1. The iterated shared memory model of computation and an enrichment with safe-consensus tasks
Rodolfo Conde
Joint work with Sergio Rajsbaum
Instituto de Matemáticas
Universidad Nacional Autónoma de México
GETCO 2010

2. Rodolfo Conde and Sergio Rajsbaum
The Model
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Rodolfo Conde and Sergio Rajsbaum

3. The Iterated Snapshot Shared Memory model
We have n processes that communicate using a memory SM[i][0…n] (i ≥ 0) of Read/Write registers
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Rodolfo Conde and Sergio Rajsbaum

4. The Iterated Snapshot Shared Memory model
The computation proceeds in rounds
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Rodolfo Conde and Sergio Rajsbaum

5. The Iterated Snapshot Shared Memory model
In each round, a process P can atomically write to SM[i][P] 1
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Rodolfo Conde and Sergio Rajsbaum

6. The Iterated Snapshot Shared Memory model
each process can atomically read all of SM[i] 1
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Rodolfo Conde and Sergio Rajsbaum

7. The Iterated Snapshot Shared Memory model
In each round, the processes use a new memory array 1 1
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Rodolfo Conde and Sergio Rajsbaum

8. Rodolfo Conde and Sergio Rajsbaum
Asynchronous
The n processes are asynchronous
Arbitrary delays of any kind
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Rodolfo Conde and Sergio Rajsbaum

9. Rodolfo Conde and Sergio Rajsbaum
Wait-Free
The protocols are wait-free
All but one process can crash
A process cannot wait to hear from another process ?
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Rodolfo Conde and Sergio Rajsbaum

10. Generic Iterated Snapshot protocol
init r := 0; sm := input, dec := NULL; loop forever r := r + 1; SM[r].write(sm); sm := SM[r].snapshot(); /* Local computing */ end loop
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Rodolfo Conde and Sergio Rajsbaum

11. Generic Iterated Snapshot protocol
init r := 0; sm := input, dec := NULL; loop forever r := r + 1; SM[r].write(sm); sm := SM[r].snapshot(); /* Local computing */ end loop
P writes sm to SM[r][P]
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Rodolfo Conde and Sergio Rajsbaum

12. Generic Iterated Snapshot protocol
init r := 0; sm := input, dec := NULL; loop forever r := r + 1; SM[r].write(sm); sm := SM[r].snapshot(); /* Local computing */ end loop
P reads all the array SM[r]
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Rodolfo Conde and Sergio Rajsbaum

13. Two processes protocol
One possible execution is the following: the two processes read and write concurrently WR RD 1 WR RD
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Rodolfo Conde and Sergio Rajsbaum

14. Two processes protocol
We can represent this execution as a 1-simplex WR RD 1 WR RD
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Rodolfo Conde and Sergio Rajsbaum

15. Two processes protocol
Each vertex represents the process view of the memory WR RD 1 WR RD 01 01
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Rodolfo Conde and Sergio Rajsbaum

16. Two processes protocol
Another possible execution: One process is faster that the other WR RD WR RD 1 WR RD 1 WR RD 01 01
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Rodolfo Conde and Sergio Rajsbaum

17. Two processes protocol
The red process only sees itself, but the green can see both of them WR RD WR RD 1 WR RD 1 WR RD 01 01
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Rodolfo Conde and Sergio Rajsbaum

18. Two processes protocol
And the last possibility 1 WR RD WR RD WR RD WR RD 1 WR RD 1 WR RD 01 01
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Rodolfo Conde and Sergio Rajsbaum

19. Two processes protocol
And the last possibility 1 WR RD WR RD WR RD WR RD 1 WR RD 1 WR RD 1 01 01
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Rodolfo Conde and Sergio Rajsbaum

20. Protocol complex (1 round)01 01
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Rodolfo Conde and Sergio Rajsbaum

21. Rodolfo Conde and Sergio Rajsbaum
The 2nd round
The input for the 2nd round is any possible state after the first round WR RD 1 WR RD 01 01
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Rodolfo Conde and Sergio Rajsbaum

22. Rodolfo Conde and Sergio Rajsbaum
The 2nd round
And the three possibilities repeat WR RD 1 WR RD WR RD 1 WR RD 01 01
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Rodolfo Conde and Sergio Rajsbaum

23. Rodolfo Conde and Sergio Rajsbaum
The 2nd round
And the three possibilities repeat WR RD 1 WR RD WR RD 1 WR RD 01 01
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Rodolfo Conde and Sergio Rajsbaum

