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The Optimal Strategyfor the Average Long-Lived Consensus

Eric Remila

Universite de Lyon

Laboratoire de l’Informatique du Parallelisme(LIP, umr 5668 CNRS - Universite Lyon 1 - ENS de Lyon)

IXXI (Institut des Systemes Complexes - Complex System Institute, Lyon)IUT Roanne, Universite de Saint-Etienne

Partially supported by Program Ecos C09E04, ANR Subtile and ANR Mint.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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The consensus problem: informal approach

What qualities are requested for a collective decision, calledconsensus?

Representativity: the consensus corresponds to a sufficientnumber of individual opinions,

Stability: the consensus is robust to individual opinionvariations.

Problem: the qualities above are often incompatible.What can we do in in the real life?

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: informal approach

What qualities are requested for a collective decision, calledconsensus?

Representativity: the consensus corresponds to a sufficientnumber of individual opinions,

Stability: the consensus is robust to individual opinionvariations.

Problem: the qualities above are often incompatible.What can we do in in the real life?

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: informal approach

What qualities are requested for a collective decision, calledconsensus?

Representativity: the consensus corresponds to a sufficientnumber of individual opinions,

Stability: the consensus is robust to individual opinionvariations.

Problem: the qualities above are often incompatible.What can we do in in the real life?

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: informal approach

What qualities are requested for a collective decision, calledconsensus?

Representativity: the consensus corresponds to a sufficientnumber of individual opinions,

Stability: the consensus is robust to individual opinionvariations.

Problem: the qualities above are often incompatible.What can we do in in the real life?

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

An illustrative example: the wedding party problem

A disc-jockey with two kinds of musics: traditional music andtechno music, people going in and out.

The music must appeal to at least some of the people in theroom (representativity).changing style after every song is frowned upon (stability).The party has to last ’till the end of the night.Another example: governments simultaneously need

stability for consistence of politics and representativity toremain.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

An illustrative example: the wedding party problem

A disc-jockey with two kinds of musics: traditional music andtechno music, people going in and out.

The music must appeal to at least some of the people in theroom (representativity).

changing style after every song is frowned upon (stability).The party has to last ’till the end of the night.Another example: governments simultaneously need

stability for consistence of politics and representativity toremain.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

An illustrative example: the wedding party problem

A disc-jockey with two kinds of musics: traditional music andtechno music, people going in and out.

The music must appeal to at least some of the people in theroom (representativity).changing style after every song is frowned upon (stability).

The party has to last ’till the end of the night.Another example: governments simultaneously need

stability for consistence of politics and representativity toremain.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

An illustrative example: the wedding party problem

A disc-jockey with two kinds of musics: traditional music andtechno music, people going in and out.

The music must appeal to at least some of the people in theroom (representativity).changing style after every song is frowned upon (stability).The party has to last ’till the end of the night.

Another example: governments simultaneously needstability for consistence of politics and representativity toremain.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

An illustrative example: the wedding party problem

A disc-jockey with two kinds of musics: traditional music andtechno music, people going in and out.

The music must appeal to at least some of the people in theroom (representativity).changing style after every song is frowned upon (stability).The party has to last ’till the end of the night.Another example: governments simultaneously need

stability for consistence of politics and representativity toremain.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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The consensus problem: system formalization

n sensors, each one is given an input component,chosen in Zm = {0, 1, ....,m − 1},

A global input: an n-uple of Vn = (Zm)n,

For each pair (x , b) of Vn ×Zm, #x(b) is the number of inputcomponents of x that are equal to b,

For each x of Vn, dom(x) is the integer b that maximizes#x(b) (the lowest one in case of ties)

Example: x = (0, 3, 1, 3, 1).

#x(0) = 1, #x(1) = 2, #x(2) = 0, #x(3) = 2.dom(x) = 1.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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The consensus problem: system formalization

n sensors, each one is given an input component,chosen in Zm = {0, 1, ....,m − 1},

A global input: an n-uple of Vn = (Zm)n,

For each pair (x , b) of Vn ×Zm, #x(b) is the number of inputcomponents of x that are equal to b,

For each x of Vn, dom(x) is the integer b that maximizes#x(b) (the lowest one in case of ties)

Example: x = (0, 3, 1, 3, 1).

