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

Leader Election in Asymmetric Labeled Unidirectional Rings

Karine Altisen1 Ajoy K. Datta2 Stéphane Devismes1 Anaïs Durand1 Lawrence L. Larmore2

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

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 1/32

Context

b a c d d b a f e a

Leader election Unidirectional rings Homonym processes Deterministic algorithm Asynchronous

message-passing

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 2/32

State of the Art - Leader Election

+

Anonymous processes Deterministic solution

Impossible

[Angluin, 80] [Lynch, 96]

Probabilistic solution

Possible

[Afek and Matias, 94] [Kutten et al., 13]

Identified processes

Possible

[LeLann, 77] [Chang and Roberts, 79] [Peterson, 82]

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 3/32

State of the Art - Leader Election

+

Anonymous processes Deterministic solution

Impossible

[Angluin, 80] [Lynch, 96]

Probabilistic solution

Possible

[Afek and Matias, 94] [Kutten et al., 13]

Identified processes

Possible

[LeLann, 77] [Chang and Roberts, 79] [Peterson, 82]

Homonym processes [Yamashita and Kameda, 89]

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 3/32

Two versions of the Leader Election problem

1 Message-terminating: Processes do not explicitly terminate but

• nly a finite number of messages are exchanged.

2 Process-terminating: Every process eventually halts.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 4/32

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.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 5/32

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.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 5/32

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.

Kk 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.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 5/32

Symmetric vs. Asymmetric

a c b a b c

+3

Figure : Symmetric Ring a a b a b b Figure : Asymmetric Ring

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 6/32

Inclusions

K1 ⊂ K2 ⊂ K3 . . . U∗ ∩ K1 ⊂ U∗ ∩ K2 ⊂ U∗ ∩ K3 . . . ⊂ U∗ K1 ⊂ U∗ U∗ ⊂ A

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 7/32

State of the Art vs. Contribution

Leader Election in Rings of Homonym Processes

PT/MT Asynch. Uni./Bi. Known Ring Class # Msg Time [Delporte et al., 14] MT Bi. # labels > greatest proper divisor of n ? ? PT n O(n log n) ? [Dobrev, Pelc, 04] PT

• Bi. + Uni.

m ≤ n Decide if inputs are unambiguous O(n log n) O(M) Bi. M ≥ n O(nM) ? [SSS 2016] PT Uni. k ∃ unique label and # proc with same label ≤ k O(kn) O(kn) O(n2 + kn) O(kn) [IPDPS 2017] PT Uni. k Asymmetric la- belling and # proc with same label ≤ k O(k2n2) O(k2n2)

Uni : Unidirectional / Bi : Bidirectional MT = Message-terminating PT = Process-terminating

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 8/32

Contributions

A ¯ A

MT-LE Impossible

MT-LE: Message-Terminating Leader Election PT-LE: Process-Terminating Leader Election A: Rings with asymmetric labelling A: Rings with symmetric labelling U∗: Rings with at least one unique label Kk: Rings with no more than k processes with the same label

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 9/32

Contributions

A ¯ A U∗

PT-LE Impossible

MT-LE: Message-Terminating Leader Election PT-LE: Process-Terminating Leader Election A: Rings with asymmetric labelling A: Rings with symmetric labelling U∗: Rings with at least one unique label Kk: Rings with no more than k processes with the same label

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 9/32

Contributions

A ¯ A U∗

PT-LE Impossible PT-LE Impossible ⇒

MT-LE: Message-Terminating Leader Election PT-LE: Process-Terminating Leader Election A: Rings with asymmetric labelling A: Rings with symmetric labelling U∗: Rings with at least one unique label Kk: Rings with no more than k processes with the same label

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 9/32

Contributions

Kk A ¯ A

MT-LE Impossible

MT-LE: Message-Terminating Leader Election PT-LE: Process-Terminating Leader Election A: Rings with asymmetric labelling A: Rings with symmetric labelling U∗: Rings with at least one unique label Kk: Rings with no more than k processes with the same label

