Optimal Rendezvous -Algorithms for Two Asynchronous Mobile Robots - PowerPoint PPT Presentation

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

Optimal Rendezvous 𝓜 -Algorithms for Two Asynchronous Mobile Robots with External-Lights

Rendezvous External-Lights 𝓜 -Algorithms

Autonomous Mobile Robots not these robots! 4 T.Okumura, K.Wada, X.Défago 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 ‣ “power of lights" and additional assumptions 
 to solve Rendezvous 5 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Outline Model(s) 
 Related Work 
 Our Results 
 Conclusion 
 6 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Model(s)


 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 T.Okumura, K.Wada, X.Défago 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 Change color ‣ Move to the next position Snapshot ! time robot L C M Look Compute Move 9 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Scheduler LCM . r Centralized t L C M L C M n e ‣ LCM atomic; 1 robot at a time C L C M L C M FSYNC c n L C M L C M L C M L C M y ‣ LCM atomic; all robots together S F L C M L C M L C M L C M SSYNC c ‣ LCM atomic; subset of robots n L C M L C M L C M y S S L C M L C M ASYNC ‣ no bounds on delays/durations c n L M C M L C y LC-Atomic ASYNC S A C L M L C M L M C ‣ LC atomic 10 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19


 Difficulty of Rendezvous Move toHalf (midpoint) 
 FSYNC execution ‣ Rendezvous SOLVED ! 
 Centralized execution ‣ Convergence achieved ‣ Rendezvous NOT SOLVED [20] I. Suzuki, M. Yamashita. Distributed anonymous mobile robots . SIAM J. Comput. , 28(4):1347–1363, 1999. 11 T.Okumura, K.Wada, X.Défago 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 => requires a Stay move 12 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Light Models ℓ ∈ L #Colors ‣ log(|L|) bits of information Full light ( ℓ ( me ), ℓ ( other ) ) ‣ can observe: own and others' color Internal light ( Fstate ) ‣ can observe: own color only ℓ ( me ) ‣ basically log(|L|) bits register External light ( Fcomm ) ℓ ( other ) ‣ can observe: others' color only [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. 13 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Example: SSYNC, Full(2) other is Black ⇒ ���� other is Black : other is White ⇒ ������� A B other is White ⇒ ������ [21] G. Viglietta. Rendezvous of two robots with visible bits . In Proc. 9th ALGOSENSORS , pp. 291–306, 2014. 14 T.Okumura, K.Wada, X.Défago 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 δ (for some unknown δ >0) Non-rigid with δ ‣ robots know the value of δ δ destination destination Movement is Non-Rigid Movement is Rigid 15 T.Okumura, K.Wada, X.Défago 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 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Related Work

Related Work full external Scheduler Mvmt not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – 0 not at all – quasi – ( 𝛆 ) SSYNC self-stabilizing rigid non Rigid – LC-atomic non Rigid with δ – ( 𝛆 ) ASYNC rigid – ASYNC – ( 𝛆 ) rigid 18 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Related Work full external Scheduler Mvmt not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – 0 – 2 𝓜 – ( 𝛆 ) SSYNC rigid – LC-atomic – ( 𝛆 ) ASYNC rigid – ASYNC – ( 𝛆 ) rigid [21] G. Viglietta. Rendezvous of two robots with visible bits . In Proc. 9th ALGOSENSORS , pp. 291–306, 2014. 19 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

╳ 
 
 Algorithm class 𝓜 Algorithm 
 observation destination ( ℓ ( me ), ℓ ( other ) ) ↦ λ ∈ ℝ colors Destination point 
 destination (1 − λ ) ⋅ me.pos λ ⋅ other.pos = + Examples ‣ toOther ( λ = 1) other ‣ toHalf ( λ = 0.5) ‣ Stay me ( λ = 0) destination [21] G. Viglietta. Rendezvous of two robots with visible bit . In Proc. 9th ALGOSENSORS , pp. 291–306, 2014. 20 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Related Work full external Scheduler Mvmt not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – 0 – 2 𝓜 – ( 𝛆 ) SSYNC rigid – LC-atomic – ( 𝛆 ) ASYNC rigid – ASYNC – ( 𝛆 ) rigid [21] G. Viglietta. Rendezvous of two robots with visible bit . In Proc. 9th ALGOSENSORS , pp. 291–306, 2014. 21 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Related Work full external Scheduler Mvmt not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – 0 – 2 𝓜 – ( 𝛆 ) SSYNC rigid – LC-atomic – ( 𝛆 ) ASYNC rigid – (3,3) 𝓜 (lower bound, upper bound) ASYNC – ( 𝛆 ) rigid [21] G. Viglietta. Rendezvous of two robots with visible bit . In Proc. 9th ALGOSENSORS , pp. 291–306, 2014. 22 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Related Work full external Scheduler Mvmt not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – 0 – 2 𝓜 – ( 𝛆 ) SSYNC rigid – 2 𝓜 LC-atomic – ( 𝛆 ) ASYNC rigid – (3,3) 𝓜 ASYNC – ( 𝛆 ) rigid 2 𝓜 [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. 23 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Related Work: not 𝓜 full external Scheduler Mvmt not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – 0 – – ( 𝛆 ) SSYNC rigid – LC-atomic – ( 𝛆 ) ASYNC rigid not class 𝓜 – 2 ASYNC – ( 𝛆 ) uses position info: 
 rigid distinct vs. gathered [11] A. Heriban, X. Défago, S. Tixeuil. Optimally gathering two robots , In Proc. 19th ICDCN , Jan. 2018. 24 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Related Work full external Scheduler Mvmt not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – 0 – 2 𝓜 – ( 𝛆 ) SSYNC rigid – 2 𝓜 LC-atomic – ( 𝛆 ) ASYNC rigid – (3,3) 𝓜 ASYNC – ( 𝛆 ) rigid 2 𝓜 25 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

External Lights

Related Work full external Scheduler Mvmt not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – 0 – 2 𝓜 – ( 𝛆 ) SSYNC rigid – 2 𝓜 LC-atomic – ( 𝛆 ) ASYNC rigid – (3,3) 𝓜 ASYNC – ( 𝛆 ) rigid 2 𝓜 27 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Related Work full external Scheduler Mvmt not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – 0 – 2 𝓜 3 𝓜 – ( 𝛆 ) SSYNC rigid – 2 𝓜 LC-atomic – ( 𝛆 ) ASYNC rigid – (3,3) 𝓜 ASYNC – ( 𝛆 ) rigid 2 𝓜 [10] P. Flocchini, N. Santoro, G. Viglietta, and M. Yamashita. Rendezvous with constant memory . Theor. Comput. Sci. , 621(C):57–72, March 2016. 28 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19

Related Work full external Scheduler Mvmt not QSS Quasi SS SS not QSS Quasi SS SS FSYNC – 0 – 2 𝓜 3 𝓜 – ( 𝛆 ) SSYNC rigid – 2 𝓜 LC-atomic – ( 𝛆 ) ASYNC rigid – (3,3) 𝓜 ∞ 𝓜 ASYNC – ( 𝛆 ) rigid 2 𝓜 [10] P. Flocchini, N. Santoro, G. Viglietta, and M. Yamashita. Rendezvous with constant memory . Theor. Comput. Sci. , 621(C):57–72, March 2016. 29 T.Okumura, K.Wada, X.Défago OPODIS 2018, Hong Kong, 2018/12/19