OPODIS 2018 Loosely-stabilizing Leader Election with Polylogarithmic Convergence Time Yuichi Sudo 1 , Fukuhito Ooshita 2 , Hirotsugu Kakugawa 1 , Toshimitsu Masuzawa 1 , Ajoy K. Datta 3 , Lawrence L. Larmore 3 1. Osaka University, Japan 2. NAIST, Japan 3. The University of Nevada, Las Vegas, USA
What is “Population Protocol Model”? 2
Population Protocol Model [AAD+06] • Represent a network of passively mobile devices • Each device moves but cannot control its movement • Basically, a network consists of vast number of tiny devices A flock of birds Molecular computing 3 [AAD+06] D. Angluin, J Aspnes, Z. Diamadi, M.J. Fischer, and R. Peralta. Computation in networks of passively mobile finite-state sensors. Distributed Computing, 18(4):235–253, 2006.
Population Protocol Model [AAD+06] • Population • Consists of a vast number of identical and anonymous finite state machines ( agents ) • Execution • At each step, one pair of agents has an interaction • one agent is an initiator , the other is a responder • Their states are updated according to transition function 1 1 2 2 2 1 1 3 3 1 1 2 2 5 5 1 3 3 3 3 1 1 1 1 1 1 1 1 5 5 4
Scheduler • The uniformly random scheduler • Chose the ordered pair to interact at each step uniformly at random 1 st step 2 nd step 3 rd step 4 th step 5 th step 1 1 2 2 2 1 1 3 3 1 1 2 2 5 5 1 3 3 3 3 1 1 1 1 1 1 1 1 5 5 5 Scheduler
What is “Leader Election Problem”? 6
Leader Election Problem Goal Electing exactly one leader single leader Elect one leader Keeps the single leader eventually leader follower 7
Time Metrics • Basically, detecting termination is impossible in the PP model • Instead, evaluate the expected convergence time during which the output of the population converges • Usually, evaluated in terms of parallel time parallel time = #steps #agents single leader If E [#steps] = 120 20 parallel time and #agents =6 8
leader Two Categories follower 1. Non-stabilizing leader election (LE) • All agents are in the same state initially 1 8 1 9 1 6 single leader 1 2 1 9 1 7 2. Self-stabilizing leader election (SS-LE) • Agents may be in different states initially 2 7 8 6 6 7 single leader 9 8 2 1 1 8 9
Two Categories 1. Non-stabilizing leader election (LE) • All agents are in the same state initially 1 8 1 9 1 6 single leader 1 2 1 9 1 7 2. Self-stabilizing leader election (SS-LE) • Agents may be in different states initially 2 7 8 6 6 7 single leader 9 8 2 1 1 8 10
Non-stabilizing Leader Election Paper Conv. Time expected or w.h.p #states ���� ��1� Expected [AAD+06] ��log � �� ��log � �� Expected [AG15] ��log � �� ��log �� Both [BKKO18] ��log � �� ��log log �� w.h.p. [GS18] ��log � ⋅ log log �� ��log log �� Expected [GSU18] ����� �� ����� �� Expected [SOI+18] [AG15] Dan Alistarh and Rati Gelashvili. Polylogarithmic-time leader election in population protocols. In Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming, pages 479–491. Springer, 2015 [BKKO18] P. Berenbrink, D. Kaaser, P. Kling, and L. Otterbach. Simple and efficient leader election. In SOSA, pages 9:1–9:11, 2018. [GS18] L. Gąsieniec and G. Stachowiak. Fast space optimal leader election in population protocols. In SODA, pages 2653–2667, 2018. [GSU18] ] L. Gąsieniec, G. Stachowiak, and P. Uznanski. Almost logarithmic expected-time space optimal leader election in population protocols, arXiv:1802.06867v2 11 [SOI+18] Y. Sudo, F. Ooshita, T. Izumi, H. Kakugawa, and T. Masuzawa. Logarithmic Expected-Time Leader Election in Population Protocol Model, to be submitted.