24. Rodolfo Conde and Sergio Rajsbaum
The 2nd round
And the three possibilities repeat WR RD 1 WR RD 1 WR RD WR RD 01 01
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Rodolfo Conde and Sergio Rajsbaum

25. Two processes protocols in the iterated model
Given a possible input:
Each execution of a round is represented as a 1-simplex
All possible executions are represented as a simplicial complex (subdivision of a line)
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Rodolfo Conde and Sergio Rajsbaum

26. Three processes protocols
The state after an execution can be described by a triangle (2-simplex) WR RD WR RD WR RD
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Rodolfo Conde and Sergio Rajsbaum

27. Three processes protocols
The state after an execution can be described by a triangle (2-simplex)
000 WR RD WR RD WR RD 00 00
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Rodolfo Conde and Sergio Rajsbaum

28. Protocol complex (1st round)
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Rodolfo Conde and Sergio Rajsbaum

29. Protocol complex (1st round)WR RD WR RD WR RD
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Rodolfo Conde and Sergio Rajsbaum

30. Protocol complex (1st round)WR RD WR RD WR RD
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Rodolfo Conde and Sergio Rajsbaum

31. Protocol complex (1st round)WR RD WR RD WR RD
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Rodolfo Conde and Sergio Rajsbaum

32. Protocol complex (1st round)WR RD WR RD WR RD
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Rodolfo Conde and Sergio Rajsbaum

33. Protocol complex (3 processes)
For the 2nd round
Each triangle (state) of the 1st round subdivides in the same way
Because we work in an iterated model
Recursive behaviour
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Rodolfo Conde and Sergio Rajsbaum

34. Rercursive behaviour (2nd round)
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Rodolfo Conde and Sergio Rajsbaum

35. Rercursive behaviour (2nd round)
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Rodolfo Conde and Sergio Rajsbaum

36. Rercursive behaviour (2nd round)
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Rodolfo Conde and Sergio Rajsbaum

37. Protocol complex (2nd round)
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Rodolfo Conde and Sergio Rajsbaum

38. Rodolfo Conde and Sergio Rajsbaum
In general
For n + 1 processes:
Each state of a protocol is represented as a n-simplex
The executions of a protocol are represented as a n-dimensional complex
A subdivision of the n-simplex !! [Gafni & Borowsky]
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Rodolfo Conde and Sergio Rajsbaum

39. The (n,k)-set agreement task [S. Chaudhuri, 90]
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Rodolfo Conde and Sergio Rajsbaum

40. The (n,k)-set agreement task [S. Chaudhuri, 90]2 7 9
Processes start with private input values from a domain I (|I| ≥ n)
Set agreement
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Rodolfo Conde and Sergio Rajsbaum

41. The (n,k)-set agreement task [S. Chaudhuri, 90]2 7 9
Their outputs must agree on at most k < n distinct values
Set agreement 7 7 2
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Rodolfo Conde and Sergio Rajsbaum

42. Impossibility of (3,2)-set agreement in the Iterated model
We can use the geometric view of distributed protocols to show this remarkable result.
The basic idea is as follows:
Assume a protocol exists.
Find an execution of this protocol (using the protocol complex) where processes decide 3 values !!!
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Rodolfo Conde and Sergio Rajsbaum

43. Suppose a protocol exists
Consider an input where processes have as input values their own ids
Run the protocol until processes decide
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Rodolfo Conde and Sergio Rajsbaum

44. Suppose a protocol exists
Because we work in the iterated model
The protocol complex is a subdivision of the triangle
We can colour the vertices with the decision each process takes
This colouring satisfies the hypothesis of Sperner’s Lemma
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Rodolfo Conde and Sergio Rajsbaum

45. We apply Sperner’s lemma to the subdivided complex
By Sperner’s Lemma, at least one simplex has all three colours
This simplex corresponds to an execution where processes decide three distinct values !!!
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Rodolfo Conde and Sergio Rajsbaum

46. Rodolfo Conde and Sergio Rajsbaum
In summary
The iterated model
Executions are represented as simplicial complexes
Simple recursive structure
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Rodolfo Conde and Sergio Rajsbaum

47. Rodolfo Conde and Sergio Rajsbaum
In summary
The set agreement task is impossible to solve [Borowsky & Gafni, Saks & Zaharoglou, Herlihy & Shavit, 93]
The iterated model is equivalent to the usual read/write model [Borowsky & Gafni, 97]
Set agreement result is valid in the usual model (but easier to prove in the iterated model)
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Rodolfo Conde and Sergio Rajsbaum

