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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
L´ elia Blin, Laurent Feuillolet et Gabriel Le Bouder
Sorbonne Universit´ e, LIP6.
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Mod` ele Problems Memory bounds Conclusion
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Mod` ele Problems Memory bounds Conclusion Syst` eme r´ eparti
Mod` ele R´ eseaux asynchrones G = (V , E) avec identifiants. Fautes transitoires (corruptions de variables).
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Mod` ele Problems Memory bounds Conclusion Mod` ele ` a Etat
En une ´ etape atomique un noeud v peut
a jour ses variables.
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Mod` ele Problems Memory bounds Conclusion Mod` ele ` a Etat
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Identifiants Identifiants deux ` a deux distincts. ∃c > 1 : ∀v ∈ V , idv ∈ [1, nc] Les identifiants ne sont pas stock´ e dans les variables, ils ne sont pas accessibles aux voisins.
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Mod` ele Problems Memory bounds Conclusion Mod` ele ` a Etat
Un noeud est activable si au moins une des r` egles de son algorithme est ex´ ecutable.
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Mod` ele Problems Memory bounds Conclusion Scheduler
In´ equitable Faiblement ´ equitable ´ equitable
Synchrone
Definition Ordonnanceur choisit parmi les noeuds activables les noeuds qui s’activent.
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Mod` ele Problems Memory bounds Conclusion 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 2 3 4 7 (a) 8 2 3 2 4 (b)
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Mod` ele Problems Memory bounds Conclusion Configurations
D´ epend du pr´ edicat correspondant ` a la tˆ ache ` a r´ esoudre. Exemple : pr´ edicat arbre couvrant.
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Mod` ele Problems Memory bounds Conclusion Configurations
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Mod` ele Problems Memory bounds Conclusion Configurations
Initial
Conf. l´ egitime Conf. l´ egitime
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Mod` ele Problems Memory bounds Conclusion Configurations
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Mod` ele Problems Memory bounds Conclusion Configurations
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Mod` ele Problems Memory bounds Conclusion D´ efinition
Dijkstra, 1974
Un algorithme auto-stabilisant r´ esolvant une tˆ ache T est un algorithme distribu´ e A satisfaisant :
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Convergence : D´ emarrant d’une configuration arbitraire, A finit par rejoindre une configuration l´ egale.
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Cloture : D´ emarrant d’une configuration l´ egale, le syst` eme reste dans une configuration l´ egale.
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Mod` ele Problems Memory bounds Conclusion Performances
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
u tout noeud activable est activ´ e ou devient non activable.
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Mod` ele Problems Memory bounds Conclusion
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Mod` ele Problems Memory bounds Conclusion (Delta+1)-coloration
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Mod` ele Problems Memory bounds Conclusion Spanning-tree construction
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Mod` ele Problems Memory bounds Conclusion Leader Election
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Mod` ele Problems Memory bounds Conclusion Complexity
Definition : Space complexity Number of bits per node Parameters n number of nodes ∆ degree of the graph
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Mod` ele Problems Memory bounds Conclusion State of the art
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.
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Mod` ele Problems Memory bounds Conclusion State of the art
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
for every legal state, the verifier returns true at each node for every illegal state, the verifier returns false at at least
Lower Bound : [BlinFP14] If there exist a silent self-stabilizing algorithm using k bits, then there exist a PLS using at most k bits.
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Mod` ele Problems Memory bounds Conclusion State of the art
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.
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Mod` ele Problems Memory bounds Conclusion Our contribution
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.
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Mod` ele Problems Memory bounds Conclusion Our contribution
We consider the n-nodes ring.
15 4 7
Idea Main algorithm : A : [nc] × {0, 1}3f (n) → {0, 1}f (n) ∀v ∈ V , with ID idv, ∃δidv : {0, 1}3f (n) → {0, 1}f (n) S denotes the set of all functions δi If f (n) = o(log log n), then we can find n different IDs that match the same function
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Mod` ele Problems Memory bounds Conclusion Our contribution
|S| = 2f (n)×23f (n) f (n) = o(log log n) ⇒ |S| = o(nc−1)
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· · ·
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· · ·
ID ∈ [nc ]
δ1 δ2 δ3 δ4
· · ·
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Mod` ele Problems Memory bounds Conclusion Our contribution
|S| = 2f (n)×23f (n) f (n) = o(log log n) ⇒ |S| = o(nc−1)
01 02 03 04 05
· · ·
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· · ·
ID ∈ [nc ]
δ1 δ2 δ3 δ4
· · ·
δ ∈ S
01 02 21 22 24 26 27 35 42 67 01 02 21 22 24 26 27 35 42 67
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Mod` ele Problems Memory bounds Conclusion Open Problem
Message passing Does the generic bound Ω(log log n) still holds in the message passing model ? Thight bound for leader election ? Does there exist an algorithm for the leader election that does not require the extra O(log ∆) bits per node quantity ?
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