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k-set agreement in communication networks with omission faults - - PowerPoint PPT Presentation

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k-set agreement in communication networks with omission faults Emmanuel Godard Eloi Perdereau Laboratoire dInformatique Fondamentale 02/10/2017 DESCARTES 2-4 octobre 2017 - Chasseneuil-du-Poitou Model and problem Distributed model The

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SLIDE 1

k-set agreement in communication networks with

  • mission faults

Emmanuel Godard Eloi Perdereau

Laboratoire d’Informatique Fondamentale

02/10/2017 DESCARTES 2-4 octobre 2017 - Chasseneuil-du-Poitou

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Model and problem

Distributed model

The distributed model : Synchronous; Message passing; Underlying connected communication graph G = (V , E) fixed; Dynamic network; f omission faults, i.e. at each round : f messages can be lost.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 2 / 24

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SLIDE 3

Model and problem

Distributed model

The distributed model : Synchronous; Message passing; Underlying connected communication graph G = (V , E) fixed; Dynamic network; f omission faults, i.e. at each round : f messages can be lost.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 2 / 24

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SLIDE 4

Model and problem

Distributed model

The distributed model : Synchronous; Message passing; Underlying connected communication graph G = (V , E) fixed; Dynamic network; f omission faults, i.e. at each round : f messages can be lost.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 2 / 24

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SLIDE 5

Model and problem

Distributed model

The distributed model : Synchronous; Message passing; Underlying connected communication graph G = (V , E) fixed; Dynamic network; f omission faults, i.e. at each round : f messages can be lost.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 2 / 24

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SLIDE 6

Model and problem

Distributed model

The distributed model : Synchronous; Message passing; Underlying connected communication graph G = (V , E) fixed; Dynamic network; f omission faults, i.e. at each round : f messages can be lost.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 2 / 24

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SLIDE 7

Model and problem

Distributed model

The distributed model : Synchronous; Message passing; Underlying connected communication graph G = (V , E) fixed; Dynamic network; f omission faults, i.e. at each round : f messages can be lost.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 2 / 24

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SLIDE 8

Model and problem

The k-set agreement problem

The k-set agreement problem : k-Agreement The set of output values contains at most k elements. Validity The decided values are ones of the initial values. Termination All processes must decide.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 3 / 24

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Model and problem

Message adversaries

First introduced by Afek & Gafni (2013) Message adversary : set of infinite sequences of digraphs (instant graphs) defining the messages received in each round. Oblivious message adversary : the set of potential graphs in each round remains constant all along the execution. ⇒ Set of instant digraphs.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 4 / 24

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Model and problem

The f -omission message adversary : Of (G)

For example, the message adversary that allows at most f faults : Definition (f -omissions message adversary) Of (G) = {G′ = (V , A′) | A′ ⊆ A ∧ |A| − |A′| ≤ f } Example of f -omission message adversary : O1(P3) P3 O1(P3)

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 5 / 24

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Model and problem

The f -omission message adversary : Of (G)

For example, the message adversary that allows at most f faults : Definition (f -omissions message adversary) Of (G) = {G′ = (V , A′) | A′ ⊆ A ∧ |A| − |A′| ≤ f } Example of f -omission message adversary : O1(P3) P3 O1(P3)

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 5 / 24

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Model and problem

Our contribution : computability of k-set agreement

How many lost messages (omissions) the k-set agreement is tolerant to ? Theorem Let k ∈ N and G = (V , E) be any communication network. The k−set agreement problem is solvable despite f omission faults if and only if f ≤ ck(G). ck(G) is the maximum number of (undirected) edges that can be removed without disconnecting G in k + 1 components. i.e. removing ck(G) edges from G keeps at most k connected components

  • n G.

⇒ the standard connectivity is c1(G) + 1.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 6 / 24

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Model and problem

Our contribution : computability of k-set agreement

How many lost messages (omissions) the k-set agreement is tolerant to ? Theorem Let k ∈ N and G = (V , E) be any communication network. The k−set agreement problem is solvable despite f omission faults if and only if f ≤ ck(G). ck(G) is the maximum number of (undirected) edges that can be removed without disconnecting G in k + 1 components. i.e. removing ck(G) edges from G keeps at most k connected components

  • n G.

