Loosely-stabilizing Leader Election with Polylogarithmic Convergence - - PowerPoint PPT Presentation

Loosely-stabilizing Leader Election with Polylogarithmic Convergence Time

Yuichi Sudo1, Fukuhito Ooshita2, Hirotsugu Kakugawa1, Toshimitsu Masuzawa1, Ajoy K. Datta3, Lawrence L. Larmore3

• 1. Osaka University, Japan
• 2. NAIST, Japan
• 3. The University of Nevada, Las Vegas, USA

OPODIS 2018

What is “Population Protocol Model”?

2

• 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

3

[AAD+06] D. Angluin, J Aspnes, Z. Diamadi, M.J. Fischer, and R. Peralta. Computation in networks

• f passively mobile finite-state sensors. Distributed Computing, 18(4):235–253, 2006.

A flock of birds Molecular computing

Population Protocol Model [AAD+06]

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

4

1 1 1 1 1 1 1 2 3 1 1 1 2 2 3 1 3 1 2 5 3 5 3 1 2 5 3 5 1 1

Scheduler

• The uniformly random scheduler
• Chose the ordered pair to interact at each step

uniformly at random

5

1 1 1 1 1 1 1 2 3 1 1 1 2 2 3 1 3 1 2 5 3 5 3 1 2 5 3 5 1 1

2nd step 1st step 3rd step 5th step 4th step Scheduler

What is “Leader Election Problem”?

6

Leader Election Problem

7 Electing exactly one leader

Goal Keeps the single leader single leader

leader follower

Elect one leader eventually

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

8

parallel time = #steps #agents If E[#steps] = 120 and #agents =6 20 parallel time single leader

Two Categories

• 1. Non-stabilizing leader election (LE)
• All agents are in the same state initially
• 2. Self-stabilizing leader election (SS-LE)
• Agents may be in different states initially

9

8 6 2 7 9 9

single leader

1 1 1 1 1 1 7 7 8 8 6 1

single leader

2 6 9 1 8 2

leader follower

Two Categories

• 1. Non-stabilizing leader election (LE)
• All agents are in the same state initially
• 2. Self-stabilizing leader election (SS-LE)
• Agents may be in different states initially

10

8 6 2 7 9 9

single leader

1 1 1 1 1 1 7 7 8 8 6 1

single leader

2 6 9 1 8 2

Non-stabilizing Leader Election

Paper

• Conv. Time

expected or w.h.p #states

[AAD+06]

• Expected

1

[AG15]

log Expected log

[BKKO18]

log Both log

[GS18]

log w.h.p. log log

[GSU18]

log ⋅ log log Expected log log

[SOI+18]

Expected

11

[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

• ptimal leader election in population protocols, arXiv:1802.06867v2

[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
• 2. Self-stabilizing leader election (SS-LE)
• Agents may be in different states initially

12

8 6 2 7 9 9

single leader

1 1 1 1 1 1 7 7 8 8 6 1

single leader

2 6 9 1 8 2

Convergence Closure

Self-stabilizing Leader Election (SS-LE)

single leader

any configuration safe configuration

leader follower

Reach a safe configuration eventually starting from any configuration Keep a single leader forever after the convergence

13

Impossibility Results [AAFJ15]

No SS-LE protocols exists unless every agent knows #agents exactly

Impossibility Theorem [AAFJ15]

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?

15

Yes! We can use loose-stabilization!

• May deviate from the specification even after the convergence
• But deviation happens only after an extremely long time

(in expectation)

Loose-stabilization [SNY+09]

single leader

any configuration safe configuration

short Extremely long

May deviate!

Loosely-stabilizing Leader Election (LS-LE)

16

The Power of Loose-stabilization

17

• Knowledge of exact #agents is no longer required
• Only an upper bound of #agents is needed

→ Practical assumption

• Linear convergence time and exponential deviation time!

Convergence Time Deviation Time expected

• r w.h.p

#states

[SNY+09]

log Ω expected

• [Izumi15]
• Ω

expected

• LS-LE protocols

Lower Bound [Izumi15]

• Linear convergence time protocol [Izumi,2015] is optimal

if we want an exponential deviation time

18

Any protocol with

deviation time

requires convergence time

Lower Bound Theorem [Izumi, 2015]

Our Contribution

• 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

is a constant parameter

Convergence Time Deviation Time expected

• r w.h.p

#states

[SNY+09]

log Ω expected

• [Izumi15]
• Ω

expected

• New!

