Colorless Wait-Free Computation

Transcription

1 Colorless Wait-Free Computation Companion slides for Distributed Computing Through Maurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum Distributed Computing through 1

2 Colorless Tasks Apr-14 2

3 Colorless Tasks Apr-14 3

4 Colorless Tasks The set of input values determines the set of output values. Number and identities irrelevant for both input and output values 19-Apr-14 4

5 Examples Consensus 32 k-set agreement Apr-14 5

6 Non-Examples Weak Symmetry-Breaking When all participate At least one on group 0, group 1 19-Apr-14 6

7 Yes No No No! No! Majority No!

8 Operational Model Road Map Combinatorial Model Main Theorem Distributed Computing through 8

9 A process is a state machine Processes Could be Turing machines or more powerful Distributed Computing though 9

10 Processes A process s state is called its view Distributed Computing though Process names taken from a domain Each process has a unique name (color) P i 2 10

11 Each process knows its own name Processes But not the names of the other processes Distributed Computing though 11

12 Often, P i is just i Processes Sometimes P i and i are distinct, and the process knows P i but not i Distributed Computing though 12

13 Processes There are n+1 processes Distributed Computing though 13

14 Shared Read-Write Memory Distributed Computing though For now, they communicate via reading & writing shared memory 14

15 Immediate Snapshot Individual reads & writes are too low-level A snapshot = atomic read of all memory We will use immediate snapshot Distributed Computing through 15

16 Immediate Snapshot write view to memory take snapshot adjacent steps! Distributed Computing through 16

17 Immediate Snapshot write view to memory take snapshot write view to memory take snapshot Distributed Computing through 17

18 Immediate Snapshot immediate mem[i] := view; snap := snapshot(mem[*]) Distributed Computing through 18

19 P Q R write snap write snap write snap {p} {p,q} {p,q,r} P Q R write snap write write snap snap {p} {p,q,r} {p,q,r} P Q R write write write snap snap snap {p,q,r} {p,q,r} {p,q,r}

20 Realistic? No! My laptop reads only a few contiguous memory words at a time Yes! Simpler lower bounds: if it s impossible with IS, it s impossible on your laptop. Can implement IS from read-write Distributed Computing through 20

21 Crashes Processes may halt without warning as many as n out of n+1 19-Apr-14 21

22 Asynchronous

23 Asynchronous Failures?? detection impossible

24 Configurations C = {s 0,, s n } set of simultaneous process states initial configurations final configurations Distributed Computing through 24

25 Executions processes that communicate C 0, S 0, C 1, S 1,,S r, C r+1 initial configuration final configuration next configuration Distributed Computing through 25

26 Executions triple is a concurrent step C 0, S 0, C 1, S 1,,S r, C r+1 Processes in S 0 said to participate in step Only P i 2 S 0 can change between C 0 and C 1 state change result of communication Distributed Computing through 26

27 Executions finite C 0, S 0, C 1, S 1,, S r, C r+1 infinite C 0, S 0, C 1, S 1,, partial C 0, S 0, C 1, S 1,, S r, C r+1 Distributed Computing through 27

28 Crashes are Implicit C 0, S 0, C 1, S 1,, S r, C r+1 If P i s state not final in the final configuration then P i has crashed. crash cannot be detected in finite time Distributed Computing through 28

29 Colorless Tasks (I, O, ) (colorless) input assignment carrier map : I! 2 O (colorless) output assignment Distributed Computing through 29

30 Input Assignments {(P j, v j ), P j 2, v j 2 V in } name, value pair domain of process names domain of input values Distributed Computing through 30

31 Colorless Input Assignments {(P j, v j ), P j 2, v j 2 V in } discard process names, keep values Distributed Computing through 31

32 (Colorless) Output Assignments {(P j, v j ), P j 2, v j 2 V out } {(P j, v j ), P j 2, v j 2 V out } Distributed Computing through 32

33 Example: Binary Consensus I = {{0}, {1}, {0,1}} All start with 0 All start with 1 Distributed Computing through start with both 33

34 Example: Binary Consensus I = {{0}, {1}, {0,1}} O = {{0}, {1}} All decide 0 All decide 1 Distributed Computing through 34