⇒ the standard connectivity is c1(G) + 1.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 6 / 24

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SLIDE 14

Model and problem

Our contribution : computability of k-set agreement

How many lost messages (omissions) the k-set agreement is tolerant to ? Theorem Let k ∈ N and G = (V , E) be any communication network. The k−set agreement problem is solvable despite f omission faults if and only if f ≤ ck(G). ck(G) is the maximum number of (undirected) edges that can be removed without disconnecting G in k + 1 components. i.e. removing ck(G) edges from G keeps at most k connected components

  • n G.

⇒ the standard connectivity is c1(G) + 1.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 6 / 24

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Model and problem

Example of ck(G)

G

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 7 / 24

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Model and problem

Example of ck(G)

G c1(G) = 0

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 7 / 24

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SLIDE 17

Model and problem

Example of ck(G)

G c1(G) = 0

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 7 / 24

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Model and problem

Example of ck(G)

G c1(G) = 0 c2(G) = 3

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 7 / 24

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Model and problem

Example of ck(G)

G c1(G) = 0 c2(G) = 3

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 7 / 24

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SLIDE 20

Model and problem

Example of ck(G)

G c1(G) = 0 c2(G) = 3

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 7 / 24

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SLIDE 21

Model and problem

Example of ck(G)

G c1(G) = 0 c2(G) = 3

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 7 / 24

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SLIDE 22

Model and problem

Example of ck(G)

G c1(G) = 0 c2(G) = 3

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 7 / 24

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SLIDE 23

Model and problem

Example of ck(G)

G c1(G) = 0 c2(G) = 3 c3(G) = 5

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 7 / 24

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Model and problem

Example of ck(G)

G c1(G) = 0 c2(G) = 3 c3(G) = 5

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 7 / 24

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Model and problem

Example of ck(G)

G c1(G) = 0 c2(G) = 3 c3(G) = 5

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 7 / 24

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Model and problem

Outline

1

Model and problem

2

Impossibility proofs Impossibility of 2-set in K3 and P3 (|G| = k + 1 = 3) Reduction from the case |G| > k + 1 Impossibility of set-agreement for generalized tournaments

3

Possibility : a priority-based algorithm

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 8 / 24

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SLIDE 27

Model and problem

Outline

1

Model and problem

2

Impossibility proofs Impossibility of 2-set in K3 and P3 (|G| = k + 1 = 3) Reduction from the case |G| > k + 1 Impossibility of set-agreement for generalized tournaments

3

Possibility : a priority-based algorithm

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 8 / 24

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SLIDE 28

Model and problem

Outline

1

Model and problem

2

Impossibility proofs Impossibility of 2-set in K3 and P3 (|G| = k + 1 = 3) Reduction from the case |G| > k + 1 Impossibility of set-agreement for generalized tournaments

3

Possibility : a priority-based algorithm

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 8 / 24

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SLIDE 29

Impossibility proofs

Outline

1

Model and problem

2

Impossibility proofs Impossibility of 2-set in K3 and P3 (|G| = k + 1 = 3) Reduction from the case |G| > k + 1 Impossibility of set-agreement for generalized tournaments

3

Possibility : a priority-based algorithm

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 9 / 24

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Impossibility proofs

Results for K3 and P3

(a) K3 (b) P3 ❍❍❍❍❍ ❍

f k 1 2 3 1 yes yes yes 2 no yes yes 3 no no yes

❍❍❍❍❍ ❍

f k 1 2 3 1 no yes yes 2 no no yes 3 no no yes

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 10 / 24

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SLIDE 31

Impossibility proofs Impossibility of 2-set in K3 and P3 (|G| = k + 1 = 3)

Classical impossibility proof in O3(K3)

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 11 / 24

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Impossibility proofs Impossibility of 2-set in K3 and P3 (|G| = k + 1 = 3)

Classical impossibility proof in O3(K3)

