Leader Election in Swarms of Deterministic Robots

Mika ¨el Rabie & Franck Petit

LiP6, Sorbonne Universit ´e

00 11

Question

00 11

Question

SoD and Chirality

[Flocchini et al., 1999]

A solution exists for any n.

SoD and No Chirality

[Flocchini et al., 2001]

A solution exists if n is odd.

SoD and Chirality

[Flocchini et al., 1999]

A solution exists for any n.

North

SoD and No Chirality

[Flocchini et al., 2001]

A solution exists if n is odd.

SoD and Chirality

[Flocchini et al., 1999]

A solution exists for any n.

North

SoD and No Chirality

[Flocchini et al., 2001]

A solution exists if n is odd.

SoD and Chirality

[Flocchini et al., 1999]

A solution exists for any n.

North

SoD and No Chirality

[Flocchini et al., 2001]

A solution exists if n is odd.

North

SoD and Chirality

[Flocchini et al., 1999]

A solution exists for any n.

North

SoD and No Chirality

[Flocchini et al., 2001]

A solution exists if n is odd.

North

No SoD

[Prencipe 2002]

Impossible in general.

No SoD

[Prencipe 2002]

Impossible in general.

In such a configuration, it is not possible to break the symmetry.

Question

Assuming no sense of direction (with or without chirality), what are the geometric conditions to be able to deterministically agree on a single robot?

To answer to this question, we need tools from the theory on

Combinatoric on Words, specifically Lyndon Words.

Question

Assuming no sense of direction (with or without chirality), what are the geometric conditions to be able to deterministically agree on a single robot?

To answer to this question, we need tools from the theory on

Combinatoric on Words, specifically Lyndon Words.

Definition (Word)

Let A = {a ₀ , a ₁ , . . . , a _n } be an alphabet. A word is a (possibly empty) sequence of letters in A.

A = {a, b, c, d }

abcc a dddddddd ≡ d ⁸

Definition (Word)

Let A = {a ₀ , a ₁ , . . . , a _n } be an alphabet. A word is a (possibly empty) sequence of letters in A.

A = {a, b, c, d }

abcc a dddddddd ≡ d ⁸

Definition (Concatenation)

Let u = a ₁ , . . . , a _i , . . . , a _k and v = b ₁ , . . . , b _j , . . . , b _` .

The concatenation of u and v , denoted uv , is equal to the word a ₁ , . . . , a _i , . . . , a _k , b ₁ , . . . , b _j , . . . , b _` .

u = UP, v = MC, uv = UPMC

Definition (Concatenation)

Let u = a ₁ , . . . , a _i , . . . , a _k and v = b ₁ , . . . , b _j , . . . , b _` .

The concatenation of u and v , denoted uv , is equal to the word a ₁ , . . . , a _i , . . . , a _k , b ₁ , . . . , b _j , . . . , b _` .

u = UP, v = MC, uv = UPMC

Definition (Lexicographic Order)

Let A be an alphabet totally ordered by ≺, i.e., a ₀ ≺ a ₁ ≺ . . . ≺ a n .

A word u = a ₀ a ₁ . . . a _s is said to be lexicographically smaller than or equal to a word v = b ₀ b ₁ . . . b _t , denoted by u v , iff:

either u is a prefix of v ,

or, ∃k : ∀i ∈ [1, . . . , k − 1], a _i = b _i and a _k ≺ b _k .

ab abc abc abc abc abc def

Definition (Lexicographic Order)

Let A be an alphabet totally ordered by ≺, i.e., a ₀ ≺ a ₁ ≺ . . . ≺ a n .

A word u = a ₀ a ₁ . . . a _s is said to be lexicographically smaller than or equal to a word v = b ₀ b ₁ . . . b _t , denoted by u v , iff:

either u is a prefix of v ,

or, ∃k : ∀i ∈ [1, . . . , k − 1], a _i = b _i and a _k ≺ b _k .

ab abc abc abc abc abc def

Definition (Primitive Word)

A word u is said to be primitive iff u = v ^k ⇒ k = 1. Otherwise, u is said to be periodic.

Primitive Words

ab dabcbc dcba

Periodic Words

d ⁸ bcbc

Definition (Primitive Word)

A word u is said to be primitive iff u = v ^k ⇒ k = 1. Otherwise, u is said to be periodic.