35 Example: Binary Consensus ({0}) = {{0}} All start with 0, all decide 0 Distributed Computing through 35

36 Example: Binary Consensus ({0}) = {{0}} ({1}) = {{1}} All start with 1, all decide 1 Distributed Computing through 36

37 Example: Binary Consensus ({0}) = {{0}} ({1}) = {{1}} ({0,1}) = {{0},{1}} with mixed inputs, all decide 0, or all decide 1 Distributed Computing through 37

38 Colorless Layered Protocol shared mem array 0..N-1,0..n of Value view := input for l := 0 to N-1 do immediate mem[l][i] := view; snap := snapshot(mem[l][*]) view := set of values in snap return δ(view) Distributed Computing through 38

39 Colorless Layered Protocol shared mem array 0..N-1,0..n of Value view := input for j := 0 to N-1 do immediate 2-dimensional memory array mem[j][i] := view; row is clean per-layer memory snap := snapshot(mem[j][*]) view := set column of values is per-process in snap word return δ(view) Distributed Computing through 39

40 Colorless Layered Protocol shared mem array 0..N-1,0..n of Value view := input for j := 0 to N-1 do immediate initial mem[j][i] view is input := view; value snap := snapshot(mem[j][*]) view := set of values in snap return δ(view) Distributed Computing through 40

41 Colorless Layered Protocol shared mem array 0..N-1,0..n of Value view := input for l := 0 to N-1 do immediate mem[j][i] := view; snap run := for snapshot(mem[j][*]) N layers view := set of values in snap return δ(view) Distributed Computing through 41

42 Colorless Layered Protocol shared mem array 0..N-1,0..n of Value layer l : immediate write & snapshot of row l view := input for j := 0 to N-1 do immediate mem[l][i] := view; snap := snapshot(mem[l][*]) view := set of values in snap return δ(view) Distributed Computing through 42

43 Colorless Layered Protocol shared mem array 0..N-1,0..n of Value view := input for j := 0 to N-1 do immediate new view is set of values seen mem[j][i] := view; snap := snapshot(mem[j][*]) view := set of values in snap return δ(view) Distributed Computing through 43

44 Colorless Layered Protocol shared mem array 0..N-1,0..n of Value view := input for j := 0 to N-1 do immediate finally mem[j][i] apply decision := view; value to final view snap := snapshot(mem[j][*]) view := set of values in snap return δ(view) Distributed Computing through 44

45 {p},{p,q,r} Q {p},q,{p,r} {p},{p,q,r},{p,r} R {p},{p,q},r {p},{p,q},{p,q,r} {p,q,r},{q} P p,{q},{q,r} {p,q,r},{q},{q,r} {p,q},{q},r R {p,q},{q},{p,q,r} {q,r},{p,q,r} {p,r},{p,q,r} {p,q},{p,q,r} {p,q,r},{r} p,{q,r},{r} P {p,q,r},{q,r},{r} {p,r},q,{r} Q {p,r},{p,q,r},{r} R {p,q,r} p,{q,r} {p,r},q {p,q},r R Q QR {p},q,r p,{q},r p,q,{r} R P PR Q P PQ Q P p,q,r PQR P Q R PQ PR QR

46 P Q R p,q,r PQ PR QR PQR {p},q,r p,{q},r p,q,{r} {p,q},r {p,q,r} QR PR PQ Q R P R P Q R {p},{p,q},r {p},q,{p,r} {p},{p,q,r} {p,q},{q},r p,{q},{q,r} {p,q,r},{q} {p,r},q,{r} p,{q,r},{r} {p,q,r},{r} {q,r},{p,q,r} R Q R P Q P {p},{p,q},{p,q,r} {p},{p,q,r},{p,r} {p,q},{q},{p,q,r} {p,q,r},{q},{q,r} {p,r},{p,q,r},{r} {p,q,r},{q,r},{r} Colorless configurations for {p,r},q {p,q},{p,q,r} processes Q P,Q,R, inputs p,q,r, p,{q,r} final P configurations {p,r},{p,q,r} in black.

