Distributed Computing through Combinatorial Topology. Maurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum

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1 Distributed Computing through Maurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum 1

2 In the Beginning a computer was just a Turing machine Distributed Computing though 2

3 Today??? Computing is co-ordination and communication Distributed Computing though 3

4 Distributed computations unfold in time! No, distributed computations are static mathematical objects! Operational versus combinatorial approaches 4

5 There are Many Models Communication? Distributed Computing though 5

6 There are Many Models Communication? Failures? Distributed Computing though 6

7 There are Many Models Communication? Failures? Timing? Distributed Computing though 7

8 Local Views 110 Each process has a 3-bit local view Distributed Computing though 8

9 Multiple Local Views local views differ by 1 bit Distributed Computing though 9

10 Multiple Local Views 110 local views differ by 1 bit but no process knows which one Distributed Computing though

11 Multiple Local Views each view is represented by a labeled vertex Distributed Computing though 11

12 Global States compatible views represented by an edge Distributed Computing though 12

13 All possible global states (where (where blue blue has has 111) 111)

14 Communication Each process sends local view to the other Distributed Computing though 14

Communication 110 111 111 Each process sends local view to the other

15 Communication Each process sends local view to the other but at most one message may be lost! Distributed Computing though 15

16 One Communication Round 110? Distributed Computing though 111? 16

17 One Communication Round 110? ? 1 Lost Distributed Computing though 17

18 One Communication Round 110? Lost ? none Lost Distributed Computing though 18

19 One Communication Round 110? Lost none Lost Distributed Computing though 111? 1 Lost 19

20 One Communication Round 110? Distributed Computing though 111? 20

21 All possible global states after one round unreliable communication 110? ?

23 Informally. Unreliable communication does not change topology of global states

24 Reliable Communication? 110? Distributed Computing though 111? 24

25 Reliable Communication? Distributed Computing though 25

27 A Classic Distributed Problem Muddy Children 11:00 Distributed Computing though 27

28 Muddy Children 11:01 Distributed Computing though 28

29 Muddy Children 12:00 At least one of you is dirty! Distributed Computing though 29

30 Muddy Children 12:00 You may not communicate! Distributed Computing though 30

31 Muddy Children 12:00 When you realize you are dirty, confess on the hour! Distributed Computing though 31

32 Muddy Children 1:00 (silence ) Distributed Computing though 32

33 Muddy Children 2:00 Me! Me! Distributed Computing though 33

34 Operational Explanation 1:00 Distributed Computing though 34

35 Operational Explanation Others are clean, so I must be dirty. Distributed Computing though 1:00 35

36 Operational Explanation Others are clean, so I must be dirty. 1:00 Me! Distributed Computing though 36

37 Operational Explanation 1:01 Distributed Computing though 37

38 Operational Explanation He was quiet, so I must be dirty. 1:01 He was quiet, so I must be dirty. Distributed Computing though 38

39 Combinatorial Explanation 12:00 Distributed Computing though 39

40 Combinatorial Explanation 12:00 Distributed Computing though 40

41 Combinatorial Explanation 12:00 Each process has its own input Distributed Computing though 41

42 Combinatorial01? Explanation?11 12:00 Each process has its own input 0?1 Distributed Computing though 42

43 Combinatorial01? Explanation?11 12:00 Global State 0?1 Distributed Computing though 43

44 11:59 all clean 00? 0? 0?00 0?1 01??1 0?01 10? 1?1?11 1? 0 11? all dirty

45 12:01 00? 0? 0?00 0?1 01??1 0?01 10? 1?1 1? 0?11 11? all dirty Distributed Computing though

46 1:01 0?1 01??1 0?01 10? 1?1 1? 0?11 11? all dirty Distributed Computing though

47 2:01 1?1?11 11? all dirty Distributed Computing though

48 Informal Task Definition Processes start with input values They communicate They halt with output values legal for those inputs. Distributed Computing through 48

49 Consensus: Each Thread has a Private Input Art of Multiprocessor Programming 49

50 They Communicate Art of Multiprocessor Programming 50

51 They Agree on One Thread s Input Art of Multiprocessor Programming 51

52 Colorless Tasks The set of input values determines the set of output values. Number and identities irrelevant for both input and output values 52

