Topics in Self-stabilization

(1)

Topics in

Self-stabilization

Franck Petit / Sebastien Tixeuil [email protected]

(2)

Robust Protocols

Nature

Extent

Time

(3)

Self-stabilizing Protocols

Nature

Extent

Time Faults

Stabilization

Legitimate

(4)

The Big Picture

Weaker Guarantees

Complexity Guarantees

Stronger Guarantees Self-

stabilization Pseudo stabilization Weak

stabilization

Fault- tolerant stabilization

Strong stabilization

k- stabilization

time- adaptive

k-time- adaptive

Predicate- preserving stabilization Probabilistic

stabilization

Snap- stabilization Probable

stabilization

Strict stabilization

Fault- containment

(5)

Snap-Stabilization

(6)

Snap-Stabilization

• Specification

• Self-stabilization: Starting from any initial configuration, the system eventually reaches a configuration from which its

behavior is correct

• Snap-stabilization: Starting from any initial configuration, the system always

behaves as its specification

(7)

Self-stabilization

(8)

Self-stabilization

(9)

Self-stabilization

(10)

Self-stabilization

(11)

Self-stabilization

(12)

Self-stabilization

(13)

Self-stabilization

(14)

Self-stabilization

?

(15)

Self-stabilization

(16)

Self-stabilization

(17)

Self-stabilization

BC

(18)

Self-stabilization

BC

(19)

Self-stabilization

BC

(20)

Self-stabilization

(21)

Self-stabilization

(22)

Self-stabilization

(23)

Self-stabilization

(24)

Self-stabilization

(25)

Self-stabilization

(26)

Snap-stabilization

(27)

Snap-stabilization

(28)

Snap-stabilization

BC

(29)

Snap-stabilization

BC

(30)

Snap-stabilization

BC

(31)

Snap-stabilization

BC

(32)

Snap-stabilization

BC

(33)

Snap-stabilization

BC

(34)

Snap-stabilization

BC

(35)

Snap-stabilization

(36)

Snap-stabilization

(37)

Non Self-*

(38)

Non Self-*

Network

(39)

Non Self-*

Network

F(.)

(40)

Non Self-*

X

Network

F(.)

(41)

Non Self-*

X F(X)

Network

F(.)

(42)

Self-stabilization

(43)

Self-stabilization

1.X _F(.) ≠^F(X)

(44)

Self-stabilization

(45)

Self-stabilization

2.X _F(.) ≠^F(X)

(46)

Self-stabilization

(47)

Self-stabilization

3.X _F(.) ≠^F(X)

(48)

Self-stabilization

(49)

Self-stabilization

4.X _F(.) ≠^F(X)

(50)

Self-stabilization

5.X _F(.) ≠^F(X)

(51)

Self-stabilization

6.X _F(.) ≠^F(X)

(52)

Self-stabilization

7.X _F(.) ≠^F(X)

(53)

Self-stabilization

8.X _F(.) ≠^F(X)

(54)

Self-stabilization

... _F(.) ≠^F(X)

(55)

Self-stabilization

(56)

Self-stabilization

k.X ^F(.) F(X)

(57)

Snap-stabilization

(58)

Snap-stabilization

1.X ^F(.)

(59)

Snap-stabilization

1.X ^F(.) F(X)

(60)

Related Works

• Bui, Datta, Petit, Villain. State-optimal snap-

stabilizing PIF in tree networks. WSS 1999.

• Cournier, Datta, Petit, Villain. Enabling snap- stabilization. SSS 2003.

• Cournier, Devismes, Villain. From Self- to Snap- Stabilization. SSS 2006.

• Bui, Datta, Petit, Villain. Snap-stabilization and PIF in tree networks. Distributed Computing 2007.

(61)

• Propagation of Information with FeedbackPIF

• Distributed-Control Problems

• Broadcast, Routing, Synchronization,

Protocol, Leader Election, Resource Sharing and Allocation, Graph Algorithms,

Termination Detection, Deadlock

Detection, Reset, Distributed Ranking, Distributed Sorting...