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 9/32

Contributions

Kk A ¯ A U∗

PT-LE Algorithm for U∗ ∩ Kk PT-LE Algorithms for A ∩ Kk

MT-LE: Message-Terminating Leader Election PT-LE: Process-Terminating Leader Election A: Rings with asymmetric labelling A: Rings with symmetric labelling U∗: Rings with at least one unique label Kk: Rings with no more than k processes with the same label

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 9/32

Lower bound for U∗ ∩ Kk

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 10/32

Lower bound

Lemma

Let k ≥ 2. Let A be an algorithm that solves the PT-LE for U∗ ∩ Kk. ∀Rn ∈ K1 of n processes, the synchronous execution of A in Rn lasts at least 1 + (k − 2) n time units.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 11/32

Proof Outline (1/3)

Ln-1 L0 Ln-2 L1 Li

Figure : Rn ∈ K1 ⊂ U∗ ∩ Kk

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 12/32

Proof Outline (1/3)

Ln-1 L0 Ln-2 L1 Li

Figure : Rn ∈ K1 ⊂ U∗ ∩ Kk

X Ln-1 L0 L0 Li Ln-1 Li Ln-1 Li L0

Figure : Rn,k ∈ U∗ ∩ Kk

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 12/32

Proof Outline (1/3)

Ln-1 L0 Ln-2 L1 Li

Figure : Rn ∈ K1 ⊂ U∗ ∩ Kk

X Ln-1 L0 L0 Li Ln-1 Li Ln-1 Li L0

Figure : Rn,k ∈ U∗ ∩ Kk

By the contradiction, assume that the synchronous execution of A on Rn terminates before time 1 + (k − 2) n.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 12/32

Proof Outline (2/3)

Ln-1 L0 Ln-2 L1 Li

Figure : Rn ∈ K1 ⊂ U∗ ∩ Kk

X Ln-1 L0 L0 Li Ln-1 Li Ln-1 Li L0

Figure : Rn,k ∈ U∗ ∩ Kk

Synchronous execution after up to T < 1 + (k − 2) n time units.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 13/32

Proof Outline (3/3)

Ln-1 L0 Ln-2 L1 Li

Figure : Rn ∈ K1 ⊂ U∗ ∩ Kk

At time T, one node is elected in Rn.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 14/32

Proof Outline (3/3)

Ln-1 L0 Ln-2 L1 Li

Figure : Rn ∈ K1 ⊂ U∗ ∩ Kk

X Ln-1 L0 L0 Li Ln-1 Li Ln-1 Li L0

Figure : Rn,k ∈ U∗ ∩ Kk

At time T, one node is elected in Rn. But, two nodes are elected in Rn,k, contradiction.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 14/32

Consequences (1/2)

Corollary

Let k ≥ 2. The time complexity of any algorithm that solves the process-terminating leader election for U∗ ∩ Kk (resp. A ∩ Kk) is Ω(k n) time units, where n is the number of processes.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 15/32

Consequences (2/2)

Theorem

There is no algorithm that solves the process-terminating leader election for U∗ (resp. A).

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 16/32

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∗.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 16/32

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∗ ∩ K3, U∗ ∩ K4, . . . Let Rn be a ring network of K1 with n processes.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 16/32

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∗ ∩ K3, U∗ ∩ K4, . . . Let Rn be a ring network of K1 with n processes. Since Rn ∈ U∗ ∩ K3, the synchronous execution of A in Rn lasts at least 1 + n time units, by Lemma 1.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 16/32

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∗ ∩ K3, U∗ ∩ K4, . . . Let Rn be a ring network of K1 with n processes. Since Rn ∈ U∗ ∩ K3, the synchronous execution of A in Rn lasts at least 1 + n time units, by Lemma 1. Since Rn ∈ U∗ ∩ K4, the synchronous execution of A in Rn lasts at least 1 + 2n time units, by Lemma 1.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 16/32

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∗ ∩ K3, U∗ ∩ K4, . . . Let Rn be a ring network of K1 with n processes. Since Rn ∈ U∗ ∩ K3, the synchronous execution of A in Rn lasts at least 1 + n time units, by Lemma 1. Since Rn ∈ U∗ ∩ K4, the synchronous execution of A in Rn lasts at least 1 + 2n time units, by Lemma 1. . . .

