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OPODIS 2018, Hong Kong, 2018/12/19 Optimal Rendezvous -Algorithms for Two Asynchronous Mobile Robots with External-Lights Takashi OKUMURA Koichi WADA Xavier DFAGO Hosei University Hosei University Tokyo Institute of Technology Japan

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SLIDE 1

December 2018

OPODIS 2018, Hong Kong, 2018/12/19

Optimal Rendezvous π“œ-Algorithms for Two Asynchronous Mobile Robots with External-Lights

Xavier DÉFAGO Tokyo Institute of Technology Japan Koichi WADA Hosei University Japan Takashi OKUMURA Hosei University Japan

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Optimal Rendezvous π“œ-Algorithms for Two Asynchronous Mobile Robots with External-Lights

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Rendezvous π“œ-Algorithms External-Lights

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Autonomous Mobile Robots

4

not these robots!

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Autonomous Mobile Robots

Theoretical model

  • Suzuki and Yamashita’s seminal work

Distributed anonymous mobile robots, by I. Suzuki and M. Yamashita, SIAM J. Computing, 28(4): 1347-1363(1999)

Coordination task by Mobile Robots

  • Rendezvous, Gathering, Convergence, Formation ...

Rendezvous

  • Reach same location in finite steps

Question

  • β€œpower of lights" and additional assumptions


to solve Rendezvous

5

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Outline

Model(s)
 Related Work
 Our Results
 Conclusion


6

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SLIDE 7

Model(s)

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Autonomous Mobile Robots (Basic model)

Robot: Point on an infinite 2D-space

  • No global coordinate system

(Local only)

  • Anonymous (No distinguished ID)
  • Oblivious (No memory)
  • Deterministic
  • Uniform (Identical algorithm)
  • No communication (Observe the environment) 





  • With lights (more later)

8

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Execution: Look-Compute-Move

Look

  • Take a snapshot of all robots' current locations (in terms of LCS)

Compute

  • Deciding the next position and color

Move

  • Move to the next position

9

L

Look Snapshot !

time robot

C

Compute Change color

M

Move

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SLIDE 10

T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Scheduler

Centralized

  • LCM atomic; 1 robot at a time

FSYNC

  • LCM atomic; all robots together

SSYNC

  • LCM atomic; subset of robots

ASYNC

  • no bounds on delays/durations

LC-Atomic ASYNC

  • LC atomic

10

L C M L C M L C M L C M L C M L C M L C M L C M

F S y n c

L C M L C M L C M L C M L C M

S S y n c

L C M C M L C M L C M L C M L

A S y n c

L C M

LCM

L C M L C M L C M

C e n t r .

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Move toHalf (midpoint)
 FSYNC execution

  • Rendezvous SOLVED! 





Centralized execution

  • Convergence achieved
  • Rendezvous NOT SOLVED

Difficulty of Rendezvous

11

[20]

  • I. Suzuki, M. Yamashita. Distributed anonymous mobile robots. SIAM J. Comput., 28(4):1347–1363, 1999.
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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Difficulty of Rendezvous

Move to Other
 Centralized execution

  • Rendezvous SOLVED! 





FSYNC execution

  • Swap places forever
  • Rendezvous NOT SOLVED

12

=> requires a Stay move

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SLIDE 13

T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Light Models

#Colors

  • log(|L|) bits of information

Full light

  • can observe: own and others' color

Internal light (Fstate)

  • can observe: own color only
  • basically log(|L|) bits register

External light (Fcomm)

  • can observe: others' color only

13

[4]

  • S. Das, P. Flocchini, G. Prencipe, N. Santoro, and M. Yamashita. Autonomous mobile robots with lights. Theor. Comput. Sci., 609:171–184, 2016

[10]

  • P. Flocchini, N. Santoro, G. Viglietta, and M. Yamashita. Rendezvous with constant memory. Theor. Comput. Sci., 621(C):57–72, 2016.

(β„“(me), β„“(other)) β„“(me) β„“(other) β„“ ∈ L

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Example: SSYNC, Full(2)

14

  • ther is Black β‡’
  • ther is White β‡’
  • ther is Black:

β‡’

  • ther is White

B A

[21]

  • G. Viglietta. Rendezvous of two robots with visible bits. In Proc. 9th ALGOSENSORS, pp. 291–306, 2014.
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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Movement Restriction

Rigid

  • Robots always reach the destination

Non-rigid

  • may stop before reaching the destination
  • guarantee to move by at least Ξ΄ (for some unknown Ξ΄>0)

Non-rigid with Ξ΄

  • robots know the value of Ξ΄

15

destination Movement is Rigid Movement is Non-Rigid destination

Ξ΄

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Algorithm Properties

Self-Stabilizing

  • arbitrary initial configurations


Quasi Self-Stabilizing

  • robots start with the same arbitrary color (whichever).


non QSS

  • robots start with some specific colors.