(62)

• Distributed-Control Problems

• Broadcast, Routing, Synchronization,

Protocol, Leader Election, Resource Sharing and Allocation, Graph Algorithms,

Termination Detection, Deadlock

Detection, Reset, Distributed Ranking, Distributed Sorting...

PIF

(63)

r

(64)

r

(65)

r

1

1 1

(66)

r

1

1 1

(67)

r

1

1 1

(68)

r

1 2

2 1

1

1 1

(69)

r

1 2

2 1

1

1 1

(70)

r

1 2

2 1

1

1 1

(71)

r

1

3 2

1 2

2 1

1 2

2

1

(72)

r

1

3 2

1 2

2 1

1 2

2

1

(73)

r

1

3 2

1 2

2 1

1 2

2

1

(74)

r

1

3 2

2 2

2 3

2 2

2

1

1 1

(75)

r

1

3 2

2 2

2 3

2 2

2

1

1 1

(76)

r

1

3 2

2 2

2 3

2 2

2

1

1 1

(77)

r

4

1

3 2

2 2

2 3

2 2

2

1

2 1

(78)

r

4

1

3 2

2 2

2 3

2 2

2

1

2 1

(79)

r

4

1

4 2

2 2

2 3

2 2

2

2 2

(80)

r

4

1

4 2

2 2

2 3

2 2

2

2 2

(81)

r

4

1

4 2

2 3

2 2

2

1

2 3

(82)

r

4

1

4 2

2 3

2 2

2

1

2 3

(83)

r

4

2

4 2

2 3

2 2

2

1

2 3

(84)

r

4

2

4 2

2 3

2 2

2

1

2 3

(85)

r

4

2

4 2

2 3

2 2

2

1

2 4

(86)

Self-Stabilizing Compiler

• ^GOAL

• «Universal» Tool to Transform (Compile)

any non self-stabilizing distributed algorithm into a self-stabilizing one

[Katz and Perry 1993]

(87)

Self-Stabilizing Compiler

• ^GOAL

any non self-stabilizing distributed algorithm into a self-stabilizing one

[Katz and Perry 1993]

T Self-Stabilizing T

(88)

Snap-Stabilizing Compiler

• ^GOAL

any non self-stabilizing distributed algorithm into a snap-stabilizing one

[Cournier, Datta, Petit, Villain 2003]

T Snap-Stabilizing T

(89)

• ^IDEA

• Snap-stabilizing leader test

• Snap-stabilizing reset

• Snap-stabilizing snapshot

• Snap-stabilizing termination detection

(90)

• ^IDEA

Sna p-sta bilizi ng

PIF

(91)

With a Single Initiator

(92)

Network

(93)

Network

Initiator

(94)

Network

Initiator

T

(95)

Network

Initiator

T T T

T

(96)

Network

Initiator

T T

T

(97)

Network

Initiator

Snapshot

T T

T

(98)

Network

Initiator

T T

T

(99)

Network

Initiator

If not OK then Reset

T T

T

(100)

Network

Initiator

(101)

Network

Initiator

T

(102)

With a Multiple Initiators

• Same principle

• Snap-stabilizing leader election

(103)

Data Links

(104)

State Model

1 3

5 5

(105)

State Model

1 3

7 5

(106)

Message Passing Model

(107)

Bootstrapping ?

1 3

5 5

(108)

Bootstrapping ?

1 3

5 5

(109)

Bootstrapping ?

1 3

5 5

(110)

Bootstrapping ?

1 3

5 5

(111)

Bootstrapping ?

1 3

5 5

(112)

Bootstrapping ?

1 3

7 5

(113)

Results

• Safety-distributed specifications admit no snap-stabilizing solution when the bound on channel capacity is unknown

• Known bounded capacity channels

enables snap-stabilization in message passing networks

(114)

Algorithms

• Complete (i.e. 1-hop) networks

• Fair lossy links of fixed capacity (e.g. 1)

• ^Protocols

• Reliable Communication

• Propagation of Information with Feedback (PIF)

• Mutual Exclusion

(115)

Reliable

Communications

(116)

Reliable

Communications

(117)

Reliable

Communications

BC

(118)

Reliable

Communications

BC

(119)

Reliable

Communications

BC

(120)

Reliable

Communications

BC

(121)

Reliable

Communications

BC

(122)

Reliable

Communications

BC

(123)

Reliable

Communications

(124)

Reliable

Communications

(125)

Reliable

Communications

(126)

Ensuring Safety

BC 1 BC 2

(127)

Ensuring Safety

BC 1 BC 2

(128)

Ensuring Safety

0

BC 1 BC 2

(129)

Ensuring Safety

0

BC 1 BC 2

(130)

Ensuring Safety

0

BC 1 BC 2

(131)

Ensuring Safety

0

BC 1 BC 2

(132)

Ensuring Safety

1

BC 1 BC 2

(133)

Ensuring Safety

1

BC 2

(134)

Ensuring Safety

1

BC 2

(135)

Ensuring Safety

1

BC 2

(136)

Ensuring Safety

2

BC 2

(137)

Ensuring Safety

2

BC 2

(138)

Ensuring Safety

2

(139)

Ensuring Safety

2

(140)

Ensuring Safety

2

(141)

Ensuring Safety

3

(142)

Ensuring Safety

3

(143)

Ensuring Safety

BC

3

(144)

Ensuring Safety

BC

3

(145)

Ensuring Safety

3

(146)

Ensuring Safety

3

(147)

Ensuring Safety

3

(148)

The Price of Safety

(149)

The Price of Safety

0

(150)

The Price of Safety

BC

0

(151)

The Price of Safety

BC

0

(152)

The Price of Safety

0

(153)

The Price of Safety

0

(154)

The Price of Safety

1

(155)

The Price of Safety

BC

1

(156)

The Price of Safety

BC

1

(157)

The Price of Safety

1

(158)

The Price of Safety

1

(159)

The Price of Safety

2

(160)

The Price of Safety

BC

2

(161)

The Price of Safety

BC

2

(162)

The Price of Safety

2

(163)

The Price of Safety

2

(164)

The Price of Safety

3

(165)

The Price of Safety

BC

3

(166)

The Price of Safety

BC

3

(167)

The Price of Safety

3

(168)

The Price of Safety

3

(169)

The Price of Safety

3

(170)

Conclusion

• Bootstrapping Snap-stabilization in realistic networks

• Enabling conditions are more restrictive than those of self-stabilization

• Safety can be ensured, yet induces some overhead

(171)

Multitolerance

(172)

Stronger Stabilization

(173)

(174)

(175)

(176)

(177)

(178)

(179)

(180)

Strict Stabilization

(181)

non (c,f)-contained configurations

(c,f)-contained configurations transient fault

(182)

• Impossibility result (known)

• r-restrictive specifications do not allow containment radius less than r

• many useful tasks are D-restrictive

• spanning tree, etc.

(183)

Strong Stabilization

non (c,f)-contained configurations

(c,f)-contained configurations transient fault

(184)

Strong Stabilization

• Our results

• It is possible to solve global tasks and maintain strong stabilization

• Spanning tree (arbitrary)

• Tree orientation

• Impossible to manage maximizing tree metrics

(185)

Conclusion

(186)

The Big Picture

Weaker Guarantees

Complexity Guarantees

Stronger Guarantees Self-

stabilization Pseudo stabilization Weak

stabilization

Fault- tolerant stabilization

Strong stabilization

k- stabilization

time- adaptive

k-time- adaptive

Predicate- preserving stabilization Probabilistic

stabilization

Snap- stabilization Probable

stabilization

Strict stabilization

Fault- containment