47 Operational Model Road Map Combinatorial Model Main Theorem Distributed Computing through 47

48 Vertex = Process State Process ID (color) 7 Value (input or output) 19-Apr-14 48

49 Simplex = Global State 19-Apr-14 49

50 Complex = Global States 19-Apr-14 50

51 Input Complex for Binary Consensus 0 All possible initial states Processes: red, green, blue 1 Independently assigned 0 or 1 19-Apr-14 51

52 Output Complex for Binary Consensus 0 0 All possible final states Output values all 0 or all 1 1 Two disconnected simplexes 19-Apr-14 52

53 Carrier Map for Consensus All 0 inputs All 0 outputs 19-Apr-14 53

54 Carrier Map for Consensus All 1 inputs All 1 outputs 19-Apr-14 54

55 Carrier Map for Consensus All 0 outputs Mixed 0-1 inputs All 1 outputs 19-Apr-14 55

56 Task Specification (I, O, ) Input complex Output complex Carrier map : I! 2 O 19-Apr-14 56

57 Colorless Tasks (I, P, ) (colorless) input complex strict carrier map : I! 2 P (colorless) protocol complex Distributed Computing through 57

58 Protocol Complex Vertex: process name, view all values read and written Simplex: compatible set of views Each execution defines a simplex 19-Apr-14 Distributed Computing through 58

59 Example: Synchronous Message-Passing Round 0 Round 1 19-Apr-14 59

60 Failures: Fail-Stop Partial broadcast 19-Apr-14 Distributed Computing through 60

61 Single Input: Round Zero 0 No messages sent View is input value 0 0 Same as input simplex 19-Apr-14 Distributed Computing through 61

62 Round Zero Protocol Complex 0 No messages sent View is input value 1 Same as input complex 19-Apr-14 Distributed Computing through 62

63 Single Input: Round One Apr-14 Distributed Computing through 63

64 Single Input: Round One no one fails 19-Apr-14 Distributed Computing through 64

65 Single Input: Round One blue fails no one fails Apr-14 Distributed Computing through 65

66 Single Input: Round One red fails green fails blue fails no one fails Apr-14 Distributed Computing through 66

67 Protocol Complex: Round One 19-Apr-14 Distributed Computing through 67

68 Protocol Complex: Round Two 19-Apr-14 Distributed Computing through 68

69 Protocol Complex Evolution zero one two 19-Apr-14 69

70 Ξ protocol complex Summary δ input complex output complex Δ 19-Apr-14 70

71 Decision Map δ Simplicial map, sending simplexes to simplexes Protocol complex Output complex 19-Apr-14 71

72 Lower Bound Strategy δ Find topological obstruction to this simplicial map Protocol complex Output complex 19-Apr-14 72

73 Consensus Example 1 0 Subcomplex of all-0 inputs δ Must map here 0 1 Protocol Output 19-Apr-14 Distributed Computing through 73

74 Consensus Example 1 0 δ 0 1 Protocol Subcomplex of all-1 inputs Must map Output here 19-Apr-14 Distributed Computing through 74

75 Consensus Example Image under δ must start here δ 0 1 Protocol Path from all-0 to all-1 and end here Output 19-Apr-14 Distributed Computing through 75

76 Consensus Example path 0 1 δ? 0 1 Output 19-Apr-14 Distributed Computing through 76

77 Consensus Example Image under δ must start here.. But this hole is δ an obstruction Protocol Path from all-0 to all-1 and end here Output 19-Apr-14 Distributed Computing through 77

78 Conjecture A protocol cannot solve consensus if its complex is path-connected Model-independent! 19-Apr-14 Distributed Computing through 78

79 If Adversary keeps Protocol Complex path-connected Forever Consensus is impossible For r rounds A round-complexity lower bound For time t A time-complexity lower bound 19-Apr-14 Distributed Computing through 79

80 Barycentric Agreement I Bary I Distributed Computing through 80

81 Barycentric Agreement (I, bary I, bary( )) arbitrary input complex subdivided output complex subdivision as carrier map Distributed Computing through 81

82 If There are n Processes (I, bary skel n I, bary skel n ) Inputs only from n-skeleton of input complex Distributed Computing through 82

83 Theorem A one-layer immediate snapshot protocol solves the n-process barycentric agreement task (I, bary skel n I, bary skel n ) Proof All input simplices belong to skel n I Immediate snapshot results are ordered Distributed Computing through 83

