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STABILIZATION & LOCALITY. Shay Kutten Technion, Israel. Recall: Traditional methods are global. * Dijkstra- 1 fault- O( n ) Time ; f faults- o( fn ) time * [Katz, Perry]- 1 st general method, 1 fault- O( n ) time (1) self stab bcast freezees all nodes

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stabilization locality
STABILIZATION & LOCALITY

Shay Kutten

Technion, Israel

recall traditional methods are global
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
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)

global effect example

8

8

8

8

8

8

8

8

8

8

7 = 8

8

7

8

7

token

7

8

7

7  8

Global effect example

leader

global effect example1

8

8

8

8

8

8

8

8

leader

8

8

7 = 8

8

7

token

8

7

7

8

8

7

Global effect example
global effect example2

8

8

8

8

8

8

8

8

leader

8

8

7 = 8

8

7

token

8

7

8

8

8

7

Global effect example
global effect example3

7

8

8

8

8

8

8

8

8

leader

8

8

8

7

8

7

token

7

8

7

Global effect example
global effect example4

7

7

8

8

8

8

8

8

leader

8

8

8

7

8

7

token

7

8

7

Global effect example
global effect example5

7

7

8

7

8

8

8

8

leader

8

8

8

7

8

7

token

7

8

7

Global effect example
global effect example6

7

7

8

7

8

7

8

8

leader

8

8

8

7

8

7

token

7

8

7

Global effect example
another example of self stab in industry

General purpose computer

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 industry
another example of self stab in industry1

A

B

E

C

D

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

received the bcast

Another example of self stab in Industry
another example of self stab in industry2

A

B

E

C

D

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

Ports of links marked tree

Another example of self stab in industry
another example of self stab in industry3

A

B

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 industry
industrial solutions to the self stab problem 1 digital s lan bridges solution1

root

A

pulse

pulse

E

B

pulse

pulse

C

D

Industrial Solutions to the self stab problem(1) Digital’s LAN Bridges Solution
industrial solutions to the self stab problem 1 digital s lan bridges solution2

root

A

pulse

pulse

E

B

pulse

pulse

C

D

pulse

Industrial Solutions to the self stab problem(1) Digital’s LAN Bridges Solution
industrial solutions to the self stab problem 2 ibm s atm solution

root

A

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 Solution
industrial solutions to the self stab problem 2 ibm s atm solution1

root

A

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 Solution
industrial solutions to the self stab problem 2 ibm s atm solution2

root

A

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 Solution
industrial solutions to the self stab problem 1 digital s lan bridges solution3

root

A

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 Solution
industrial solutions to the self stab problem 1 digital s lan bridges solution4

root

A

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 solution
scallability
Scallability
  • Industrial solutions are global. They do not scale well
  • Conventional self stab protocols are global. They do not scale well
local detection afek kutten yung

(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]
local checking of a spanning tree with a root

2(A)

1(A)

D

B

root

A

0(C)

C

0(A)

root

G

E

2(A)

1(A)

H

F

1(C)

0(A)

Local checking of a spanning tree, with a root
local checking of other functions

Any graph marking function is locally checkable

  • Any algorithm global state is locally checkable
  • For any bit complexity Cfunction with local checking bit complexity O(C)
  • Some complexities for interesting marking functions are known
  • Many other problems are open
Local checking of other functions
from local checking to local correction

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 correction
from local checking to local correction1

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

(“impossible” if faulty majority)

25

24

5

10

6

7

Possibly consistent faults

From local checking to local correction
slide31

For example: need to prevent expansion of

Faulty area: recall Dijkstra’s algorithm

slide33

T

T

T

f faults create “gap” or “bump” of length f.

another example of expanding faulty area stable value problem

val2

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 Problem
stable value problem

val1

val1

A

B

val5

D

C

E

val1

val1

val’1 val1

val’1

Stable Value Problem

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

Alg. Recovers val1 everywhere.

reducing a general problem to the stable value problem

val2

val1

A

B

val5

D

C

E

val3

val4

Reducing a general problem to the Stable Value Problem

B can compute any func(val1, val2, val3,val4, val5) if it receives every correct vali.