53 Road Map Operational Model Combinatorial Model Main Theorem Distributed Computing through 53

54 Processes A process is a state machine Could be Turing machines or more powerful Distributed Computing though 54

55 Processes A process s state is called its view Process names taken from a domain Each process has a unique name (color) Pii 2 Distributed Computing though 55

56 Processes Each process knows its own name But not the names of the other processes Distributed Computing though 56

57 Processes Often, Pii is just i Sometimes Pii and i are distinct, and the process knows Pii but not i Distributed Computing though 57

58 Processes There are n+1 processes Distributed Computing though 58

59 Shared Read-Write Memory Distributed Computing though For now, they communicate via reading & writing shared memory 59

60 Immediate Snapshot Individual reads & writes are too low-level A snapshot = atomic read of all memory We will use immediate snapshot Distributed Computing through 60

61 Immediate Snapshot write view to memory take snapshot Distributed Computing through adjacent steps! 61

62 Immediate Snapshot Con curr ent step s write view to memory write view to memory take snapshot take snapshot Distributed Computing through 62

63 Immediate Snapshot immediate mem[i] := view; snap := snapshot(mem[*]) Distributed Computing through 63

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

65 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 65

66 Crashes Processes may halt without warning as many as n out of n+1 5/17/1 66

67 Asynchronous

68 Asynchronous Failures?? detection impossible

69 Configurations C = {s0,, sn} set of simultaneous process states initial configurations final configurations Distributed Computing through 69

70 Executions processes that communicate C0, S0, C1, S1,,Sr, Cr+1 initial configuration final configuration next configuration Distributed Computing through 70

71 Executions triple is a concurrent step C0, S0, C1, S1,,Sr, Cr+1 Processes in S00 said to participate in step Only Pii 2 S00 can change between C00 and C11 state change result of communication Distributed Computing through 71

72 Executions finite C0, S0, C1, S1,, Sr, Cr+1 infinite C0, S0, C1, S1,, partial C0, S0, C1, S1,, Sr, Cr+1 Distributed Computing through 72

73 Crashes are Implicit C0, S0, C1, S1,, Sr, Cr+1 If Pii s state not final in the final configuration then Pii has crashed. crash cannot be detected in finite time Distributed Computing through 73

74 Colorless Tasks (I, O, ) (colorless) input assignment carrier map : I! 2OO (colorless) output assignment Distributed Computing through 74

75 Example: Binary Consensus I = {{0}, {1}, {0,1}} All start with 0 All start with 1 start with both Distributed Computing through 75

76 Example: Binary Consensus I = {{0}, {1}, {0,1}} O = {{0}, {1}} All decide 0 All decide 1 Distributed Computing through 76

77 Example: Binary Consensus ({0}) = {{0}} All start with 0, all decide 0 Distributed Computing through 77

78 Example: Binary Consensus ({0}) = {{0}} ({1}) = {{1}} All start with 1, all decide 1 Distributed Computing through 78

79 Example: Binary Consensus ({0}) = {{0}} ({1}) = {{1}} ({0,1}) = {{0},{1}} with mixed inputs, all decide 0, or all decide 1 Distributed Computing through 79

80 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 80

81 Colorless Layered Protocol shared mem array 0..N-1,0..n of Value view := input for j := 0 to N-1 do 2-dimensional memory array immediate mem[j][i] := view; row is clean per-layer memory snap := snapshot(mem[j][*]) view := setcolumn of values in snap is per-process word return δ(view) Distributed Computing through 81

82 Colorless Layered Protocol shared mem array 0..N-1,0..n of Value view := input for j := 0 to N-1 do immediate initial view is input value mem[j][i] := view; snap := snapshot(mem[j][*]) view := set of values in snap return δ(view) Distributed Computing through 82

83 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; N layers snap run := for snapshot(mem[j][*]) view := set of values in snap return δ(view) Distributed Computing through 83

84 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 84

85 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 85

86 Colorless Layered Protocol shared mem array 0..N-1,0..n of Value view := input for j := 0 to N-1 do immediate finally apply decision value to final view mem[j][i] := view; snap := snapshot(mem[j][*]) view := set of values in snap return δ(view) Distributed Computing through 86

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

88 Road Map Operational Model Combinatorial Model Main Theorem Distributed Computing through 88

89 Vertex = Process State Process ID (color) 7 Value (input or output) 89

90 Simplex = Global State 90

91 Complex = Global States 91

92 Input Complex for Binary Consensus 0 All possible initial states 0 1 Processes: red, green, blue 0 1 Independently assigned 0 or 1 92