84 Barycentric Agreement Protocol {v 0 } {v 0,v 1,v 2 } {v 0, v 2 } v 0 v 1 v 2 Snapshots are ordered

85 Barycentric Agreement Protocol {v 0,v 1,v 2 } {v 0 } {v 0, v 2 } Each view is a face of ¾

86 Barycentric Agreement Protocol {v 0 } {v 0,v 1,v 2 } {v 0, v 2 } Each view is a face of ¾

87 Barycentric Agreement Protocol {v 0 } {v 0,v 1,v 2 } {v 0, v 2 } Each face of ¾ is a vertex of Bary ¾

88 Barycentric Agreement Protocol Ordered faces ) simplex of Bary ¾

89 Iterated Barycentric Agreement skel n I bary N skel n I

90 Iterated Barycentric Agreement (I, bary N skel n I, bary N skel n ) Distributed Computing through 90

91 One-Layer Immediate Snapshot Protocol Complex {p}{pq}{pqr} {p}{pqr}{pqr} {pqr}{pqr}{pqr} {pqr}{qr}{qr} Distributed Computing through

92 Compare Views P Q R p,q,r PQ P R Q PQR R {p},q,r Q {p},{p,q},r {p},{p,q},{p,q,r} R R {p},q,{p,r} {p},{p,q,r},{p,r} Q QR {p},{p,q,r} p,{q},r P {p,q},{q},r {p,q},{q},{p,q,r} R R p,{q},{q,r} {p,q,r},{q},{q,r} P PR {p,q,r},{q} p,q,{r} P {p,r},q,{r} {p,r},{p,q,r},{r} Q Q p,{q,r},{r} {p,q,r},{q,r},{r} P PQ {p,q},r {p,q,r},{r} R {p,r},q {p,q},{p,q,r} Q p,{q,r} {p,r},{p,q,r} P {p,q,r} {q,r},{p,q,r} Operational view {p}{pq}{pqr} {p}{pqr}{pqr} {pqr}{pqr}{pqr} {pqr}{qr}{qr} Combinatorial view Distributed Computing through

93 Compositions Given protocols (I,P, ) (I,P, ) where P µ I their composition is (I,P, ) where = and P = (I) Distributed Computing through 93

94 Operational Model Road Map Combinatorial Model Main Theorem Distributed Computing through 94

95 Fundamental Theorem Theorem (I,O, ) has a wait-free (n+1)-process layered protocol iff there is a continuous map f: skel n I O... carried by 19-Apr-14 Distributed Computing through

96 Proof Outline Lemma If there is a WF layered protocol for then (I,O, ) there is a continuous f: skel n I! O carried by. 19-Apr-14 Distributed Computing through 96

97 Map ) Protocol Hypothesis f Continuous Bary n I O Distributed Computing through 97

98 Map ) Protocol Á f Simplicial approximation Bary n I O 19-Apr-14 Distributed Computing through 98

99 Map ) Protocol Á Run barycentric Agreement Bary n I Apply Á O 19-Apr-14 Distributed Computing through 99 QED

100 Protocol ) Map Lemma Theorem If there is a WF-RW protocol for then if and only if (I,O, ) there is a continuous f: skel n I! O carried by. 19-Apr-14 Distributed Computing through 100

101 Proof strategy Protocol ) Map Inductive construction g d : skel d I! (I). Base d = 0 Define g 0 : skel 0 I! (I) Let g 0 (v) be any vertex in ({v}) 19-Apr-14 Distributed Computing through 101

102 Induction Hypothesis Protocol ) Map g d-1 : skel d-1 I! (I) For all ¾ in skel d-1 I g d-1 sends skel d-1 ¾ (¾) But (¾) is (d-1)-connected (earlier theorem) Can extend to g d : ¾ (¾) Yielding g d : skel d I! (I) 19-Apr-14 Distributed Computing through 102

103 Constructed Protocol ) Map g: skel n I (I) Simplicial decision map δ: (skel n I) O δ : (skel n I) O Composition f = δ g yields f: skel n I! O carried by. QED 19-Apr-14 Distributed Computing through 103

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