorithme auto-stabilisant r´ esolvant une tˆ ache T est un algorithme distribu´ e A satisfaisant : Convergence : D´ emarrant d’une configuration arbitraire, 1 A finit par rejoindre une configuration l´ egale. Cloture : D´ emarrant d’une configuration l´ egale, le 2 syst` eme reste dans une configuration l´ egale. L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 14 / 27

Mod` ele Problems Memory bounds Conclusion Performances Complexit´ es Espace m´ emoire Espace m´ emoire maximum utilis´ e par l’en- semble des variables (en binaire). Temps : nombre d’´ etapes Une ´ etape est une transition d’une configuration vers une autre. Temps : Le nombre de rondes Une ronde est la plus petite portion d’ex´ ecution o` u tout noeud activable est activ´ e ou devient non activable. L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 15 / 27

Mod` ele Problems Memory bounds Conclusion Fondamental Problems L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 16 / 27

Mod` ele Problems Memory bounds Conclusion (Delta+1)-coloration (∆ + 1)-coloration 48 33 18 71 17 12 10 13 94 04 22 19 L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 17 / 27

Mod` ele Problems Memory bounds Conclusion Spanning-tree construction Spanning-tree construction 48 33 18 71 17 12 10 13 94 04 22 19 L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 18 / 27

Mod` ele Problems Memory bounds Conclusion Leader Election Leader Election 48 33 18 71 17 12 10 13 94 04 22 19 L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 19 / 27

Mod` ele Problems Memory bounds Conclusion Complexity Performances Definition : Space complexity Number of bits per node Parameters n number of nodes ∆ degree of the graph L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 20 / 27

Mod` ele Problems Memory bounds Conclusion State of the art State of the art, Silent algorithms (1/2) Memory requirements for silent stabilization : lower bound [DolevGS99] prove that the leader election, the spanning tree construction, and the identification of the centers of a graph, require Ω(log n ) bits per edge. Memory requirements for silent stabilization : upper bound There exist algorithms for the leader election, the spanning tree construction, the identification of the centers of a graph, and the coloration, that use only O (log n ) bits per node. L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 21 / 27

Mod` ele Problems Memory bounds Conclusion State of the art State of the art : Silent algorithms (2/2) Definition : Proof Labelling Scheme (PLS) : [KormanKP07] An oracle , assigning to every node v a label l ( v ) based on its local state, and a verifier , a distributed predicate that, on node v , reads both the states and the labels of v and the label of its neighboors, such that for every legal state, the verifier returns true at each node for every illegal state, the verifier returns false at at least one node Lower Bound : [BlinFP14] If there exist a silent self-stabilizing algorithm using k bits, then there exist a PLS using at most k bits. L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 22 / 27

Mod` ele Problems Memory bounds Conclusion State of the art State of the art : General case Best complexity achieved [BlinT18] provide algorithm that require O (log log n + log ∆) bits per node for the problems of (∆ + 1)-coloration, spanning tree, and leader election. Lower Bound [BeauquierGJ99] prove that the leader election cannot be solved with a constant number of bit per node. L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 23 / 27

Mod` ele Problems Memory bounds Conclusion Our contribution Our result Space complexity of the (∆ + 1)-coloration The space complexity of the (∆ + 1)-coloring problem is Θ(log log n + log ∆) bits per node. Space complexity of the spanning tree construction The space complexity of the spanning tree construction problem is Θ(log log n + log ∆) bits per node. Space complexity of the leader election The leader election problem requires Ω(log log n ) bits per node. L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 24 / 27

Mod` ele Problems Memory bounds Conclusion Our contribution Sketch of proof 1/2 We consider the n -nodes ring. 4 15 7 Idea Main algorithm : A : [ n c ] × { 0 , 1 } 3 f ( n ) → { 0 , 1 } f ( n ) ∀ v ∈ V , with ID id v , ∃ δ id v : { 0 , 1 } 3 f ( n ) → { 0 , 1 } f ( n ) S denotes the set of all functions δ i If f ( n ) = o (log log n ), then we can find n different IDs that match the same function L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 25 / 27

Mod` ele Problems Memory bounds Conclusion Our contribution Sketch of proof 2/2 ID ∈ [ n c ] 01 02 03 04 05 δ ∈ S |S| = 2 f ( n ) × 2 3 f ( n ) · · · 21 δ 1 22 δ 2 23 f ( n ) = o (log log n ) ⇒ |S| = o ( n c − 1 ) 24 δ 3 25 δ 4 26 · · · 27 · · · L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 26 / 27

Mod` ele Problems Memory bounds Conclusion Our contribution Sketch of proof 2/2 ID ∈ [ n c ] 01 02 03 04 05 δ ∈ S |S| = 2 f ( n ) × 2 3 f ( n ) · · · 21 δ 1 22 δ 2 23 f ( n ) = o (log log n ) ⇒ |S| = o ( n c − 1 ) 24 δ 3 25 δ 4 26 · · · 27 · · · 22 21 22 21 24 02 24 02 26 01 26 01 27 67 27 67 35 42 35 42 L´ elia Blin Memory Lower Bounds for Deterministic Self-Stabilization 26 / 27