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 11 / 24

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Impossibility proofs Impossibility of 2-set in K3 and P3 (|G| = k + 1 = 3)

Half-duplex message adversary

Remark : only “half-duplex” graphs are depicted, i.e. we allow only one arc to be removed between two nodes, not both. Definition (Half-Duplex graphs) HDf (G) = {G′ = (V , A′) | G′ ∈ Of (G)∧ ∀p, q ∈ V {p, q} ∈ E ∧ (p, q) / ∈ A′ ⇒ (q, p) ∈ A′} ⇒ HDf (G) ⊆ Of (G) Proposition The 2-set agreement problem is impossible in HD2(P3).

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 12 / 24

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Impossibility proofs Impossibility of 2-set in K3 and P3 (|G| = k + 1 = 3)

Half-duplex message adversary

Remark : only “half-duplex” graphs are depicted, i.e. we allow only one arc to be removed between two nodes, not both. Definition (Half-Duplex graphs) HDf (G) = {G′ = (V , A′) | G′ ∈ Of (G)∧ ∀p, q ∈ V {p, q} ∈ E ∧ (p, q) / ∈ A′ ⇒ (q, p) ∈ A′} ⇒ HDf (G) ⊆ Of (G) Proposition The 2-set agreement problem is impossible in HD2(P3).

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 12 / 24

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Impossibility proofs Impossibility of 2-set in K3 and P3 (|G| = k + 1 = 3)

Half-duplex message adversary

Remark : only “half-duplex” graphs are depicted, i.e. we allow only one arc to be removed between two nodes, not both. Definition (Half-Duplex graphs) HDf (G) = {G′ = (V , A′) | G′ ∈ Of (G)∧ ∀p, q ∈ V {p, q} ∈ E ∧ (p, q) / ∈ A′ ⇒ (q, p) ∈ A′} ⇒ HDf (G) ⊆ Of (G) Proposition The 2-set agreement problem is impossible in HD2(P3).

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 12 / 24

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Impossibility proofs Impossibility of 2-set in K3 and P3 (|G| = k + 1 = 3)

Impossibility of (2-)set agreement in P3

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 13 / 24

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SLIDE 37

Impossibility proofs Impossibility of 2-set in K3 and P3 (|G| = k + 1 = 3)

Impossibility of (2-)set agreement in P3

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 13 / 24

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Impossibility proofs Reduction from the case |G| > k + 1

Reduction to HD3(K3) or HD2(P3) for the 2-set (1)

For arbitrary G = (V , E) : Suppose f > c2(G); Partition V in 3 sets V1, V2 and V3; An algorithm A for G would solve the 2-set agreement for all M ⊆ Of (G); In particular in S ⊆ HDf (G) where omissions are synchronized between Vis; Reduce to HD3(K3) or HD2(P3) by syncing omissions.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 14 / 24

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Impossibility proofs Reduction from the case |G| > k + 1

Reduction to HD3(K3) or HD2(P3) for the 2-set (2)

f > c2(G) f1 + f2 + f3 = f

V1 V2 V3

f1 . . . f2 . . . f3 . . . H

⇒ Generalizable for all k: k-set reduced to set agreement in H (|H| = k + 1).

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 15 / 24

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SLIDE 40

Impossibility proofs Reduction from the case |G| > k + 1

Reduction to HD3(K3) or HD2(P3) for the 2-set (2)

f > c2(G) f1 + f2 + f3 = f

V1

simulated by

V2

simulated by

V3

simulated by . . . . . . . . . H

⇒ Generalizable for all k: k-set reduced to set agreement in H (|H| = k + 1).

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 15 / 24

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SLIDE 41

Impossibility proofs Reduction from the case |G| > k + 1

Reduction to HD3(K3) or HD2(P3) for the 2-set (2)

f > c2(G) f1 + f2 + f3 = f

V1

simulated by

V2

simulated by

V3

simulated by . . . . . . . . . H

⇒ Generalizable for all k: k-set reduced to set agreement in H (|H| = k + 1).