#x(0) = 1, #x(1) = 2, #x(2) = 0, #x(3) = 2.dom(x) = 1.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: system formalization

n sensors, each one is given an input component,chosen in Zm = {0, 1, ....,m − 1},

A global input: an n-uple of Vn = (Zm)n,

For each pair (x , b) of Vn ×Zm, #x(b) is the number of inputcomponents of x that are equal to b,

For each x of Vn, dom(x) is the integer b that maximizes#x(b) (the lowest one in case of ties)

Example: x = (0, 3, 1, 3, 1).

#x(0) = 1, #x(1) = 2, #x(2) = 0, #x(3) = 2.dom(x) = 1.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: system formalization

n sensors, each one is given an input component,chosen in Zm = {0, 1, ....,m − 1},

A global input: an n-uple of Vn = (Zm)n,

For each pair (x , b) of Vn ×Zm, #x(b) is the number of inputcomponents of x that are equal to b,

For each x of Vn, dom(x) is the integer b that maximizes#x(b) (the lowest one in case of ties)

Example: x = (0, 3, 1, 3, 1).

#x(0) = 1, #x(1) = 2, #x(2) = 0, #x(3) = 2.dom(x) = 1.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: system formalization

n sensors, each one is given an input component,chosen in Zm = {0, 1, ....,m − 1},

A global input: an n-uple of Vn = (Zm)n,

For each pair (x , b) of Vn ×Zm, #x(b) is the number of inputcomponents of x that are equal to b,

For each x of Vn, dom(x) is the integer b that maximizes#x(b) (the lowest one in case of ties)

Example: x = (0, 3, 1, 3, 1).

#x(0) = 1, #x(1) = 2, #x(2) = 0, #x(3) = 2.dom(x) = 1.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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The consensus problem: evolution formalization

The input graph (Vn,En): the undirected graph whose edgesare input pairs only differing in a unique sensor.

A trajectory: a sequence x0 → x1 → x2 → .... with∀k ∈ N, {xk+1, xk} ∈ En.At each step: change of one input component.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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The consensus problem: evolution formalization

The input graph (Vn,En): the undirected graph whose edgesare input pairs only differing in a unique sensor.

A trajectory: a sequence x0 → x1 → x2 → .... with∀k ∈ N, {xk+1, xk} ∈ En.At each step: change of one input component.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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The consensus problem: adding a memory

a memory: a set Q of states,

Vn × Q: configuration set,

A consensus protocol:

A memory evolution function τ : Vn × Q → Q,A consensus function f : Vn × Q → Zm,

An execution: sequence (x0, s0)→ (x1, s1)→ (x2, s2)→ ....with

x0 → x1 → x2 → .... is a trajectory,sk+1 = τ(xk , sk),s0 = ⊥, the empty memory state

Each trajectory induces a unique execution,

At each time step: change (or not) of the memory and of theconsensus value f (xk , sk).

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: adding a memory

a memory: a set Q of states,

Vn × Q: configuration set,

A consensus protocol:

A memory evolution function τ : Vn × Q → Q,A consensus function f : Vn × Q → Zm,

An execution: sequence (x0, s0)→ (x1, s1)→ (x2, s2)→ ....with

x0 → x1 → x2 → .... is a trajectory,sk+1 = τ(xk , sk),s0 = ⊥, the empty memory state

Each trajectory induces a unique execution,

At each time step: change (or not) of the memory and of theconsensus value f (xk , sk).

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: adding a memory

a memory: a set Q of states,

Vn × Q: configuration set,

A consensus protocol:

A memory evolution function τ : Vn × Q → Q,A consensus function f : Vn × Q → Zm,

An execution: sequence (x0, s0)→ (x1, s1)→ (x2, s2)→ ....with

x0 → x1 → x2 → .... is a trajectory,sk+1 = τ(xk , sk),s0 = ⊥, the empty memory state

Each trajectory induces a unique execution,

At each time step: change (or not) of the memory and of theconsensus value f (xk , sk).