simple but global solution consensus voting

val1

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 voting
slide38

val1

A

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
Local voting does not solve

Minority faulty notes but local majority everywhere

idea 1 for unknown f faults get votes values from radius 2 f

In O(f) time

get >2f votes

so majority is

non faulty

A

2f

Idea 1: For (unknown) f faults get votes (values) from radius 2f
idea 1 for unknown f faults get votes values from radius 2 f1

In O(f) time

get >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 2f
bcast vote sending spreading faults problem under state faults2

1

A said: “1”

D

A

B

C

A said: “0”

1

E

Bcast (vote sending) spreading faults problem under state faults
bcast vote sending spreading faults problem under state faults3

1

A said: “1”

D

A

B

C

A said: “0”

1

A

said:

“0”

E

Bcast (vote sending) spreading faults problem under state faults
bcast vote sending spreading faults problem under state faults4

1

A said: “1”

D

A

B

C

A said: “0”

1

A

said:

“0”

E

Bcast (vote sending) spreading faults problem under state faults
bcast vote sending spreading faults problem under state faults5

1

A said: “1”

D

A

B

C

A said: “0”

1

A

said:

“0”

E

Bcast (vote sending) spreading faults problem under state faults
bcast vote sending spreading faults problem under state faults6

1

A said: “1”

D

A

B

C

A said: “0”

1

A

said:

“0”

E

Bcast (vote sending) spreading faults problem under state faults
bcast vote sending spreading faults problem under state faults7

1

A said: “1”

D

A

B

C

A said: “0”

1

A

said:

“0”

E

Bcast (vote sending) spreading faults problem under state faults
bcast vote sending spreading faults problem under state faults8

Actually,

A said: “1”

1

A said: “1”

D

A

B

C

1

A

said:

“0”

E

Bcast (vote sending) spreading faults problem under state faults
bcast vote sending spreading faults problem under state faults9

Actually,

A said: “1”

1

A said: “1”

D

A

B

C

1

Actually,

A said:

“1”

E

Bcast (vote sending) spreading faults problem under state faults
bcast vote sending spreading faults problem under state faults10

Actually,

A 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 faults
bcast vote sending spreading faults problem under state faults11

Actually,

A 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 faults
bcast vote sending spreading faults problem under state faults12

Actually,

A 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 faults
bcast vote sending spreading faults problem under state faults13

Actually,

A 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 faults
another bcast spreading fault problem

F

A

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 problem
sample technique solving the bcast authenticity in o f time1

Regulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

1

A

B

C

D

A

1111111111111111111111111111111111111111111111111111111111

Sample technique: solving the bcast authenticity in O(f) time
sample technique solving the bcast authenticity in o f time2

Regulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

A

f= five faults hit

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

1111111111111111111111111111111111111111111111111111111111

111111111111111000001111111111111111111111111111111111111

sample technique solving the bcast authenticity in o f time3

Regulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

0

1

111111110000  11111111111111111111111

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

A

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

sample technique solving the bcast authenticity in o f time4

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) time

After two time units, the  spreads twice as fast as

the bcast (“0”s and “1”s).

sample technique solving the bcast authenticity in o f time5

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) time

After Three time units, the  spreads twice as fast as

the bcast (“0”s and “1”s).

sample technique solving the bcast authenticity in o f time6

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) time

After Four time units, the  spreads twice as fast as

the bcast (“0”s and “1”s).

sample technique solving the bcast authenticity in o f time7

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) time
sample technique solving the bcast authenticity in o f time8

Regulated Bcast [Kutten, Patt-Shamir]

Power Supply [Afek, Bremler]

0

1

11111111111  0 0 11111111111

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

A

After Six time units, the  spreads twice as fast as

the bcast (“0”s and “1”s).

sample technique solving the bcast authenticity in o f time9

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) time

A

sample technique solving the bcast authenticity in o f time10

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) time

A

sample technique solving the bcast authenticity in o f time11

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) time
sample technique solving the bcast authenticity in o f time12

Regulated 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) time
recall the stable value problem

Impossible

after 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
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