Primitive Words

ab dabcbc dcba

Periodic Words

d ⁸ bcbc

Definition (Rotation)

A word u is said to be a rotation of a word v iff there exists two words x, y such that u = xy and v = yx.

u = abcd and v = cd ab u = abcd and v = bcd a

Definition (Minimality)

A word u is said to be a minimal iff u is lexicographically smaller

than any of its rotations.

Definition (Rotation)

A word u is said to be a rotation of a word v iff there exists two words x, y such that u = xy and v = yx.

u = abcd and v = cd ab u = abcd and v = bcd a Definition (Minimality)

A word u is said to be a minimal iff u is lexicographically smaller

than any of its rotations.

Definition (Rotation)

A word u is said to be a rotation of a word v iff there exists two words x, y such that u = xy and v = yx.

u = abcd and v = cd ab u = abcd and v = bcd a Definition (Minimality)

A word u is said to be a minimal iff u is lexicographically smaller

than any of its rotations.

Definition (Lyndon Word)

A word u is a Lyndon word iff u is primitive and minimal.

Lyndon Word

abc (abc cab and abc bca)

Not a Lyndon Word

bca (bca abc)

Definition (Lyndon Word)

A word u is a Lyndon word iff u is primitive and minimal.

Lyndon Word

abc (abc cab and abc bca)

Not a Lyndon Word

bca (bca abc)

O

c ^a b

d

d e

f

O d

d d

d

α

b d

d e

f d d d

d c

β

β

O

α γ

a

ρ

b d

d e

f d d d

d c

β

β O

α ρ₂ γ

α

1

a

W (ρ

) = (abd , β)(d

, α)

(d , α)(f, β)(ec, γ) W (ρ

) = (d

, α)

(d , α)(f, β)(ec, γ)(abc, β)

Lemma

If two distinct radii ρ 1 and ρ 2 exist such that both W (ρ ₁ ) and W(ρ ₂ ) are Lyn- don words, then ∀ρ, W (ρ) = (0, 0).

Lemma (⇒)

If there exists a radius ρ such that

ρ

b d

d e

f d d d

d c

β

β O

α ρ₂ γ

α

1

a

W (ρ

) = (abd , β)(d

, α)

(d , α)(f, β)(ec, γ) W (ρ

) = (d

, α)

(d , α)(f, β)(ec, γ)(abc, β)

Lemma

If two distinct radii ρ 1 and ρ 2 exist such that both W (ρ ₁ ) and W(ρ ₂ ) are Lyn- don words, then ∀ρ, W (ρ) = (0, 0).

Lemma (⇒)

If there exists a radius ρ such that

ρ

b d

d e

f d d d

d c

β

β O

α ρ₂ γ

α

1

a

W (ρ

) = (abd , β)(d

, α)

(d , α)(f, β)(ec, γ) W (ρ

) = (d

, α)

(d , α)(f, β)(ec, γ)(abc, β)

Lemma

If two distinct radii ρ 1 and ρ 2 exist such that both W (ρ ₁ ) and W(ρ ₂ ) are Lyn- don words, then ∀ρ, W (ρ) = (0, 0).

Lemma (⇒)

If there exists a radius ρ such that

Lemma (⇐)

If there exists no radius ρ such that W (ρ) is a Lyndon word, then the robots are not able to deterministically agree on the same leader.

Property

[Lothaire 1983]

If no rotation of a word u is a

Lyndon word, then u is periodic.

Lemma (⇐)

If there exists no radius ρ such that W (ρ) is a Lyndon word, then the robots are not able to deterministically agree on the same leader.

Property

[Lothaire 1983]

If no rotation of a word u is a

Lyndon word, then u is periodic.

Lemma (⇐)

If there exists no radius ρ such that W (ρ) is a Lyndon word, then the robots are not able to deterministically agree on the same leader.

Property

[Lothaire 1983]

If no rotation of a word u is a

x

x

x x

α α

α α

x

b a a

d

b

d

Theorem

Assuming chirality, a swarm of robots is able to deterministically

agree on the same leader if and only if there exists a radius ρ

such that W (ρ) is a Lyndon word.

c

β

β O

α α

ρ₂ γ

α ρ1

a b

c

d e

f d d d

d

c

β

β O

α α

ρ₂ γ

α ρ1

a b

c

d e

f d d d

d

For each ρ, there are 2 ways to com- pute W (ρ)

W (ρ

) = either

(abc, β)(d

, α)

(d, α)(f , β)(ec, γ) or

(abc, γ)(ec, β)(f , α)(d, α)(d

, α)(d

, β)

depending on either or , respectively.