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 16/32

Algorithm for U∗ ∩ Kk

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 17/32

PT-LE Algorithm Uk for U∗ ∩ Kk

b a c a a

b, 0 a, 0 a, 0 c, 0 a, 0

Lowest unique label

Counter = rough estimation of the

predominance

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 18/32

PT-LE Algorithm Uk for U∗ ∩ Kk

b a c a a

1

a, 0 b, 0 a, 1 a, 0 c, 0

Lowest unique label

Counter = rough estimation of the

predominance

Process elimination: ◮ Lower counter, = label → not unique

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 18/32

PT-LE Algorithm Uk for U∗ ∩ Kk

b a c a a

1 1 1

c, 0 a, 1 b, 0 a, 1 a, 1

Lowest unique label

Counter = rough estimation of the

predominance

Process elimination: ◮ Lower counter, = label → not unique Message elimination: ◮ Passive, same ID → not relevant

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 18/32

PT-LE Algorithm Uk for U∗ ∩ Kk

b a c a a

2 1 1

a, 1 c, 0 b, 0 a, 2

Lowest unique label

Counter = rough estimation of the

predominance

Process elimination: ◮ Lower counter, = label → not unique Message elimination: ◮ Passive, same ID → not relevant

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 18/32

PT-LE Algorithm Uk for U∗ ∩ Kk

b a c a a

2 1 1

a, 2 c, 0 b, 0

Lowest unique label

Counter = rough estimation of the

predominance

Process elimination: ◮ Lower counter, = label → not unique Message elimination: ◮ Passive, same ID → not relevant

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 18/32

PT-LE Algorithm Uk for U∗ ∩ Kk

b a c a a

1 2 1 1 1

c, 1 b, 1

Lowest unique label

Counter = rough estimation of the

predominance

Process elimination: ◮ Lower counter, = label → not unique ◮ Same counter = 0, lower label → not

lowest unique

Message elimination: ◮ Passive, same ID → not relevant

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 18/32

PT-LE Algorithm Uk for U∗ ∩ Kk

b a c a a

1 2 1 1 1

c, 1 b, 1

Lowest unique label

Counter = rough estimation of the

predominance

Process elimination: ◮ Lower counter, = label → not unique ◮ Same counter = 0, lower label → not

lowest unique

Message elimination: ◮ Passive, same ID → not relevant

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 18/32

PT-LE Algorithm Uk for U∗ ∩ Kk

b a c a a

k 2 1 1 1

b, k

Lowest unique label

Counter = rough estimation of the

predominance

Process elimination: ◮ Lower counter, = label → not unique ◮ Same counter = 0, lower label → not

lowest unique

Message elimination: ◮ Passive, same ID → not relevant Phases: ◮ 1st traversal: no more active

non-unique labels

◮ 2nd traversal: no more active

non-lowest unique labels

◮ Election detection: receiving id, k

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 18/32

PT-LE Algorithm Uk for U∗ ∩ Kk

b a c a a

k+1

2 1 1 1

b, k+1

Lowest unique label

Counter = rough estimation of the

predominance

Process elimination: ◮ Lower counter, = label → not unique ◮ Same counter = 0, lower label → not

lowest unique

Message elimination: ◮ Passive, same ID → not relevant Phases: ◮ 1st traversal: no more active

non-unique labels

◮ 2nd traversal: no more active

non-lowest unique labels

◮ Election detection: receiving id, k

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 18/32

PT-LE Algorithm for U∗ ∩ Kk

Time complexity: at most n(k + 2)

Asymptotically optimal (work under submission)