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 15 / 24

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SLIDE 42

Impossibility proofs Reduction from the case |G| > k + 1

Reduction to HD3(K3) or HD2(P3) for the 2-set (2)

f > c2(G) f1 + f2 + f3 = f

V1

simulated by

V2

simulated by

V3

simulated by . . . . . . . . . H

⇒ Generalizable for all k: k-set reduced to set agreement in H (|H| = k + 1).

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 15 / 24

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SLIDE 43

Impossibility proofs Impossibility of set-agreement for generalized tournaments

How to generalize

we had to consider both HD(K3, 3) and HD(P3, 2) we need a way to handle all HD(G, f ) we are reducing to (generalized tournaments) => subdivision diagram

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 16 / 24

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SLIDE 44

Impossibility proofs Impossibility of set-agreement for generalized tournaments

Construction of the subdivision diagram of S3

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 17 / 24

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SLIDE 45

Impossibility proofs Impossibility of set-agreement for generalized tournaments

Construction of the subdivision diagram of S3

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 17 / 24

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SLIDE 46

Impossibility proofs Impossibility of set-agreement for generalized tournaments

Construction of the subdivision diagram of S3

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 17 / 24

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SLIDE 47

Impossibility proofs Impossibility of set-agreement for generalized tournaments

Construction of the subdivision diagram of S3

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 17 / 24

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SLIDE 48

Possibility : a priority-based algorithm

Outline

1

Model and problem

2

Impossibility proofs Impossibility of 2-set in K3 and P3 (|G| = k + 1 = 3) Reduction from the case |G| > k + 1 Impossibility of set-agreement for generalized tournaments

3

Possibility : a priority-based algorithm

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 18 / 24

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SLIDE 49

Possibility : a priority-based algorithm

An algorithm for the k-set when f ≤ ck(G)

Priority order π on {p1, p2, . . . , pn} : π(pi) = i Algorithm: Algorithm for the k-set agreement in G for process pi known ← i ; for T rounds do send known to all neighbors ; known ← known ∪ received ids ; end decide max(known) ;

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 19 / 24

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SLIDE 50

Possibility : a priority-based algorithm

Proof of possibility when f = c2(G) (1)

pn is the process with the highest priority. Let I : the set of process informed by pn; Consider the processes eventually in I; If I = V , the consensus is solved; Otherwise, I = V \ I form a strongly connected component because f = c2(G); After enough rounds, processes in I have the same information.

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 20 / 24

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SLIDE 51

Possibility : a priority-based algorithm

Proof of possibility when f = c2(G) (2) I :

processes informed by pn

V \ I

. . . ≤ f

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 21 / 24

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SLIDE 52

Possibility : a priority-based algorithm

Conclusion and perspectives

Complete characterization of k-set agreement in oblivious message adversaries with f omission faults : solvable iff f ≤ ck(G). ⇒ New proof technique for impossibility with subdivision diagrams. Perspectives of generalization : Managing partioning vs uncertainty, Find a generic method to construct a subdivision diagram for arbitrary oblivious message adversaries;

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 22 / 24

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SLIDE 53

Possibility : a priority-based algorithm

Conclusion and perspectives

Complete characterization of k-set agreement in oblivious message adversaries with f omission faults : solvable iff f ≤ ck(G). ⇒ New proof technique for impossibility with subdivision diagrams. Perspectives of generalization : Managing partioning vs uncertainty, Find a generic method to construct a subdivision diagram for arbitrary oblivious message adversaries;

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 22 / 24

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SLIDE 54

Possibility : a priority-based algorithm

Contribution to DESCARTES

HD(G, f ) can be simulated by HD(Kκ, κ(κ − 1)/2) which is equivalent to the wait-free RW model for κ processes where κ is the smallest integer such that cκ(G) ≥ f . Colorless Computability RWwaitfree(κ) ⊆ HD(G, f ) RWwaitfree(κ − 1)

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 23 / 24

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Possibility : a priority-based algorithm

Thank you

  • E. Godard

(L.I.F) k-set agreement with omission faults 02/10/2017 24 / 24