48. We can enrich the Iterated model with more powerful objects
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Rodolfo Conde and Sergio Rajsbaum

49. The safe-consensus task
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Rodolfo Conde and Sergio Rajsbaum

50. The safe-consensus task [Afek, Gafni & Lieber, 09]
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Rodolfo Conde and Sergio Rajsbaum

51. The safe-consensus task [Afek, Gafni & Lieber, 09]2 7 9
Processes start with private input values from a domain I
Safe-consensus
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Rodolfo Conde and Sergio Rajsbaum

52. The safe-consensus task [Afek, Gafni & Lieber, 09]2 7 9
Their outputs values must be the same
Safe-consensus 2 2 2
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Rodolfo Conde and Sergio Rajsbaum

53. The safe-consensus task [Afek, Gafni & Lieber, 09]2 7 9
The safe-consensus has two special rules
Safe-consensus
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Rodolfo Conde and Sergio Rajsbaum

54. The safe-consensus task [Afek, Gafni & Lieber, 09]2 7 9
(1) If a process starts executing the task and outputs before any other process starts executing the task
Safe-consensus 7
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Rodolfo Conde and Sergio Rajsbaum

55. The safe-consensus task [Afek, Gafni & Lieber, 09]2 7 9
the task’s output is that process proposed input value.
Safe-consensus 7 7 7
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Rodolfo Conde and Sergio Rajsbaum

56. The safe-consensus task [Afek, Gafni & Lieber, 09]2 7 9
(2) Otherwise, if two or more processes initially access the task concurrently
Safe-consensus
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Rodolfo Conde and Sergio Rajsbaum

57. The safe-consensus task [Afek, Gafni & Lieber, 09]2 7 9
it can return any value.
(even invalid values)
Safe-consensus α α α
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Rodolfo Conde and Sergio Rajsbaum

58. Rodolfo Conde and Sergio Rajsbaum
What happens if we enrich the iterated model with safe-consensus tasks ?
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Rodolfo Conde and Sergio Rajsbaum

59. Rodolfo Conde and Sergio Rajsbaum
The enriched Model
init r := 0; sm, input, scret, dec := NULL; loop forever r := r + 1; SM[r].write(sm, scret); scret := safe-consensus[h(sm, scret)](id); sm := SM[r].snapshot(); /* Local computing */ end loop
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Rodolfo Conde and Sergio Rajsbaum

60. Rodolfo Conde and Sergio Rajsbaum
The enriched Model
init r := 0; sm, input, scret, dec := NULL; loop forever r := r + 1; SM[r].write(sm, scret); scret := safe-consensus[h(sm, scret)](id); sm := SM[r].snapshot(); /* Local computing */ end loop
Process access the object
indicated by h(sm, scret)
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Rodolfo Conde and Sergio Rajsbaum

61. What happens to the protocol complex ?
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Rodolfo Conde and Sergio Rajsbaum

62. Protocol complex with safe-consensus
1 round
3 processes
All processes invoke the safe-consensus
Input values: Ids
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Rodolfo Conde and Sergio Rajsbaum

63. Rodolfo Conde and Sergio Rajsbaum
A closer look
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

64. Rodolfo Conde and Sergio Rajsbaum
A closer look
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

65. Rodolfo Conde and Sergio Rajsbaum
A closer look
Which executions are represented in this complex ?
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

66. Rodolfo Conde and Sergio Rajsbaum
A closer look
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

67. Rodolfo Conde and Sergio Rajsbaum
A closer look
Why do we have only these executions ?
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

68. Rodolfo Conde and Sergio Rajsbaum
A closer look WR SC RD WR SC RD WR SC RD
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

69. Rodolfo Conde and Sergio Rajsbaum
A closer look WR SC RD
Safe-consensus = WR SC RD WR SC RD
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

70. Rodolfo Conde and Sergio Rajsbaum
A closer look
Executions where the safe-consensus returns green
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

71. Rodolfo Conde and Sergio Rajsbaum
A closer look
Why there cannot be more adjacent simplexes ?
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

72. Rodolfo Conde and Sergio Rajsbaum
A closer look
Why there cannot be more adjacent simplexes ?
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

73. Rodolfo Conde and Sergio Rajsbaum
A closer look
Why there cannot be more adjacent simplexes ? WR SC RD WR SC RD
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value WR SC RD
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Rodolfo Conde and Sergio Rajsbaum