# messages: O(n2 + kn) Memory requirement: ⌈log(k + 1)⌉ + log(n) + 4

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 19/32

Algorithms for A ∩ Kk

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 20/32

Ak, first PT-LE Algorithm for A ∩ Kk

Chosen Leader:

process whose LabelSequence = LyndonWord(LabelSequence) Lyndon Word = smallest rotation in lexicographic order

1 2 2 1 2

p3 p4 p5 p1 p2

Label Sequence at p1:

LSp1 = 12212 Rotations: 12212 (= LSp1) 21221 (= LSp2) 12122 (= LSp3) LW = LSp1 21212 (= LSp4) 22121 (= LSp5)

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 21/32

Ak, first PT-LE Algorithm for A ∩ Kk

Chosen Leader:

process whose LabelSequence = LyndonWord(LabelSequence) Lyndon Word = smallest rotation in lexicographic order

1 2 2 1 2

p3 p4 p5 p1 p2

1 2 2 1 2 1 2 2 1 2 Local label aggregation

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 22/32

Ak, first PT-LE Algorithm for A ∩ Kk

Chosen Leader:

process whose LabelSequence = LyndonWord(LabelSequence) Lyndon Word = smallest rotation in lexicographic order

1 2 2 1 2

p3 p4 p5 p1 p2

12 21 22 12 21 2 2 1 2 1 Local label aggregation

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 22/32

Ak, first PT-LE Algorithm for A ∩ Kk

Chosen Leader:

process whose LabelSequence = LyndonWord(LabelSequence) Lyndon Word = smallest rotation in lexicographic order

1 2 2 1 2

p3 p4 p5 p1 p2

121 212 221 122 212 2 1 2 1 2 Local label aggregation

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 22/32

Ak, first PT-LE Algorithm for A ∩ Kk

Chosen Leader:

process whose LabelSequence = LyndonWord(LabelSequence) Lyndon Word = smallest rotation in lexicographic order

1 2 2 1 2

p3 p4 p5 p1 p2

12122 21212 22121 12212 21221 2 1 2 2 1 Local label aggregation

• Do not know n

⇒ Leader cannot detect its election

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 22/32

Ak, first PT-LE Algorithm for A ∩ Kk

Chosen Leader:

process whose LabelSequence = LyndonWord(LabelSequence) Lyndon Word = smallest rotation in lexicographic order

1 2 2 1 2

p3 p4 p5 p1 p2

12122 1212212 Smallest repeating prefix = LabelSequence = LyndonWord(Smallest repeating prefix)

k = 3

Local label aggregation

• Do not know n

⇒ Leader cannot detect its election

Termination detection =

(2k + 1) × the same label ⇒ at least 2 times the sequence of labels

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 22/32

Ak, first PT-LE Algorithm for A ∩ Kk

Time complexity: at most (2k + 2)n time units Message complexity: at most n2(2k + 1) messages Memory: (2k + 1)nb + 2b + 3 bits,

where b = number of bits to store an ID

Asymptotically optimal time complexity

but

Large memory requirement

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 23/32

Bk, second PT-LE Algorithm for A ∩ Kk

Decrease memory usage ⇒ Peterson principle with radix sort

1 2 2 1 2

p3 p4 p5 p1 p2

1 2122 2 1212 2 2121 1 2212 2 1221

1 2 2 1 2

Known

Phase 1

During a phase, Known values of active processes circulate clockwise End of phase: each still active process received its Known value k + 1 times

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 24/32

Bk, second PT-LE Algorithm for A ∩ Kk

Decrease memory usage ⇒ Peterson principle with radix sort

1 2 2 1 2

p3 p4 p5 p1 p2

1 2122 2 1212 2 2121 1 2212 2 1221

1 2 2 1 2

Known

Phase 1

During a phase, Known values of active processes circulate clockwise End of phase: each still active process received its Known value k + 1 times