16

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Related Work

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

18

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – – (𝛆) rigid LC-atomic ASYNC – – (𝛆) rigid ASYNC – – (𝛆) rigid

non Rigid with Ξ΄ non Rigid self-stabilizing quasi not at all

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

19

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ – (𝛆) rigid LC-atomic ASYNC – – (𝛆) rigid ASYNC – – (𝛆) rigid

[21]

  • G. Viglietta. Rendezvous of two robots with visible bits. In Proc. 9th ALGOSENSORS, pp. 291–306, 2014.
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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Algorithm class π“œ

Algorithm
 
 Destination point
 
 Examples

  • toOther
  • toHalf
  • Stay

20

destination = (1 βˆ’ Ξ») β‹… me.pos + Ξ» β‹… other.pos (Ξ» = 1) (Ξ» = 0.5) (Ξ» = 0) (β„“(me), β„“(other)) ↦ Ξ»

  • bservation

destination

colors βˆˆβ„

me

  • ther

β•³ destination

[21]

  • G. Viglietta. Rendezvous of two robots with visible bit. In Proc. 9th ALGOSENSORS, pp. 291–306, 2014.
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SLIDE 21

T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

21

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ – (𝛆) rigid LC-atomic ASYNC – – (𝛆) rigid ASYNC – – (𝛆) rigid

[21]

  • G. Viglietta. Rendezvous of two robots with visible bit. In Proc. 9th ALGOSENSORS, pp. 291–306, 2014.
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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

22

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ – (𝛆) rigid LC-atomic ASYNC – – (𝛆) rigid ASYNC – (3,3)π“œ – (𝛆) rigid

[21]

  • G. Viglietta. Rendezvous of two robots with visible bit. In Proc. 9th ALGOSENSORS, pp. 291–306, 2014.

(lower bound, upper bound)

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

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Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ – (𝛆) rigid LC-atomic ASYNC – 2π“œ – (𝛆) rigid ASYNC – (3,3)π“œ – (𝛆) rigid 2π“œ

[21]

  • G. Viglietta. Rendezvous of two robots with visible bit. In Proc. 9th ALGOSENSORS, pp. 291–306, 2014.

[17]

  • T. Okumura, K. Wada, Y. Katayama. Optimal asynchronous rendezvous for mobile robots with lights, In Proc. 19th SSS, Nov. 2017.
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SLIDE 24

T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work: not π“œ

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Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – – (𝛆) rigid LC-atomic ASYNC – – (𝛆) rigid ASYNC – 2 – (𝛆) rigid

[11]

  • A. Heriban, X. DΓ©fago, S. Tixeuil. Optimally gathering two robots, In Proc. 19th ICDCN, Jan. 2018.

not class π“œ uses position info:
 distinct vs. gathered

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

25

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ – (𝛆) rigid LC-atomic ASYNC – 2π“œ – (𝛆) rigid ASYNC – (3,3)π“œ – (𝛆) rigid 2π“œ

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SLIDE 26

External Lights

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

27

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ – (𝛆) rigid LC-atomic ASYNC – 2π“œ – (𝛆) rigid ASYNC – (3,3)π“œ – (𝛆) rigid 2π“œ

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

28

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid LC-atomic ASYNC – 2π“œ – (𝛆) rigid ASYNC – (3,3)π“œ – (𝛆) rigid 2π“œ

[10]

  • P. Flocchini, N. Santoro, G. Viglietta, and M. Yamashita. Rendezvous with constant memory. Theor. Comput. Sci., 621(C):57–72, March 2016.
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SLIDE 29

T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

29

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid LC-atomic ASYNC – 2π“œ – (𝛆) rigid ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

[10]

  • P. Flocchini, N. Santoro, G. Viglietta, and M. Yamashita. Rendezvous with constant memory. Theor. Comput. Sci., 621(C):57–72, March 2016.
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SLIDE 30

T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

30

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid LC-atomic ASYNC – 2π“œ – (𝛆) rigid ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) 12 rigid 2π“œ 3

[10]

  • P. Flocchini, N. Santoro, G. Viglietta, and M. Yamashita. Rendezvous with constant memory. Theor. Comput. Sci., 621(C):57–72, March 2016.
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SLIDE 31

T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

31

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid LC-atomic ASYNC – 2π“œ – (𝛆) rigid ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

[10]

  • P. Flocchini, N. Santoro, G. Viglietta, and M. Yamashita. Rendezvous with constant memory. Theor. Comput. Sci., 621(C):57–72, March 2016.
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SLIDE 32

T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

32

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid LC-atomic ASYNC – 2π“œ – (𝛆) rigid ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

?