74. Rodolfo Conde and Sergio Rajsbaum
A closer look
Why there cannot be more adjacent simplexes ?
Safe-consensus = WR SC RD WR SC RD
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value WR SC RD
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Rodolfo Conde and Sergio Rajsbaum

75. Rodolfo Conde and Sergio Rajsbaum
A closer look
Why there cannot be more adjacent simplexes ?
Safe-consensus =
Safe-consensus = WR SC RD WR SC RD
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value WR SC RD
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Rodolfo Conde and Sergio Rajsbaum

76. Rodolfo Conde and Sergio Rajsbaum
A closer look
Because the safe-consensus does not allow it
Safe-consensus =
Safe-consensus =
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

77. Rodolfo Conde and Sergio Rajsbaum
A closer look
Similar argument for other executions
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

78. Rodolfo Conde and Sergio Rajsbaum
A closer look
Similary for other executions WR SC RD
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value WR SC RD WR SC RD
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Rodolfo Conde and Sergio Rajsbaum

79. Rodolfo Conde and Sergio Rajsbaum
A closer look
Similary for other executions
Safe-consensus = WR SC RD
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value WR SC RD WR SC RD
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Rodolfo Conde and Sergio Rajsbaum

80. Rodolfo Conde and Sergio Rajsbaum
A closer look
Similary for other executions
Safe-consensus =
Safe-consensus = WR SC RD
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value WR SC RD WR SC RD
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Rodolfo Conde and Sergio Rajsbaum

81. Rodolfo Conde and Sergio Rajsbaum
A closer look
Similary for other executions
Safe-consensus =
Safe-consensus =
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

82. Rodolfo Conde and Sergio Rajsbaum
A closer look
Safe-consensus =
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

83. Rodolfo Conde and Sergio Rajsbaum
A closer look
Safe-consensus =
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

84. Rodolfo Conde and Sergio Rajsbaum
A closer look
Safe-consensus =
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

85. Rodolfo Conde and Sergio Rajsbaum
And the black complex
It represents executions where the safe-consensus returns an invalid value
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

86. Rodolfo Conde and Sergio Rajsbaum
And the black complex
At least two processes invoke the safe-consensus concurrently WR SC RD WR SC RD
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value WR SC RD
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Rodolfo Conde and Sergio Rajsbaum

87. Rodolfo Conde and Sergio Rajsbaum
And the black complex
At least two processes invoke the safe-consensus concurrently WR SC RD WR SC RD WR SC RD
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

88. A closer look Safe-consensus = a value different from valid ids
Each component represents a set of executions of the protocol where the safe-consensus is allowed to return the indicated value
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Rodolfo Conde and Sergio Rajsbaum

89. Rodolfo Conde and Sergio Rajsbaum
And again…
Because we work in the iterated model
In the 2nd round
This behaviour is going to repeat
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Rodolfo Conde and Sergio Rajsbaum

90. Remember, Iterated model
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Rodolfo Conde and Sergio Rajsbaum

91. Remember, Iterated model
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Rodolfo Conde and Sergio Rajsbaum

92. Remember, Iterated model
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Rodolfo Conde and Sergio Rajsbaum

93. Some results for set agreement
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Rodolfo Conde and Sergio Rajsbaum

94. (k+1,k)-set agreement is solvable in this model
proc (k+1,k)-set-agreement(val) SM.write(val); sc := safe-consensus(id); sm := SM.snapshot(); if (sc is in {1, …, k+1} Λ sm[sc] ≠ NULL) then dec := sm[sc]; else dec := sm[j] with j := min{ m | sm[m] ≠ NULL }; end if decide dec; end proc
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Rodolfo Conde and Sergio Rajsbaum

95. Rodolfo Conde and Sergio Rajsbaum
Notice
We omit here the correctness proof of the protocol
Not difficult, but tedious
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Rodolfo Conde and Sergio Rajsbaum

96. Rodolfo Conde and Sergio Rajsbaum
In particular
(3,2)-set agreement is solvable in the iterated model with safe-consensus.
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Rodolfo Conde and Sergio Rajsbaum

97. Rodolfo Conde and Sergio Rajsbaum
In particular
(3,2)-set agreement is solvable in the iterated model with safe-consensus. But we can prove that (3,1)-set agreement (3-consensus) is not solvable in the Iterated model with safe-consensus.
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Rodolfo Conde and Sergio Rajsbaum