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 24/32

Bk, second PT-LE Algorithm for A ∩ Kk

Decrease memory usage ⇒ Peterson principle with radix sort

1 2 2 1 2

p3 p4 p5 p1 p2

1 2 122 2 1 212 2 2 121 1 2 212 2 1 221

2 1 2 2 1

Known

Phase 2

During a phase, Known values of active processes circulate clockwise End of phase: each still active process received its Known value k + 1 times

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 24/32

Bk, second PT-LE Algorithm for A ∩ Kk

Decrease memory usage ⇒ Peterson principle with radix sort

1 2 2 1 2

p3 p4 p5 p1 p2

12 1 22 21 2 12 22 1 21 12 2 12 21 2 21

1 2 1 2 2

Known

Phase 3

During a phase, Known values of active processes circulate clockwise End of phase: each still active process received its Known value k + 1 times

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 24/32

Bk, second PT-LE Algorithm for A ∩ Kk

Decrease memory usage ⇒ Peterson principle with radix sort

1 2 2 1 2

p3 p4 p5 p1 p2

12 1 22 21 2 12 22 1 21 12 2 12 21 2 21

1 2 1 2 2

Known

Phase 3

During a phase, Known values of active processes circulate clockwise End of phase: each still active process received its Known value k + 1 times

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 24/32

Bk, second PT-LE Algorithm for A ∩ Kk

Phase Shift

1 2 2 1 2

p3 p4 p5 p1 p2

1 2122 2 1212 2 2121 1 2212 2 1221

1 2 2 1 2

Known

Phase 1

1 2 2 1 2

p3 p4 p5 p1 p2

1 2 122 2 1 212 2 2 121 1 2 212 2 1 221

2 1 2 2 1

Known

Phase 2

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 25/32

Bk, second PT-LE Algorithm for A ∩ Kk

Execution

Shift Shift

S y n c h r

• n

i z a t i

• n

S y n c h r

• n

i z a t i

• n

Phase 1 Phase 2 Phase 3 · · ·

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 26/32

Bk, second PT-LE Algorithm for A ∩ Kk

Termination Detection: count = k+1

count = # phases where Known = Label

1 2 2 1 2

p3 p4 p5 p1 p2

1 2122

1 2 2 1 2

Known

Phase 1

count = 1

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 27/32

Bk, second PT-LE Algorithm for A ∩ Kk

Termination Detection: count = k+1

count = # phases where Known = Label

1 2 2 1 2

p3 p4 p5 p1 p2

1 2 122

2 1 2 2 1

Known

Phase 2

count = 1

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 27/32

Bk, second PT-LE Algorithm for A ∩ Kk

Termination Detection: count = k+1

count = # phases where Known = Label

1 2 2 1 2

p3 p4 p5 p1 p2

12 1 22

1 2 1 2 2

Known

Phase 3

count = 2

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 27/32

Bk, second PT-LE Algorithm for A ∩ Kk

Termination Detection: count = k+1

count = # phases where Known = Label

1 2 2 1 2

p3 p4 p5 p1 p2

121 2 2

2 1 2 1 2

Known

Phase 4

count = 2

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 27/32

Bk, second PT-LE Algorithm for A ∩ Kk

Termination Detection: count = k+1

count = # phases where Known = Label

1 2 2 1 2

p3 p4 p5 p1 p2

1212 2

2 2 1 2 1

Known

Phase 5

count = 2

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 27/32

Bk, second PT-LE Algorithm for A ∩ Kk

Termination Detection: count = k+1

count = # phases where Known = Label

1 2 2 1 2

p3 p4 p5 p1 p2

1 2122

1 2 2 1 2

Known

Phase 6

count = 3

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 27/32

Bk, second PT-LE Algorithm for A ∩ Kk

Termination Detection: count = k+1

count = # phases where Known = Label

1 2 2 1 2

p3 p4 p5 p1 p2

1 2 122

2 1 2 2 1

Known

Phase 7

count = 3

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 27/32

Bk, second PT-LE Algorithm for A ∩ Kk

Termination Detection: count = k+1

count = # phases where Known = Label

1 2 2 1 2

p3 p4 p5 p1 p2

12 1 22

1 2 1 2 2

Known

Phase 8

count = 4 = k+1 k = 3

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 27/32