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SLIDE 33

Our Results

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Our Results

34

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid LC-atomic ASYNC – 2π“œ – (𝛆) rigid ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

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SLIDE 35

T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Lower Bounds

35

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid (β‰₯3,_)π“œ LC-atomic ASYNC – 2π“œ – (𝛆) rigid ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

(lower bound, upper bound)

Theorem 10

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Lower Bounds

36

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid (β‰₯3,_)π“œ LC-atomic ASYNC – 2π“œ – (𝛆) rigid ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

(lower bound, upper bound)

Theorem 10 Necessary movements FSYNC: toHalf + Stay Centralized: toOther 3 distinct movements

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Lower Bounds

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Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid (β‰₯3,_)π“œ LC-atomic ASYNC – 2π“œ – (𝛆) (>3,_)π“œ rigid (>3,_)π“œ ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

(lower bound, upper bound)

Theorem 13

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Lower Bounds

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Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid (β‰₯3,_)π“œ LC-atomic ASYNC – 2π“œ – (𝛆) (>3,_)π“œ rigid (>3,_)π“œ ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

(lower bound, upper bound)

Theorem 13

r s

t1

LC

t2

LC

t3 t4

move move

r s

LC

t1

LC

t2

move move

alt sim

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SLIDE 39

T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Lower Bounds

39

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid (β‰₯3,_)π“œ LC-atomic ASYNC – 2π“œ – (𝛆) (>3,_)π“œ rigid (>3,_)π“œ (>4,_)π“œ ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

(lower bound, upper bound)

Theorem 15

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Upper Bounds

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Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid (β‰₯3,_)π“œ LC-atomic ASYNC – 2π“œ – (𝛆) (>3,_)π“œ rigid 3π“œ (>3,_)π“œ (>4,_)π“œ ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

(lower bound, upper bound)

Theorem 8

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Upper Bounds

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Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid (β‰₯3,_)π“œ LC-atomic ASYNC – 2π“œ – (𝛆) (>3,_)π“œ rigid (3,3)π“œ (>3,_)π“œ (>4,_)π“œ ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

(lower bound, upper bound)

Theorem 8

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Upper Bounds

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Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid (β‰₯3,_)π“œ LC-atomic ASYNC – 2π“œ 4π“œ – (𝛆) (>3,_)π“œ rigid (3,3)π“œ (>3,_)π“œ (>4,_)π“œ ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

(lower bound, upper bound)

Theorem 24

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Upper Bounds

43

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid (β‰₯3,_)π“œ LC-atomic ASYNC – 2π“œ (4,4)π“œ – (𝛆) rigid (3,3)π“œ (>4,_)π“œ ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

(lower bound, upper bound)

Theorem 24

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Upper Bounds

44

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid (β‰₯3,_)π“œ LC-atomic ASYNC – 2π“œ (4,4)π“œ 5π“œ – (𝛆) rigid (3,3)π“œ (>4,_)π“œ ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

(lower bound, upper bound)

Theorem 34

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Upper Bounds

45

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid (β‰₯3,_)π“œ LC-atomic ASYNC – 2π“œ (4,4)π“œ (5,5)π“œ – (𝛆) rigid (3,3)π“œ ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

(lower bound, upper bound)

Theorem 34

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SLIDE 46

Conclusion

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T.Okumura, K.Wada, X.DΓ©fago OPODIS 2018, Hong Kong, 2018/12/19

Our Results

47

Scheduler Mvmt full external not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – SSYNC – 2π“œ 3π“œ – (𝛆) rigid LC-atomic ASYNC – 2π“œ (4,4)π“œ (5,5)π“œ – (𝛆) rigid (3,3)π“œ ASYNC – (3,3)π“œ βˆžπ“œ – (𝛆) rigid 2π“œ

(lower bound, upper bound)

(β‰₯3,_)π“œ