Leader Election in Asymmetric Labeled Unidirectional Rings Karine - PowerPoint PPT Presentation

Leader Election in Asymmetric Labeled Unidirectional Rings Karine Altisen 1 Ajoy K. Datta 2 Stéphane Devismes 1 Anaïs Durand 1 Lawrence L. Larmore 2 1 Univ. Grenoble Alpes, CNRS, Grenoble INP, VERIMAG, 38000 Grenoble, France 2 University of Nevada Las Vegas, USA Meeting DESCARTES, October 2-4 2017, Poitiers 1 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Context c d � Leader election a d � Unidirectional rings � Homonym processes b b � Deterministic algorithm � Asynchronous a a message-passing e f 2 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

State of the Art - Leader Election Deterministic Probabilistic + solution solution Impossible Possible Anonymous [ Angluin, 80 ] [ Afek and Matias, 94 ] processes [ Lynch, 96 ] [ Kutten et al. , 13 ] Possible Identified [ LeLann, 77 ] processes [ Chang and Roberts, 79 ] [ Peterson, 82 ] 3 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

State of the Art - Leader Election Deterministic Probabilistic + solution solution Impossible Possible Anonymous [ Angluin, 80 ] [ Afek and Matias, 94 ] processes [ Lynch, 96 ] [ Kutten et al. , 13 ] Homonym processes [ Yamashita and Kameda, 89 ] Possible Identified [ LeLann, 77 ] processes [ Chang and Roberts, 79 ] [ Peterson, 82 ] 3 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Two versions of the Leader Election problem 1 Message-terminating: Processes do not explicitly terminate but only a finite number of messages are exchanged. 2 Process-terminating: Every process eventually halts. 4 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Ring Classes An algorithm A solves the leader election for the class of ring network R if A solves the leader election for every network R ∈ R . 5 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Ring Classes An algorithm A solves the leader election for the class of ring network R if A solves the leader election for every network R ∈ R . A cannot be given any specific information about the network unless that information holds for all members of R . 5 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Ring Classes An algorithm A solves the leader election for the class of ring network R if A solves the leader election for every network R ∈ R . A cannot be given any specific information about the network unless that information holds for all members of R . We consider three important classes of ring networks. � K k is the class of all ring networks such that no label occurs more than k times. � A is the class of all asymmetric ring networks: rings with no non-trivial rotational symmetry. � U ∗ is the class of all rings in which at least one label is unique. 5 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Symmetric vs. Asymmetric a a a b c b +3 b b b c a a Figure : Asymmetric Ring Figure : Symmetric Ring 6 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Inclusions � K 1 ⊂ K 2 ⊂ K 3 . . . � U ∗ ∩ K 1 ⊂ U ∗ ∩ K 2 ⊂ U ∗ ∩ K 3 . . . ⊂ U ∗ � K 1 ⊂ U ∗ � U ∗ ⊂ A 7 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