98. Rodolfo Conde and Sergio Rajsbaum
Proof’s idea
Suppose a protocol exists.
Consider an input where processes propose their ids
Take the gray subcomplex
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Rodolfo Conde and Sergio Rajsbaum

99. In the protocol’s 1st roundWR SC RD
Notamos el camino que hay de la ejecución en solo del proceso verde a la ejecución de los procesos rojo y amarillo (sin el verde) WR SC RD WR SC RD
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Rodolfo Conde and Sergio Rajsbaum

100. In the protocol’s 1st round
Notamos el camino que hay de la ejecución en solo del proceso verde a la ejecución de los procesos rojo y amarillo (sin el verde) WR SC RD WR SC RD WR SC RD
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Rodolfo Conde and Sergio Rajsbaum

101. In the protocol’s 1st round
Notamos el camino que hay de la ejecución en solo del proceso verde a la ejecución de los procesos rojo y amarillo (sin el verde)
There exists a path here
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Rodolfo Conde and Sergio Rajsbaum

102. In the protocol’s 2nd round
And because we work in an iterated model
El camino entre una ejecucion en solo del proceso verde y una ejecucion en solo del rojo y el amarillo aun existe…
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Rodolfo Conde and Sergio Rajsbaum

103. Rodolfo Conde and Sergio Rajsbaum
2nd round
El camino entre una ejecucion en solo del proceso verde y una ejecucion en solo del rojo y el amarillo aun existe…
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Rodolfo Conde and Sergio Rajsbaum

104. Rodolfo Conde and Sergio Rajsbaum
2nd round
El camino entre una ejecucion en solo del proceso verde y una ejecucion en solo del rojo y el amarillo aun existe…
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Rodolfo Conde and Sergio Rajsbaum

105. Rodolfo Conde and Sergio Rajsbaum
2nd round
El camino entre una ejecucion en solo del proceso verde y una ejecucion en solo del rojo y el amarillo aun existe…
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Rodolfo Conde and Sergio Rajsbaum

106. Rodolfo Conde and Sergio Rajsbaum
2nd round WR SC RD
El camino entre una ejecucion en solo del proceso verde y una ejecucion en solo del rojo y el amarillo aun existe… WR SC RD WR SC RD
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Rodolfo Conde and Sergio Rajsbaum

107. Rodolfo Conde and Sergio Rajsbaum
2nd round
El camino entre una ejecucion en solo del proceso verde y una ejecucion en solo del rojo y el amarillo aun existe… WR SC RD WR SC RD WR SC RD
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Rodolfo Conde and Sergio Rajsbaum

108. Rodolfo Conde and Sergio Rajsbaum
2nd round
There’s also a path
El camino entre una ejecucion en solo del proceso verde y una ejecucion en solo del rojo y el amarillo aun existe… …
And so on…
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Rodolfo Conde and Sergio Rajsbaum

109. Because of the iterated model
In any r-round partial execution:
a solo execution of
is “conected” to a execution of and without
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Rodolfo Conde and Sergio Rajsbaum

110. When the protocol finishes
must decide green
Argumentamos que esa conectividad entre esas ejecuciones generan una contradiccion
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Rodolfo Conde and Sergio Rajsbaum

111. When the protocol finishes
must decide green
Argumentamos que esa conectividad entre esas ejecuciones generan una contradiccion
must decide red or yellow
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Rodolfo Conde and Sergio Rajsbaum

112. Rodolfo Conde and Sergio Rajsbaum
Contradiction
Argumentamos que esa conectividad entre esas ejecuciones generan una contradiccion
This “connectivity” in all rounds lead us to a contradiction, so no such protocol can exists
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Rodolfo Conde and Sergio Rajsbaum

113. Rodolfo Conde and Sergio Rajsbaum
In general
There is no protocol in the Iterated Snapshot model with safe-consensus objects that can solve the (k, 1)-set agreement problem (k ≥ 3).
Generalizacion del resultado anterior
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Rodolfo Conde and Sergio Rajsbaum

114. Rodolfo Conde and Sergio Rajsbaum
Summary
There’s a deep conection between Distributed computing and Topology
Impossibility results arise from this conection
We can derive algorithms by looking at the geometric structure of protocol complexes
Shared objects can affect the topology of the protocol complex (safe-consensus)
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Rodolfo Conde and Sergio Rajsbaum

115. Rodolfo Conde and Sergio Rajsbaum
Thank you
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Rodolfo Conde and Sergio Rajsbaum