⋅

• expected

LS-LE protocols

• 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

20

SINGLE LEADER

any configuration safe configuration

• leader

follower

Our Contribution

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

22

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

Multiple leaders → Decrease #leaders to one

Goal

Basic Idea

Basic Idea

24

10 leaders 10 leaders 6 leaders 6 leaders 3 leaders Create Create Mask breaks

Implementation (1/3)

• Every leader has variable . timer ∈ 0,1, … ,
• Every time interacts, . timer decreases
• When . timer reaches 0, resets its timer and…
• If it is an initiator, it creates virus and wears a mask
• If it is a responder, its mask breaks

25

38 37 1 100 1 100 same prob.

Implementation (2/3)

• Each virus has its TTL (Time To Live)
• The TTL of a new virus is
• Viruses are propagated with decreasing TTL

26

e.g.) 100 100 100 77 77 49 49 78 15 50 1 Propagate Replace Dissapear

Implementation (3/3)

• A leader without a mask is killed by a virus
• A leader with a mask is never killed

27

93 92 92 93 92 92

Become a follower Remain a leader

• Provided that we set

Analysis (1/3)

28

• Every time a new virus is created, it propagates to the

whole population and disappears in ⋅ time Created Pervaded Disappears

Lifecycle of viruses

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

29

• Smaller leads to

faster convergence

Analysis (3/3)

• But, we cannot set arbitrarily small
• A single leader may kill itself if
• i.e., the single leader is killed by a virus that it created itself
• If we set ⋅ ,

Chernoff bound guarantees that suicides never happens for an exponentially long time in () time)

30

• Viruses disappear

Set Parameter

• We must satisfy
• so that a virus propagates to the whole

population

• ⋅ so that a single leader never

kills itself

• If we set Θlog and Θ log ,

31

NO KILL

Summary

32

SINGLE LEADER

• No timeout to create a leader

& No suicide of a single leader

Conclusion

• Break through the lower bound of liner conv. time, and

achieve polylog convergence time

• Deviation time is no longer exponential,

but arbitrarily large polynomial

33

is a constant parameter

Convergence Time Deviation Time expected

• r w.h.p

#states

[SNY+09]

log Ω expected

• [Izumi15]
• Ω

expected

• New!

⋅

• expected

LS-LE protocols

34

35

Strategy

1) No leader → Create a leader by timeout mechanism

• Thereafter, do not create more leaders (for sufficiently long time)

2) Multiple leaders → Decrease #agents to one by virus war mechanism

• Thereafter, do not delete the single leader (for sufficiently long time)

36

Timeout Mechanism

Basic Idea

• Each agent has a countdown timer

to detect the absence of a leader

• The value of the timer is decreasing but

it is reset if an agent meets a leader

• A follower becomes a leader

when it observes timeout!

37

No leader → Create a leader

Goal

Implementation

• Every agent have variable . time ∈ 0,1, … ,
• A leader always keeps the full timer value
• If a follower interacts with an agent ,

then . timer ← max 0, . timer 1, u.timer1

• When timeout happens, an agent becomes a leader

38

83 100 3 97 100 e.g.) 100 6 82 82 1 1 100 100 25 56

Analysis

• When no leader exists,

→ The maximum timer value decreases by one within expected time → Timeout happens within ⋅ expected time

39

90 89 88 78 83 30 32 30 26 26 1 1 2 3 2 100 100 98 98 99

• A leader is created!

Question

• When no leader exists, a leader is created within

⋅ log expected time

• Hence, smaller leads to faster convergence
• However, too small leads to frequent timeout even

when a leader exists → short deviation time

40

2 2 2 2 2 2 2 1 1 1 1 2 2 1 2 1 2 2 2 2

timeout! timeout! timeout!

How large

is enough

to avoid short deviation time?

e.g.) 2

Answer

• When at least one leader exists and Ωlog ,

→ All agents have high timer value most of the time → Timeout does not happens for time

41 Logarithmic value is enough for to keep the timers of all agents high

NO TIMEOUT

• When we assign log for parameter …