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: adding a memory

a memory: a set Q of states,

Vn × Q: configuration set,

A consensus protocol:

A memory evolution function τ : Vn × Q → Q,

A consensus function f : Vn × Q → Zm,

An execution: sequence (x0, s0)→ (x1, s1)→ (x2, s2)→ ....with

x0 → x1 → x2 → .... is a trajectory,sk+1 = τ(xk , sk),s0 = ⊥, the empty memory state

Each trajectory induces a unique execution,

At each time step: change (or not) of the memory and of theconsensus value f (xk , sk).

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: adding a memory

a memory: a set Q of states,

Vn × Q: configuration set,

A consensus protocol:

A memory evolution function τ : Vn × Q → Q,A consensus function f : Vn × Q → Zm,

An execution: sequence (x0, s0)→ (x1, s1)→ (x2, s2)→ ....with

x0 → x1 → x2 → .... is a trajectory,sk+1 = τ(xk , sk),s0 = ⊥, the empty memory state

Each trajectory induces a unique execution,

At each time step: change (or not) of the memory and of theconsensus value f (xk , sk).

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: adding a memory

a memory: a set Q of states,

Vn × Q: configuration set,

A consensus protocol:

A memory evolution function τ : Vn × Q → Q,A consensus function f : Vn × Q → Zm,

An execution: sequence (x0, s0)→ (x1, s1)→ (x2, s2)→ ....with

x0 → x1 → x2 → .... is a trajectory,sk+1 = τ(xk , sk),s0 = ⊥, the empty memory state

Each trajectory induces a unique execution,

At each time step: change (or not) of the memory and of theconsensus value f (xk , sk).

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: adding a memory

a memory: a set Q of states,

Vn × Q: configuration set,

A consensus protocol:

A memory evolution function τ : Vn × Q → Q,A consensus function f : Vn × Q → Zm,

An execution: sequence (x0, s0)→ (x1, s1)→ (x2, s2)→ ....with

x0 → x1 → x2 → .... is a trajectory,

sk+1 = τ(xk , sk),s0 = ⊥, the empty memory state

Each trajectory induces a unique execution,

At each time step: change (or not) of the memory and of theconsensus value f (xk , sk).

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: adding a memory

a memory: a set Q of states,

Vn × Q: configuration set,

A consensus protocol:

A memory evolution function τ : Vn × Q → Q,A consensus function f : Vn × Q → Zm,

An execution: sequence (x0, s0)→ (x1, s1)→ (x2, s2)→ ....with

x0 → x1 → x2 → .... is a trajectory,sk+1 = τ(xk , sk),

s0 = ⊥, the empty memory state

Each trajectory induces a unique execution,

At each time step: change (or not) of the memory and of theconsensus value f (xk , sk).

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: adding a memory

a memory: a set Q of states,

Vn × Q: configuration set,

A consensus protocol:

A memory evolution function τ : Vn × Q → Q,A consensus function f : Vn × Q → Zm,

An execution: sequence (x0, s0)→ (x1, s1)→ (x2, s2)→ ....with

x0 → x1 → x2 → .... is a trajectory,sk+1 = τ(xk , sk),s0 = ⊥, the empty memory state

Each trajectory induces a unique execution,

At each time step: change (or not) of the memory and of theconsensus value f (xk , sk).

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: adding a memory

a memory: a set Q of states,

Vn × Q: configuration set,

A consensus protocol:

A memory evolution function τ : Vn × Q → Q,A consensus function f : Vn × Q → Zm,

An execution: sequence (x0, s0)→ (x1, s1)→ (x2, s2)→ ....with

x0 → x1 → x2 → .... is a trajectory,sk+1 = τ(xk , sk),s0 = ⊥, the empty memory state

Each trajectory induces a unique execution,

At each time step: change (or not) of the memory and of theconsensus value f (xk , sk).

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The consensus problem: adding a memory

a memory: a set Q of states,

Vn × Q: configuration set,

A consensus protocol:

A memory evolution function τ : Vn × Q → Q,A consensus function f : Vn × Q → Zm,

An execution: sequence (x0, s0)→ (x1, s1)→ (x2, s2)→ ....with

x0 → x1 → x2 → .... is a trajectory,sk+1 = τ(xk , sk),s0 = ⊥, the empty memory state

Each trajectory induces a unique execution,

At each time step: change (or not) of the memory and of theconsensus value f (xk , sk).

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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Representativity formalization

We fix a threshold t.