β

α

O a a

b b

r r₂

3

β

β

α

O a a

b b

r r₂

3

β The word

W (ρ

)

= W (ρ

) = (ab, α)(ab, β)(c, β) is a

Lyndon word.

β

α

O a a

b b

r r₂

3

β

Definition (Type of Symmetry) A radius ρ _i is of Type (of symmetry) 0 if there exists no radius ρ _j such that W (ρ _i ) = W (ρ _j ) . Otherwise, ρ _i is said to be of Type 1.

A radius of Type t is said to be t- symmetric.

ρ 1 is 0-symmetric.

ρ 2 and ρ 3 are 1-symmetric.

β

α

O a a

b b

r r₂

3

β

Definition (Type of Symmetry) A radius ρ _i is of Type (of symmetry) 0 if there exists no radius ρ _j such that W (ρ _i ) = W (ρ _j ) . Otherwise, ρ _i is said to be of Type 1.

A radius of Type t is said to be t- symmetric.

ρ 1 is 0-symmetric.

ρ 2 and ρ 3 are 1-symmetric.

c

β

β O

α α

ρ₂ γ

α ρ1

a b

c

d e

f d d d

d

For each radius ρ _i , every robot com-

putes W (ρ _i ) and W (ρ _i ) of the form

(type, radiusword , angle).

c

β

β O

α α

ρ₂ γ

α ρ1

a b

c

d e

f d d d

d

For each radius ρ _i , every robot com- putes W (ρ _i ) and W (ρ _i ) of the form (type, radiusword , angle).

W(ρ )

= (0,abc, β)(0, d

, α)

(0, d, α)(0, f , β)(0,ec, γ)

c

β

β O

α α

ρ₂ γ

α ρ1

a b

c

d e

f d d d

d

Lemma

If two distinct radii ρ ₁ and ρ ₂ exist such that both W (ρ 1 ) and W(ρ ₂ ) are Lyn- don words, then ∀ρ, W (ρ) = (0, 0, 0).

Lemma

If there exists a pair of radii {ρ ₁ , ρ ₂ }

so that W (ρ _i ) or W (ρ _i ) is a Lyndon

word (i ∈ {1, 2}), then the robots are

able to deterministically agree on the

c

β

β O

α α

ρ₂ γ

α ρ1

a b

c

d e

f d d d

d

Lemma

If two distinct radii ρ ₁ and ρ ₂ exist such that both W (ρ 1 ) and W(ρ ₂ ) are Lyn- don words, then ∀ρ, W (ρ) = (0, 0, 0).

Lemma

If there exists a pair of radii {ρ ₁ , ρ ₂ }

so that W (ρ _i ) or W (ρ _i ) is a Lyndon

word (i ∈ {1, 2}), then the robots are

able to deterministically agree on the

x x

x x

α

β

γ β

d

d d

d

Lemma

If two distinct radii ρ ₁ and ρ ₂ exist such that both W (ρ 1 ) and W(ρ ₂ ) are Lyn- don words, then ∀ρ, W (ρ) = (0, 0, 0).

Lemma

If there exists a pair of radii {ρ ₁ , ρ ₂ }

so that W (ρ _i ) or W (ρ _i ) is a Lyndon

word (i ∈ {1, 2}), then the robots are

able to deterministically agree on the

β

α c r

1

O a a

b b

r r₂

3

β

Lemma

If two distinct radii ρ ₁ and ρ ₂ exist such that both W (ρ 1 ) and W(ρ ₂ ) are Lyn- don words, then ∀ρ, W (ρ) = (0, 0, 0).

Lemma

If there exists a pair of radii {ρ ₁ , ρ ₂ }

so that W (ρ _i ) or W (ρ _i ) is a Lyndon

word (i ∈ {1, 2}), then the robots are

able to deterministically agree on the

Theorem

Assuming no chirality, a swarm of robots is able to deterministi-

cally agree on the same leader if and only if there exists a radius

ρ such that W (ρ) is a 0-symmetric Lyndon word.