93 Output Complex for Binary Consensus All possible final states Output values all 0 or all 1 1 Two disconnected simplexes 93

94 Carrier Map for Consensus All 0 outputs All 0 inputs 94

95 Carrier Map for Consensus All 1 inputs All 1 outputs 95

96 Carrier Map for Consensus All 0 outputs Mixed 0-1 inputs All 1 outputs 96

97 Task Specification (I, O, ) Input complex Output complex Carrier map : I! 2O 97

98 Colorless Tasks (I, P, ) (colorless) input complex strict carrier map : I! 2PP (colorless) protocol complex 98

99 Protocol Complex Vertex: process name, view all values read and written Simplex: compatible set of views Each execution defines a simplex Distributed Computing through 99

100 Example: Synchronous Message-Passing Round 0 Round 1 100

101 Failures: Fail-Stop Partial broadcast Distributed Computing through 101

102 Single Input: Round Zero No messages sent 0 View is input value 0 0 Same as input simplex Distributed Computing through 102

103 Round Zero Protocol Complex 0 No messages sent 0 1 View is input value 0 1 Same as input complex Distributed Computing through 103

104 Single Input: Round One Distributed Computing through 104

105 Single Input: Round One no one fails 00 Distributed Computing through

106 Single Input: Round One blue fails no one fails 00 Distributed Computing through

107 Single Input: Round One red fails green fails blue fails no one fails 00 Distributed Computing through

108 Protocol Complex: Round One Distributed Computing through 108

109 Protocol Complex: Round Two Distributed Computing through 109

110 Protocol Complex Evolution zero one two 110

111 protocol complex Summary output complex input complex Δ 111

112 Decision Map Simplicial map, sending simplexes to simplexes Protocol complex 5/17/ Output complex 112

113 Lower Bound Strategy Find topological obstruction to this simplicial map Protocol complex Output complex 113

114 Consensus Example Subcomplex of all-0 inputs Must map here 0 1 Protocol 0 1 Output Distributed Computing through 114

115 Consensus Example 0 1 Protocol Subcomplex of all-1 inputs Distributed Computing through 0 1 Must map Output here 115

116 Consensus Example Image under must start here Protocol Path from all-0 to all and end here Distributed Computing through Output 116

117 Consensus Example path 1 0 0? 1 Output Distributed Computing through 117

118 Consensus Example Image under must start here.. But this hole is an obstruction Protocol Path from all-0 to all-1 and end here Distributed Computing through Output 118

119 Conjecture A protocol cannot solve consensus if its complex is path-connected Model-independent! Distributed Computing through 119

120 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 Distributed Computing through 120

121 Barycentric Subdivision ¾ 121

122 Barycentric Subdivision Geometric Definition barycenter barycenter bary ¾ barycenter 122

123 Barycentric Subdivision Each vertex of Bary ¾ is a face of ¾ Simplex = set of faces ordered by inclusion bary ¾ 123

124 Barycentric Agreement I BaryI Distributed Computing through 124

125 Barycentric Agreement (I, bary I, bary( )) arbitrary input complex subdivided output complex subdivision as carrier map Distributed Computing through 125

126 If There are n+1 Processes (I, bary skel I, baryskel ) n n Inputs only from n-skeleton of input complex Distributed Computing through 126

127 Theorem A one-layer immediate snapshot protocol solves the n-process barycentric agreement task (I, bary skelnn I, baryskelnn) Proof All input simplices belong to skelnn I Immediate snapshot returns Set of input vertices (face of input simplex) Faces ordered wrt one another Distributed Computing through 127