State of the Art vs. Contribution Leader Election in Rings of Homonym Processes PT/MT Asynch. Uni./Bi. Known Ring Class # Msg Time MT ? ? [Delporte # labels > greatest Bi. et al. , 14] proper divisor of n PT n O ( n log n ) ? Bi. + Uni. m ≤ n O ( n log n ) O ( M ) [Dobrev, Decide if inputs are PT Pelc, 04] unambiguous Bi. M ≥ n O ( nM ) ? ∃ unique label and [SSS 2016] PT Uni. k # proc with same O ( kn ) O ( kn ) label ≤ k O ( n 2 + kn ) Asymmetric la- O ( kn ) [IPDPS 2017] PT Uni. k belling and # proc with same label ≤ k O ( k 2 n 2 ) O ( k 2 n 2 ) � Uni : Unidirectional / Bi : Bidirectional � MT = Message-terminating � PT = Process-terminating 8 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Contributions ¯ A A MT-LE Impossible � A : Rings with asymmetric labelling � MT-LE: Message-Terminating � A : Rings with symmetric labelling Leader Election � U ∗ : Rings with at least one unique label � PT-LE: Process-Terminating � K k : Rings with no more than k processes with the Leader Election same label 9 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Contributions U ∗ ¯ A A PT-LE Impossible � A : Rings with asymmetric labelling � MT-LE: Message-Terminating � A : Rings with symmetric labelling Leader Election � U ∗ : Rings with at least one unique label � PT-LE: Process-Terminating � K k : Rings with no more than k processes with the Leader Election same label 9 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Contributions U ∗ ¯ A A ⇒ PT-LE Impossible PT-LE Impossible � A : Rings with asymmetric labelling � MT-LE: Message-Terminating � A : Rings with symmetric labelling Leader Election � U ∗ : Rings with at least one unique label � PT-LE: Process-Terminating � K k : Rings with no more than k processes with the Leader Election same label 9 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Contributions K k ¯ A A MT-LE Impossible � A : Rings with asymmetric labelling � MT-LE: Message-Terminating � A : Rings with symmetric labelling Leader Election � U ∗ : Rings with at least one unique label � PT-LE: Process-Terminating � K k : Rings with no more than k processes with the Leader Election same label 9 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Contributions K k U ∗ ¯ A A PT-LE Algorithms for A ∩ K k PT-LE Algorithm for U ∗ ∩ K k � A : Rings with asymmetric labelling � MT-LE: Message-Terminating � A : Rings with symmetric labelling Leader Election � U ∗ : Rings with at least one unique label � PT-LE: Process-Terminating � K k : Rings with no more than k processes with the Leader Election same label 9 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Lower bound for U ∗ ∩ K k 10 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Lower bound Lemma Let k ≥ 2 . Let A be an algorithm that solves the PT-LE for U ∗ ∩ K k . ∀ R n ∈ K 1 of n processes, the synchronous execution of A in R n lasts at least 1 + ( k − 2 ) n time units. 11 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Proof Outline (1/3) Ln-1 L0 Ln-2 L1 Li Figure : R n ∈ K 1 ⊂ U ∗ ∩ K k 12 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Proof Outline (1/3) X L0 Ln-1 Ln-1 L0 Li Li Ln-2 L1 Ln-1 L0 Li Ln-1 L0 Li Figure : R n ∈ K 1 ⊂ U ∗ ∩ K k Figure : R n , k ∈ U ∗ ∩ K k 12 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Proof Outline (1/3) X L0 Ln-1 Ln-1 L0 Li Li Ln-2 L1 Ln-1 L0 Li Ln-1 L0 Li Figure : R n ∈ K 1 ⊂ U ∗ ∩ K k Figure : R n , k ∈ U ∗ ∩ K k By the contradiction, assume that the synchronous execution of A on R n terminates before time 1 + ( k − 2 ) n . 12 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Proof Outline (2/3) X L0 Ln-1 Ln-1 L0 Li Li Ln-2 L1 Ln-1 L0 Li Ln-1 L0 Li Figure : R n ∈ K 1 ⊂ U ∗ ∩ K k Figure : R n , k ∈ U ∗ ∩ K k Synchronous execution after up to T < 1 + ( k − 2 ) n time units. 13 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Proof Outline (3/3) Ln-1 L0 Ln-2 L1 Li Figure : R n ∈ K 1 ⊂ U ∗ ∩ K k At time T , one node is elected in R n . 14 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Proof Outline (3/3) X L0 Ln-1 Ln-1 L0 Li Li Ln-2 L1 Ln-1 L0 Li Ln-1 L0 Li Figure : R n ∈ K 1 ⊂ U ∗ ∩ K k Figure : R n , k ∈ U ∗ ∩ K k At time T , one node is elected in R n . But, two nodes are elected in R n , k , contradiction. 14 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Consequences (1/2) Corollary Let k ≥ 2 . The time complexity of any algorithm that solves the process-terminating leader election for U ∗ ∩ K k (resp. A ∩ K k ) is Ω( k n ) time units, where n is the number of processes. 15 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Consequences (2/2) Theorem There is no algorithm that solves the process-terminating leader election for U ∗ (resp. A ). 16 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Consequences (2/2) Theorem There is no algorithm that solves the process-terminating leader election for U ∗ (resp. A ). By the contradiction, let A be a PT-LE algorithm for U ∗ . 16 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings

Consequences (2/2) Theorem There is no algorithm that solves the process-terminating leader election for U ∗ (resp. A ). By the contradiction, let A be a PT-LE algorithm for U ∗ . By definition, A solves PT-LE in U ∗ ∩ K 3 , U ∗ ∩ K 4 , . . . Let R n be a ring network of K 1 with n processes. 16 / 32 Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings