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## STABILIZATION & LOCALITY

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Recall: Traditional methods are global

* Dijkstra- 1 fault-O(n) Time; f faults- o(fn) time

* [Katz, Perry]- 1st general method, 1 fault- O(n) time

(1) self stab bcast freezees all nodes

(2) global self-stab snapshot to a leaser

(3) leader checks global state.

(4) Leader initializes every node (if faulty)

(5) bcast unfreezes nodes

Recall: Traditional methods are global(continuted)

* Global reset (general) methods, 1 fault- O(n) time

[Afek, Kutten, Yung],

[Awerbuch, Patt-Shamir, Vargheses],

[Awerbuch, K., Mansour, Patt-Shamir, Varghese]

(1) bcast freezes all nodes

(2) reset to a specific initial state

[Dolev, Herman] superstabilizing global reset:

(2’) reset to a state “nearest” to current state

(still 1 fault- O(n) time)

A

B

Fast stupid switch

E

C

D

Fast route from A to C, passing only B’s stupid switch, not B’s

general purpose computer. Possible since route is preset.

Another example of self stab in industryB

E

C

D

In bcast how does A’s stupid switch detect that it already

received the bcast

Another example of self stab in IndustryB

E

C

D

(non- self stab) solution- stupid switch forwards only over ports

Ports of links marked tree

Another example of self stab in industryB

fault

E

C

D

(non- self stab) solution- (stupid) switch forwards only over ports

Ports of links marked tree.

Vulnerable to state fault: suppose the “tree” is really a cycle.

Another example of self stab in industryA

pulse

pulse

E

B

pulse

pulse

C

D

Industrial Solutions to the self stab problem(1) Digital’s LAN Bridges SolutionA

pulse

pulse

E

B

pulse

pulse

C

D

pulse

Industrial Solutions to the self stab problem(1) Digital’s LAN Bridges SolutionA

6

6

E

B

C

D

Hop counter decreased. When hop counter reaches 0, discard

message.

Industrial Solutions to the self stab problem(2) IBM’s ATM SolutionA

6

6

E

B

5

5

C

D

Hop counter decreased. When hop counter reaches 0, discard

message.

Industrial Solutions to the self stab problem (2) IBM’s ATM SolutionA

E

B

5

5

C

D

4

4

4

Hop counter decreased. When hop counter reaches 0, discard

message.

Industrial Solutions to the self stab problem (2) IBM’s ATM SolutionA

E

B

3

3

3

C

D

4

4

4

3

Hop counter decreased. When hop counter reaches 0, discard

message.

Industrial Solutions to the self stab problem(1) Digital’s LAN Bridges SolutionA

2

2

E

B

2

3

3

2

C

D

3

Hop counter decreased. When hop counter reaches 0, discard

message.

Industrial solutions to the self stab problem(1) Digital’s LAN bridges solutionScallability

- Industrial solutions are global. They do not scale well
- Conventional self stab protocols are global. They do not scale well

(Local “checking” [Awerbuch, Patt-Shamir, Varghese])

root

0

1

14

2

14

13

3

A cycle will be detected by a node seeing

Only its parent’s state and its own state.

12

4

5

11

6

10

7

Local detection [Afek, Kutten, Yung]Any graph marking function is locally checkable

- Any algorithm global state is locally checkable
- For any bit complexity Cfunction with local checking bit complexity O(C)
- Some complexities for interesting marking functions are known
- Many other problems are open

Goal: (unknown)f faults O(f) time for correction

Idea: Diameter of faults is f. Prevent faults expansion, shrink faulty area.

0

1

14

2

14

13

3

fault

12

4

22

11

23

25

24

5

10

6

7

Possibly consistent faults

From local checking to local correctionGoal: (unknown)f faults O(f) time for correction

Idea: Diameter of faults is f. Prevent faults expansion, shrink faulty area.

0

1

14

2

14

13

3

fault

12

4

22

11

23

(“impossible” if faulty majority)

25

24

5

10

6

7

Possibly consistent faults

From local checking to local correctionFor example: need to prevent expansion of

Faulty area: recall Dijkstra’s algorithm

val1

Adversary spoils (ones) minority of replicated val1 and of states.

Alg. Recovers val1 everywhere.

A

B

val5

D

C

E

val3

val4

Another example of expanding faulty area: Stable Value Problemval1

A

B

val5

D

C

E

val1

val1

val’1 val1

val’1

Stable Value ProblemAdversary spoils (ones) minority of replicated val1 and of states.

Alg. Recovers val1 everywhere.

val1

A

B

val5

D

C

E

val3

val4

Reducing a general problem to the Stable Value ProblemB can compute any func(val1, val2, val3,val4, val5) if it receives every correct vali.

val1

A

B

“Simple” since E’s vote can be spoiled in D on the way to B.

val1

D

C

E

val1

val1

“Simple” but global solution:consensus votingA

B

Time = O(diameter) even for one fault.

Desired: few faults short time

val1

D

C

E

val1

val1

“Simple” but global solution:consensus voting

val1

Local voting does not solve

Minority faulty notes but local majority everywhere

get >2f votes

so majority is

non faulty

A

2f

Idea 1: For (unknown) f faults get votes (values) from radius 2fget >2f authentic

votes

so majority is

non faulty

A

2f

C

C’s vote at A is

authentic

(though faulty)

B’s vote at A is

NOT authentic

B

Idea 1: For (unknown) f faults get votes (values) from radius 2fA said: “1”

D

A

B

C

A said: “0”

1

A

said:

“0”

E

Bcast (vote sending) spreading faults problem under state faultsA said: “1”

D

A

B

C

A said: “0”

1

A

said:

“0”

E

Bcast (vote sending) spreading faults problem under state faultsA said: “1”

D

A

B

C

A said: “0”

1

A

said:

“0”

E

Bcast (vote sending) spreading faults problem under state faultsA said: “1”

D

A

B

C

A said: “0”

1

A

said:

“0”

E

Bcast (vote sending) spreading faults problem under state faultsA said: “1”

D

A

B

C

A said: “0”

1

A

said:

“0”

E

Bcast (vote sending) spreading faults problem under state faultsA said: “1”

1

A said: “1”

D

A

B

C

1

A

said:

“0”

E

Bcast (vote sending) spreading faults problem under state faultsA said: “1”

1

A said: “1”

D

A

B

C

1

Actually,

A said:

“1”

E

Bcast (vote sending) spreading faults problem under state faultsA said: “1”

1

A said: “1”

D

A

B

C

1

Actually,

A said:

“1”

E

Actually,

A said:

“1”

Bcast (vote sending) spreading faults problem under state faultsA said: “1”

1

A said: “1”

D

A

B

C

1

Actually,

A said:

“1”

E

Actually,

A said:

“1”

Actually,

A said: “1”

Bcast (vote sending) spreading faults problem under state faultsA said: “1”

1

A said: “1”

D

A

B

C

1

Actually,

A said:

“1”

Here, the vote of A at Z is not authentic even

After a long time, even with 1 fault

E

Z

Actually,

A said:

“1”

Actually,

A said: “1”

Actually…

Bcast (vote sending) spreading faults problem under state faultsA said: “1”

1

A said: “1”

D

A

B

C

1

Actually,

A said:

“1”

Here, the vote of A at Z is not authentic even

After a long time, even with 1 fault

E

Global effect for onefault

Z

Actually,

A said:

“1”

Actually,

A said: “1”

Actually…

Bcast (vote sending) spreading faults problem under state faultsA

B

C

B, C,D,D,F

All said: “0”

D

E

One faulty node can make the cotes of a majority non-authentic

Another bcast spreading fault problemRegulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

1

…

A

B

C

D

Sample technique: solving the bcast authenticity in O(f) timeRegulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

1

…

A

B

C

D

A

1111111111111111111111111111111111111111111111111111111111

Sample technique: solving the bcast authenticity in O(f) timeRegulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

A

f= five faults hit

Sample technique: solving the bcast authenticity in O(f) time1111111111111111111111111111111111111111111111111111111111

111111111111111000001111111111111111111111111111111111111

Regulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

0

1

111111110000 11111111111111111111111

Sample technique: solving the bcast authenticity in O(f) timeA

After One time unit, the first 0 notices inconsistency and resets A’s

Bcast value to bottom. The first “1” after the zeros does the same

Regulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

0

1

A

111111111 000 0 1111111111111111111

Sample technique: solving the bcast authenticity in O(f) timeAfter two time units, the spreads twice as fast as

the bcast (“0”s and “1”s).

Regulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

0

1

A

111111111 00 0 111111111111111111

Sample technique: solving the bcast authenticity in O(f) timeAfter Three time units, the spreads twice as fast as

the bcast (“0”s and “1”s).

Regulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

0

1

A

1111111111 0 0 0 111111111111111

Sample technique: solving the bcast authenticity in O(f) timeAfter Four time units, the spreads twice as fast as

the bcast (“0”s and “1”s).

Regulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

0

1

A

1111111111 0 0 11111111111111

After Five time units, the spreads twice as fast as

the bcast (“0”s and “1”s).

Sample technique: solving the bcast authenticity in O(f) timeRegulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

0

1

11111111111 0 0 11111111111

Sample technique: solving the bcast authenticity in O(f) timeA

After Six time units, the spreads twice as fast as

the bcast (“0”s and “1”s).

Regulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

0

1

11111111111 0 11111111

After seven time units, the spreads twice as fast as

the bcast (“0”s and “1”s).

Sample technique: solving the bcast authenticity in O(f) timeA

Regulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

0

1

111111111111 0 1111

After eight time units, the spreads twice as fast as

the bcast (“0”s and “1”s).

Sample technique: solving the bcast authenticity in O(f) timeA

Regulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

0

A

111111111111 11111111

After nine time units, the spreads twice as fast as

the bcast (“0”s and “1”s).

Sample technique: solving the bcast authenticity in O(f) timeRegulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

A

1111111111111

After 10 time units, the replaces the un-authentic vote, while

the authentic vote “1” continues to proceed at half speed.

Sample technique: solving the bcast authenticity in O(f) timeafter O(f)

A

2f

(3) Majority of these votes are

not faulty.

Recall the Stable Value Problem- In O(f) time all non authentic values disappear in A.

(2)In another O(f) time A

knows the authentic votes

of 2f +1.

A lot of recent related work. A very partial bibliography:

[Kutten, Peleg], [KP1]

[Ghosh, Gupta, Herman, Pamaraju]

[Afek, Dolev]

[Chlamtac,Pinter], [Dolev, Herman], [Naor, Stockmeyer]

[Arora, Zhang]

Beauquier, Genolini, Cournier, Datta, Petit, Viliain, Xin He

Error Confinement, Time Adaptive, Fault Local, Mending,

Fault Containment, Snap Stabilization, Local Stabilization

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