Bk, second PT-LE Algorithm for A ∩ Kk

Memory: 2 ⌈log k⌉ + 3b + 5 bits,

where b = number of bits to store an ID

Time complexity: O(k2n2) time units Message complexity: O(k2n2) messages

Asymptotically optimal memory requirement

but

Large time complexity

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 28/32

Conclusion

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 29/32

Contributions Summary

Impossibility results: ¯

A, A, U∗, and Kk

Lower bounds: ◮ on the time in U∗ ∩ Kk and A ∩ Kk: Ω(kn) ◮ on the # bits exchanged in U∗ ∩ Kk and A ∩ Kk: Ω(n2 + kn) Algorithms:

Uk Ak Bk Rings U∗ ∩ Kk A ∩ Kk Time O(kn) O(kn) O(k2n2) # Messages O(kn) O(n2 + kn) O(k2n2) Bits/process O(log k + b) O(knb) O(log k + b)

Key:

asymptotically optimal

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 30/32

Perspectives

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 31/32

Perspectives

Leader election possible in A ∩ Kk A, but impossible in A: where is

the boundary ?

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 31/32

Perspectives

Leader election possible in A ∩ Kk A, but impossible in A: where is

the boundary ?

Find a best trade-off leader election algorithm for A ∩ Kk

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 31/32

Perspectives

Leader election possible in A ∩ Kk A, but impossible in A: where is

the boundary ?

Find a best trade-off leader election algorithm for A ∩ Kk In A, the knowledge of k and n is computationaly equivalent. Is-it

still true in bidirectional rings? What about time complexity?

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 31/32

Perspectives

Leader election possible in A ∩ Kk A, but impossible in A: where is

the boundary ?

Find a best trade-off leader election algorithm for A ∩ Kk In A, the knowledge of k and n is computationaly equivalent. Is-it

still true in bidirectional rings? What about time complexity?

Self-stabilizing leader election in U∗ ∩ Kk and A ∩ Kk. (research line:

adapting self-stabilizing census algorithms?)

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 31/32

Perspectives

Leader election possible in A ∩ Kk A, but impossible in A: where is

the boundary ?

Find a best trade-off leader election algorithm for A ∩ Kk In A, the knowledge of k and n is computationaly equivalent. Is-it

still true in bidirectional rings? What about time complexity?

Self-stabilizing leader election in U∗ ∩ Kk and A ∩ Kk. (research line:

adapting self-stabilizing census algorithms?)

Other topologies: regular graphs, grids, torii, arbitrary connected . . .

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 31/32

Perspectives

Leader election possible in A ∩ Kk A, but impossible in A: where is

the boundary ?

Find a best trade-off leader election algorithm for A ∩ Kk In A, the knowledge of k and n is computationaly equivalent. Is-it

still true in bidirectional rings? What about time complexity?

Self-stabilizing leader election in U∗ ∩ Kk and A ∩ Kk. (research line:

adapting self-stabilizing census algorithms?)

Other topologies: regular graphs, grids, torii, arbitrary connected . . . Other problems (solutions exist for the consensus problem with

permanent failures)

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 31/32

Thank you for your attention

• K. Altisen, A. K. Datta, S. Devismes, A. Durand, and L. L. Larmore.

Leader Election in Rings with Bounded Multiplicity (Short Paper). SSS 2016, pp. 1-6, Lyon, France, Nov. 7-10, 2016.

• K. Altisen, A. K. Datta, S. Devismes, A. Durand, and L. L. Larmore.

Leader Election in Asymmetric Labeled Unidirectional Rings. IPDPS 2017, pp. 182-191, Orlando, Florida, USA, May 29 - June 2, 2017.

Altisen et al Leader Election in Asymmetric Labeled Unidirectional Rings 32/32