The consensus function f is representative when:

f (x , s) = k =⇒ #x(k) > t.

Remark: we need to have:

n > k t,

in order to be able to satisfy the threshold condition in any case.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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Stability formalization, geodesic criterion

A geodoesic: a sequence(xp, sp)→ (xp+1, sp+1)→ ...→ (xp′ , sp′), with

xp → xp+1 → ...→ xp′ is a trajectory,sk+1 = τ(xk , sk),each input component is changed at most once.

Remark: necessarily, p′ − p ≤ n

instability: maximal number of consensus changes in ageodesic

A worst case criterion,

which considers geodesics which may never appear(or with very low probability)and leads to non intuitive results

Not a good criterion. Forget it !!

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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Stability formalization, geodesic criterion

A geodoesic: a sequence(xp, sp)→ (xp+1, sp+1)→ ...→ (xp′ , sp′), with

xp → xp+1 → ...→ xp′ is a trajectory,

sk+1 = τ(xk , sk),each input component is changed at most once.

Remark: necessarily, p′ − p ≤ n

instability: maximal number of consensus changes in ageodesic

A worst case criterion,

which considers geodesics which may never appear(or with very low probability)and leads to non intuitive results

Not a good criterion. Forget it !!

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Stability formalization, geodesic criterion

A geodoesic: a sequence(xp, sp)→ (xp+1, sp+1)→ ...→ (xp′ , sp′), with

xp → xp+1 → ...→ xp′ is a trajectory,sk+1 = τ(xk , sk),

each input component is changed at most once.

Remark: necessarily, p′ − p ≤ n

instability: maximal number of consensus changes in ageodesic

A worst case criterion,

which considers geodesics which may never appear(or with very low probability)and leads to non intuitive results

Not a good criterion. Forget it !!

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Stability formalization, geodesic criterion

A geodoesic: a sequence(xp, sp)→ (xp+1, sp+1)→ ...→ (xp′ , sp′), with

xp → xp+1 → ...→ xp′ is a trajectory,sk+1 = τ(xk , sk),each input component is changed at most once.

Remark: necessarily, p′ − p ≤ n

instability: maximal number of consensus changes in ageodesic

A worst case criterion,

which considers geodesics which may never appear(or with very low probability)and leads to non intuitive results

Not a good criterion. Forget it !!

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Stability formalization, geodesic criterion

A geodoesic: a sequence(xp, sp)→ (xp+1, sp+1)→ ...→ (xp′ , sp′), with

xp → xp+1 → ...→ xp′ is a trajectory,sk+1 = τ(xk , sk),each input component is changed at most once.

Remark: necessarily, p′ − p ≤ n

instability: maximal number of consensus changes in ageodesic

A worst case criterion,

which considers geodesics which may never appear(or with very low probability)and leads to non intuitive results

Not a good criterion. Forget it !!

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Stability formalization, geodesic criterion

A geodoesic: a sequence(xp, sp)→ (xp+1, sp+1)→ ...→ (xp′ , sp′), with

xp → xp+1 → ...→ xp′ is a trajectory,sk+1 = τ(xk , sk),each input component is changed at most once.

Remark: necessarily, p′ − p ≤ n

instability: maximal number of consensus changes in ageodesic

A worst case criterion,

which considers geodesics which may never appear(or with very low probability)and leads to non intuitive results

Not a good criterion. Forget it !!

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Stability formalization, geodesic criterion

A geodoesic: a sequence(xp, sp)→ (xp+1, sp+1)→ ...→ (xp′ , sp′), with

xp → xp+1 → ...→ xp′ is a trajectory,sk+1 = τ(xk , sk),each input component is changed at most once.

Remark: necessarily, p′ − p ≤ n

instability: maximal number of consensus changes in ageodesic

A worst case criterion,

which considers geodesics which may never appear(or with very low probability)and leads to non intuitive results

Not a good criterion. Forget it !!

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Stability formalization, geodesic criterion

A geodoesic: a sequence(xp, sp)→ (xp+1, sp+1)→ ...→ (xp′ , sp′), with

xp → xp+1 → ...→ xp′ is a trajectory,sk+1 = τ(xk , sk),each input component is changed at most once.

Remark: necessarily, p′ − p ≤ n

instability: maximal number of consensus changes in ageodesic

A worst case criterion,

which considers geodesics which may never appear(or with very low probability)

and leads to non intuitive results

Not a good criterion. Forget it !!

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Stability formalization, geodesic criterion

A geodoesic: a sequence(xp, sp)→ (xp+1, sp+1)→ ...→ (xp′ , sp′), with

xp → xp+1 → ...→ xp′ is a trajectory,sk+1 = τ(xk , sk),each input component is changed at most once.

Remark: necessarily, p′ − p ≤ n

instability: maximal number of consensus changes in ageodesic

A worst case criterion,

which considers geodesics which may never appear(or with very low probability)and leads to non intuitive results

Not a good criterion. Forget it !!

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Stability formalization, geodesic criterion

A geodoesic: a sequence(xp, sp)→ (xp+1, sp+1)→ ...→ (xp′ , sp′), with

xp → xp+1 → ...→ xp′ is a trajectory,sk+1 = τ(xk , sk),each input component is changed at most once.

Remark: necessarily, p′ − p ≤ n

instability: maximal number of consensus changes in ageodesic

A worst case criterion,

which considers geodesics which may never appear(or with very low probability)and leads to non intuitive results

Not a good criterion. Forget it !!

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Stability formalization, average criterion

A uniform random walk S0,S1,S2.... on (Vn,En).

Np: the number of consensus changes during the first pexecution steps.

Np = card{k ∈ N | (0 ≤ k < p)∧ (f (Sk ,Xk) 6= f (Sk+1,Xk+1))}

Definition: instability = limp→∞

(E (Np

p))

Proposition: For Q finite, instability is well defined, doesnot depend on the origin distribution of S0, and we have:

instability = E( limp→∞

Np

p)

Proof: Consequence of the Ergodic Theorem and the Lebesgue’s

dominated convergence Theorem.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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Stability formalization, average criterion

A uniform random walk S0,S1,S2.... on (Vn,En).

Np: the number of consensus changes during the first pexecution steps.

Np = card{k ∈ N | (0 ≤ k < p)∧ (f (Sk ,Xk) 6= f (Sk+1,Xk+1))}

Definition: instability = limp→∞

(E (Np

p))

Proposition: For Q finite, instability is well defined, doesnot depend on the origin distribution of S0, and we have:

instability = E( limp→∞

Np

p)

Proof: Consequence of the Ergodic Theorem and the Lebesgue’s

dominated convergence Theorem.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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Stability formalization, average criterion

A uniform random walk S0,S1,S2.... on (Vn,En).

Np: the number of consensus changes during the first pexecution steps.

Np = card{k ∈ N | (0 ≤ k < p)∧ (f (Sk ,Xk) 6= f (Sk+1,Xk+1))}

Definition: instability = limp→∞

(E (Np

p))

Proposition: For Q finite, instability is well defined, doesnot depend on the origin distribution of S0, and we have:

instability = E( limp→∞

Np

p)

Proof: Consequence of the Ergodic Theorem and the Lebesgue’s

dominated convergence Theorem.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Stability formalization, average criterion

A uniform random walk S0,S1,S2.... on (Vn,En).

Np: the number of consensus changes during the first pexecution steps.

Np = card{k ∈ N | (0 ≤ k < p)∧ (f (Sk ,Xk) 6= f (Sk+1,Xk+1))}

Definition: instability = limp→∞

(E (Np

p))

Proposition: For Q finite, instability is well defined, doesnot depend on the origin distribution of S0, and we have:

instability = E( limp→∞

Np

p)

Proof: Consequence of the Ergodic Theorem and the Lebesgue’s

dominated convergence Theorem.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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The result

Problem. Find a protocol (a pair (f , τ))

satisfying representativity for a fixed t,which minimizes the instability.

Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by:- f (x , s) = s if s > t,- f (x , s) = dom(x) otherwise.

The pair (f , f ) is a solution the problem

Interpretation. Optimal protocol:

when it is possible,do not change the consensus value(forced change rule)when a change is forced,change to the dominating value dom(x) of the input x(dominating value rule).only store the current consensus value.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The result

Problem. Find a protocol (a pair (f , τ))

satisfying representativity for a fixed t,

which minimizes the instability.

Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by:- f (x , s) = s if s > t,- f (x , s) = dom(x) otherwise.

The pair (f , f ) is a solution the problem

Interpretation. Optimal protocol:

when it is possible,do not change the consensus value(forced change rule)when a change is forced,change to the dominating value dom(x) of the input x(dominating value rule).only store the current consensus value.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The result

Problem. Find a protocol (a pair (f , τ))

satisfying representativity for a fixed t,which minimizes the instability.

Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by:- f (x , s) = s if s > t,- f (x , s) = dom(x) otherwise.

The pair (f , f ) is a solution the problem

Interpretation. Optimal protocol:

when it is possible,do not change the consensus value(forced change rule)when a change is forced,change to the dominating value dom(x) of the input x(dominating value rule).only store the current consensus value.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The result

Problem. Find a protocol (a pair (f , τ))

satisfying representativity for a fixed t,which minimizes the instability.

Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by:- f (x , s) = s if s > t,- f (x , s) = dom(x) otherwise.

The pair (f , f ) is a solution the problem

Interpretation. Optimal protocol:

when it is possible,do not change the consensus value(forced change rule)when a change is forced,change to the dominating value dom(x) of the input x(dominating value rule).only store the current consensus value.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The result

Problem. Find a protocol (a pair (f , τ))

satisfying representativity for a fixed t,which minimizes the instability.

Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by:- f (x , s) = s if s > t,- f (x , s) = dom(x) otherwise.

The pair (f , f ) is a solution the problem

Interpretation. Optimal protocol:

when it is possible,do not change the consensus value(forced change rule)when a change is forced,change to the dominating value dom(x) of the input x(dominating value rule).only store the current consensus value.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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The result

Problem. Find a protocol (a pair (f , τ))

satisfying representativity for a fixed t,which minimizes the instability.

Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by:- f (x , s) = s if s > t,- f (x , s) = dom(x) otherwise.

The pair (f , f ) is a solution the problem

Interpretation. Optimal protocol:

when it is possible,do not change the consensus value(forced change rule)

when a change is forced,change to the dominating value dom(x) of the input x(dominating value rule).only store the current consensus value.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

The result

Problem. Find a protocol (a pair (f , τ))

satisfying representativity for a fixed t,which minimizes the instability.

Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by:- f (x , s) = s if s > t,- f (x , s) = dom(x) otherwise.

The pair (f , f ) is a solution the problem

Interpretation. Optimal protocol:

when it is possible,do not change the consensus value(forced change rule)when a change is forced,change to the dominating value dom(x) of the input x(dominating value rule).

only store the current consensus value.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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The result

Problem. Find a protocol (a pair (f , τ))

satisfying representativity for a fixed t,which minimizes the instability.

Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by:- f (x , s) = s if s > t,- f (x , s) = dom(x) otherwise.

The pair (f , f ) is a solution the problem

Interpretation. Optimal protocol:

when it is possible,do not change the consensus value(forced change rule)when a change is forced,change to the dominating value dom(x) of the input x(dominating value rule).only store the current consensus value.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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Proof. Main ideas

There exists an optimal protocol with forced changes,by postponing unforced changes method.

What happens after a forced change in (xk , sk)?Consider the part of the execution until the next forcedchange.

Let b be the chosen consensus value in (xk , sk).We have: #xk (b) ≤ #xk (dom(xk))xk

if, at any time, #(b) ≤ #(dom(xk)),then b can be replaced by dom(xk), with no loss of stability.if it happens that #(b) > #(dom(xk)),then it happens before that #(b) = #(dom(xk)) (symmetry).b ≡ dom(xk) for the future stability.

it is never worse to choose dom(xk) than b. QED

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Proof. Main ideas

There exists an optimal protocol with forced changes,by postponing unforced changes method.

What happens after a forced change in (xk , sk)?Consider the part of the execution until the next forcedchange.

Let b be the chosen consensus value in (xk , sk).We have: #xk (b) ≤ #xk (dom(xk))xk

if, at any time, #(b) ≤ #(dom(xk)),then b can be replaced by dom(xk), with no loss of stability.if it happens that #(b) > #(dom(xk)),then it happens before that #(b) = #(dom(xk)) (symmetry).b ≡ dom(xk) for the future stability.

it is never worse to choose dom(xk) than b. QED

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Proof. Main ideas

There exists an optimal protocol with forced changes,by postponing unforced changes method.