Two Categories 1. Non-stabilizing leader election (LE) • All agents are in the same state initially 1 8 1 9 1 6 single leader 1 2 1 9 1 7 2. Self-stabilizing leader election (SS-LE) • Agents may be in different states initially 2 7 8 6 6 7 single leader 9 8 2 1 1 8 12
Self-stabilizing Leader Election (SS-LE) leader follower any configuration safe configuration single leader Closure Convergence Keep a single leader Reach a safe configuration forever after the eventually starting from any configuration convergence 13
Impossibility Results [AAFJ15] Impossibility Theorem [AAFJ15] No SS-LE protocols exists unless every agent knows #agents exactly We cannot design any practical SS-LE protocol because knowledge of exact is not practical for many applications… 14
Is there any way to design practical and fault tolerant leader election protocol? Yes! We can use loose-stabilization ! 15
Loose-stabilization [SNY+09] • May deviate from the specification even after the convergence • But deviation happens only after an extremely long time (in expectation) Loosely-stabilizing Leader Election (LS-LE) any configuration May deviate! safe configuration single leader short Extremely long 16
The Power of Loose-stabilization • Knowledge of exact #agents is no longer required • Only an upper bound � of #agents is needed → Practical assumption LS-LE protocols Convergence Deviation expected #states Time Time or w.h.p Ω � � ��� log �� ���� [SNY+09] expected Ω�� � � ���� ���� [Izumi15] expected • Linear convergence time and exponential deviation time ! 17
Lower Bound [Izumi15] Lower Bound Theorem [Izumi, 2015] � deviation time Any protocol with requires convergence time • Linear convergence time protocol [Izumi,2015] is optimal if we want an exponential deviation time 18
Our Contribution LS-LE protocols Convergence Deviation expected #states Time Time or w.h.p Ω � � ��� log �� ���� expected [SNY+09] Ω�� � � ���� ���� expected [Izumi15] ��� ⋅ ��� � �� expected ��� � ��� � �� ���� ��� � New! is a constant parameter • Break through the lower bound of liner conv. time, and achieve polylog convergence time • Deviation time is no longer exponential, but sufficiently large polynomial 19
Our Contribution • We can arbitrarily increase deviation time with a constant parameter • e.g., if we assign � � 10 , ��� then we get deviation time and ��log � �� convergence time leader any configuration safe configuration follower SINGLE LEADER ��� � 20
Strategy 1) No leader → Create a leader by timeout mechanism • Thereafter, do not create more leaders (for sufficiently long time) 2) Multiple leaders → Decrease #leaders to one by virus war mechanism • Thereafter, do not delete the single leader (for sufficiently long time) 21
Virus War Mechanism Goal Multiple leaders → Decrease #leaders to one Basic Idea • Leaders try to kill each other by VIRUSES • Each leader periodically makes a coin flip • With probability ½, it make a virus and wears a new mask • With probability ½, its mask breaks if it has • Viruses propagate to the whole population killing all unmasked leaders and disappears thereafter • At each interval, #leaders decreases by half 22
Basic Idea � � � � � �
Basic Idea Create 10 leaders 10 leaders 6 leaders Create Mask breaks 24 6 leaders 3 leaders
Implementation (1/3) • Every leader � has variable �. timer � ∈ �0,1, … , � ���� � • Every time � interacts, �. timer � decreases 37 38 • When �. timer � reaches 0, � resets its timer and… • If it is an initiator , it creates virus and wears a mask 100 1 • If it is a responder , its mask breaks 100 1 same prob. 25
Implementation (2/3) • Each virus has its TTL (Time To Live) • The TTL of a new virus is � ����� e.g.) � ����� � 100 100 100 • Viruses are propagated with decreasing TTL 78 77 77 Propagate 50 15 49 49 Replace 1 Dissapear 26
Implementation (3/3) • A leader without a mask is killed by a virus 93 92 92 Become a follower • A leader with a mask is never killed 93 92 92 Remain a leader 27
Analysis (1/3) Every time a new virus is created, it propagates to the whole population and disappears in ��� ����� ⋅ ��� �� time • Provided that we set � ����� � ����� �� Lifecycle of viruses Created Pervaded Disappears 28 �����
Analysis (2/3) • Every ��� ���� � time, every leader tries to make a virus, and fails with probability ½, which breaks its mask • If it fails, it is killed in the next ��� ���� � time with constant probability if another leader exists • Hence, #agents decreases almost by half in every ��� ���� � time → the unique leader is elected in ��� ���� ⋅ log �� Smaller � ���� leads to faster convergence ���� 29
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