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Introduction to the Gathering Problem

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RES/SEC 1

Introduction to the

Gathering Problem

(2)

Gathering  -­‐  Unlimited  Visibility  

Initially the robots are in arbitrary distinct positions.

(3)

RES/SEC 3

Gathering  

Unlimited  Visibility  -­‐  SYm  

  Istantaneous activities

  n=2, the problem is unsolvable

Ando, Oasa, Suzuki, Yamashita

Siam Journal Of Computing, 1999

(4)

Gathering,  n=2  

Unlimited  Visibility  -­‐  SYm  

In fact, since the robots have no dimension…

…moving them towards each other is not useful…

…but it works if they can bump!

…and cannot bump...

(5)

RES/SEC 5

Gathering  

Unlimited  Visibility  -­‐  SYm  

  Istantaneous activities

  n=2, the problem is unsolvable

  n>2, they provide an oblivious algorithm that let the robots gather in finite time

Ando, Oasa, Suzuki, Yamashita

Siam Journal Of Computing, 1999

(6)

Gathering,  ASYNC  

•  In  spite  of  its  apparent  simplicity,  this  problem   has  been  tackled  in  several  studies  

•  In  fact,  several  factors  render  this  problem   difficult  to  solve    

– Major  problems  arise  from  symmetric   configuraHons....  

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DifficulHes  

RES/SEC 7

If at the beginning....

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DifficulHes  

Not symmetric....

General Algorithm!!

If at the beginning....

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DifficulHes  

RES/SEC 9

....after a while....

Impossible d'afficher l'image. Votre ordinateur manque peut-être de mémoire pour ouvrir l'image ou l'image est endommagée.

Redémarrez l'ordinateur, Impossible d'afficher l'image. Votre ordinateur manque peut- être de mémoire pour

(10)

DifficulHes  

Symmetric…

go to c General Algorithm…

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Gathering—easy  soluHon  

RES/SEC 11

Easy Solution: Weber Point (Weiszfeld, ‘36)!

ℜ

∑

∈

=

i p i

r p

WP arg min dist( , )

2

Given r1,…,rn:

1. It is unique (Weiszfeld, ‘36)

WP

(12)

Gathering—easy  soluHon  

2. WP is Weber Point of points on [ri,WP] (Weiszfeld, ‘36)

r4

r2 r1

r5

r

1. It is unique (Weiszfeld, ‘36)

Easy Solution: Weber Point (Weiszfeld, ‘36)!

ℜ

∑

∈

=

i p i

r p

WP arg min dist( , )

2

Given r1,…,rn:

WP

(13)

Gathering—easy  soluHon  

RES/SEC 13

r4

r2 r1

r5

r3

2. WP is Weber Point of points on [ri,WP] (Weiszfeld, ‘36)

1. It is unique (Weiszfeld, ‘36)

Easy Solution: Weber Point (Weiszfeld, ‘36)!

ℜ

∑

∈

=

i p i

r p

WP arg min dist( , )

2

Given r1,…,rn:

WP

(14)

Gathering—easy  soluHon  

r4

r2 r1

r5

r

2. WP is Weber Point of points on [ri,WP] (Weiszfeld, ‘36)

1. It is unique (Weiszfeld, ‘36)

Easy Solution: Weber Point (Weiszfeld, ‘36)!

ℜ

∑

∈

=

i p i

r p

WP arg min dist( , )

2

Given r1,…,rn:

WP

(15)

Gathering—easy  soluHon  

RES/SEC 15

2. WP is Weber Point of points on [ri,WP] (Weiszfeld, ‘36)

1. It is unique (Weiszfeld, ‘36)

Easy Solution: Weber Point (Weiszfeld, ‘36)!

ℜ

∑

∈

=

i p i

r p

WP arg min dist( , )

2

Given r1,…,rn:

WP

(16)

Gathering—easy  soluHon  

Easy Solution: Weber Point (Weiszfeld, ‘36)!

Algorithm:

1. Compute WP

2. Move Towards WP

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(18)
(19)

Gathering  

(Unlimited  Visibility,  no  agreement)

 

RES/SEC 19

(20)

Gathering  

(Unlimited  Visibility,  no  agreement)

 

(21)

Gathering,  n=3  

(Unlimited  Visibility,  no  agreement)

 

RES/SEC 21

r2

r1 r3

p p

r2

r3 r1

p

≥120° p

(22)

General  Schema  

•  Use  of  MulHplicity  DetecHon  

– n=3,4  (and  even  with  the  use  of  Weber  Point)  

Is  there  1  or  more  than  1  robot  at  a  point?  

>1

>1

>1

(23)

General  Schema  

RES/SEC 23

1. At the beginning, robots on distinct positions 2. Get a scenario where there is only one point p with multiplicity greater than one

3. All robots move towards p

(24)

MulHplicity  DetecHon  

p

P’

p

P’

(25)

MulHplicity  DetecHon  

RES/SEC 25

For n=2, the problem is not solvable (Suzuki et al., 1999)!

p

P’

It is possible to design an adversary that lets the robots occupy two distinct positions on the plane in a finite number of cycles…

(26)

MulHplicity  DetecHon  

Problem not solvable with n=2

No multiplicity detection

(27)

Let’s  design  the  adversary    

•  We  divide  the  robots  in  two  sets  

– n-­‐1  robots  (r1,  ...,  rn-­‐1)  are  the  blue  robots  

– There  is  one  red  robot  (rn)  (same  direcHon  but   opposite  orientaHon)  

RES/SEC 27

(28)

E-­‐configuraHons  

E1-configuration

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E-­‐configuraHons  

RES/SEC 29

(30)

E-­‐configuraHons  

(31)

E-­‐configuraHons  

RES/SEC 31

Time: 1

(32)

E-­‐configuraHons  

(33)

E-­‐configuraHons  

RES/SEC 33

Time: ....

(34)

E-­‐configuraHons  

(35)

E-­‐configuraHons  

RES/SEC 35

Time: t

(36)

E-­‐configuraHons  

In the example: E2-conf

(37)

E1-­‐configuraHon  

RES/SEC 37

(38)

E1-­‐configuraHon  

(39)

E1-­‐configuraHon  

RES/SEC 39

Therefore, for a while the configuration stays E1

(40)

E1-­‐configuraHon  

(41)

E1-­‐configuraHon  

RES/SEC 41

(42)

E1-­‐configuraHon  

(43)

E1-­‐configuraHon  

RES/SEC 43

(44)

E1-­‐configuraHon  

(45)

E1-­‐configuraHon  

RES/SEC 45

(46)

E1-­‐configuraHon  

Again in a E1

(47)

E2-­‐configuraHon  

RES/SEC 47

(48)

Finally....  

(49)

Gathering  

RES/SEC 49

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