What happens after a forced change in (xk , sk)?Consider the part of the execution until the next forcedchange.

Let b be the chosen consensus value in (xk , sk).We have: #xk (b) ≤ #xk (dom(xk))xk

if, at any time, #(b) ≤ #(dom(xk)),then b can be replaced by dom(xk), with no loss of stability.if it happens that #(b) > #(dom(xk)),then it happens before that #(b) = #(dom(xk)) (symmetry).b ≡ dom(xk) for the future stability.

it is never worse to choose dom(xk) than b. QED

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Proof. Main ideas

There exists an optimal protocol with forced changes,by postponing unforced changes method.

What happens after a forced change in (xk , sk)?Consider the part of the execution until the next forcedchange.

Let b be the chosen consensus value in (xk , sk).We have: #xk (b) ≤ #xk (dom(xk))xk

if, at any time, #(b) ≤ #(dom(xk)),then b can be replaced by dom(xk), with no loss of stability.

if it happens that #(b) > #(dom(xk)),then it happens before that #(b) = #(dom(xk)) (symmetry).b ≡ dom(xk) for the future stability.

it is never worse to choose dom(xk) than b. QED

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Proof. Main ideas

There exists an optimal protocol with forced changes,by postponing unforced changes method.

What happens after a forced change in (xk , sk)?Consider the part of the execution until the next forcedchange.

Let b be the chosen consensus value in (xk , sk).We have: #xk (b) ≤ #xk (dom(xk))xk

if, at any time, #(b) ≤ #(dom(xk)),then b can be replaced by dom(xk), with no loss of stability.if it happens that #(b) > #(dom(xk)),then it happens before that #(b) = #(dom(xk)) (symmetry).b ≡ dom(xk) for the future stability.

it is never worse to choose dom(xk) than b. QED

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

Proof. Main ideas

There exists an optimal protocol with forced changes,by postponing unforced changes method.

What happens after a forced change in (xk , sk)?Consider the part of the execution until the next forcedchange.

Let b be the chosen consensus value in (xk , sk).We have: #xk (b) ≤ #xk (dom(xk))xk

if, at any time, #(b) ≤ #(dom(xk)),then b can be replaced by dom(xk), with no loss of stability.if it happens that #(b) > #(dom(xk)),then it happens before that #(b) = #(dom(xk)) (symmetry).b ≡ dom(xk) for the future stability.

it is never worse to choose dom(xk) than b. QED

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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But .... we have cheated.

If it happens that #(b) > #(dom(xk)),

then it happens that #(b) = #(dom(xk))± 1

The patch: put an artificial intermediary state on each edge.

The given protocol is optimal in the modified process(even for a finite fixed time).

It can be deduced thatthe protocol is asymptotically optimal for the original process.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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But .... we have cheated.

If it happens that #(b) > #(dom(xk)),

then it happens that #(b) = #(dom(xk))± 1

The patch: put an artificial intermediary state on each edge.

The given protocol is optimal in the modified process(even for a finite fixed time).

It can be deduced thatthe protocol is asymptotically optimal for the original process.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

But .... we have cheated.

If it happens that #(b) > #(dom(xk)),

then it happens that #(b) = #(dom(xk))± 1

The patch: put an artificial intermediary state on each edge.

The given protocol is optimal in the modified process(even for a finite fixed time).

It can be deduced thatthe protocol is asymptotically optimal for the original process.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

university-logo

But .... we have cheated.

If it happens that #(b) > #(dom(xk)),

then it happens that #(b) = #(dom(xk))± 1

The patch: put an artificial intermediary state on each edge.

The given protocol is optimal in the modified process(even for a finite fixed time).

It can be deduced thatthe protocol is asymptotically optimal for the original process.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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Conclusion

We have exhibited an optimal protocol for the averageconsensus stability.

This protocol only uses a small memory: Zm.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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Conclusion

We have exhibited an optimal protocol for the averageconsensus stability.

This protocol only uses a small memory: Zm.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)

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The end

Thanks.

Eric Remila CSR, 2011 June, St-Petersburg (Russia)