128 Barycentric Agreement Protocol {v } {v,v,v } {v, v } 0 v0 0 v1 1 2 v2 Snapshots are ordered 0 2

129 Barycentric Agreement Protocol {v0,v1,v2} {v0} Each view is a face of ¾ {v0, v2}

130 Barycentric Agreement Protocol {v0} {v0,v1,v2} {v0, v2} Each view is a face of ¾

131 Barycentric Agreement Protocol {v0} {v0,v1,v2} {v0, v2} Each face of ¾ is a vertex of Bary ¾

132 Barycentric Agreement Protocol Ordered faces ) simplex of Bary ¾

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

134 Iterated Barycentric Agreement (I, bary skel I, bary skel ) N n Distributed Computing through N n 134

135 One-round IS executions Q R {p},q,r {p},{p,q},r {p},q,{p,r} R Q {p},{p,q},{p,q,r} {p},{p,q,r},{p,r} QR {p},{p,q,r} P p,{q},r Q P Q p,q,{r} {p,q},r R {p,r},q Q p,{q,r} P {p,q,r} {p,r},q,{r} P {p,q},{q},{p,q,r} {p,q,r},{q},{q,r} Q {p,r},{p,q,r},{r} P p,{q,r},{r} PQ PR PQR R {p,q,r},{q} PQ QR {p,q},{q},r p,{q},{q,r} PR R p,q,r P R {p,q,r},{r} {p,q},{p,q,r} {p,r},{p,q,r} {q,r},{p,q,r} {p,q,r},{q,r},{r}

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

137 Compare Views P Q R p,q,r PQ P Q R 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},{q},r {p,q},{q},{p,q,r} R p,{q},r P R p,{q},{q,r} {p,q,r},{q},{q,r} P PR{p,q,r},{q} Q p,q,{r} P{p,r},q,{r}{p,r},{p,q,r},{r} {pqr}{pqr}{pqr} Q P p,{q,r},{r}{p,q,r},{q,r},{r} 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}{qr}{qr} Combinatorial view Distributed Computing through

138 Compositions Given protocols (I,P, ) (I,P, ) where P µ I their composition is (I,P, ) where = and P = (I) Distributed Computing through 138

139 Theorem The protocol complex for a singlelayer IS protocol (I,P, ) is Bary I Distributed Computing through 139

140 Theorem The protocol complex for an N-layer IS protocol (I,P, ) is BaryNN I Distributed Computing through 140

141 Corollary The protocol complex for an N-layer IS protocol (I,P, ) is n-connected Distributed Computing through 141

142 Road Map Operational Model Combinatorial Model Main Theorem Distributed Computing through 142

143 Fundamental Theorem Theorem (I,O, ) has a wait-free (n+1)-process layered IS protocol iff there is a continuous map f: skelnn I O... carried by Distributed Computing through

144 Proof Outline Lemma If there is a WF layered protocol for (I,O, ) then there is a continuous f: skelnn I! O carried by. Distributed Computing through 144

145 Map ) Protocol Hypothesis f Continuous BarynI Distributed Computing through O 145

146 Map ) Protocol Áf Simplicial approximation BarynI Distributed Computing through O 146

147 Map ) Protocol Á Run barycentric BarynI Agreement Apply Á Distributed Computing through O QED 147

148 Protocol ) Map Lemma Theorem If there is a WF-RW protocol for (I,O, ) if and only if then there is a continuous f: skelnn I! O carried by. Distributed Computing through 148

149 Protocol ) Map Proof strategy Inductive construction gdd: skeldd I! (I). Base d = 0 Define g00: skel00 I! (I) Let g00(v) be any vertex in ({v}) Distributed Computing through 149

150 Protocol ) Map Induction Hypothesis d-1 d-1 I! (I) gd-1 : skel d-1 d-1 I For all ¾ in skeld-1 d-1 d-1 ¾ (¾) gd-1 sends skel d-1 But (¾) is (d-1)-connected (earlier theorem) Can extend to gdd: ¾ (¾) Yielding gdd: skeldd I! (I) Distributed Computing through 150

151 Protocol ) Map Constructed g: skelnn I (I) Simplicial decision map δ: (skelnn I) O δ : (skelnn I) O Composition f = δ g yields f: skelnn I! O carried by. QED Distributed Computing through 151

Operational Reasoning http://commons.wikimedia.

152 Operational Reasoning Distributed Computing though 152

153 Combinatorial Reasoning 153

154 Combinatorial Reasoning Model-independent properties 154

155 Combinatorial Reasoning Model-independent properties restricted model-dependent reasoning 155

156 This work is licensed under a Creative Commons AttributionShareAlike 2.5 License. You are free: to Share to copy, distribute and transmit the work to Remix to adapt the work Under the following conditions: Attribution. You must attribute the work to Distributed Computing Through (but not in any way that suggests that the authors endorse you or your use of the work). Share Alike. If you alter, transform, or build upon this work, you may distribute the resulting work only under the same, similar or a compatible license. For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this license impairs or restricts the author's moral rights. Distributed Computing though

Distributed Computing through Combinatorial Topology MITRO207, P4, 2017

Distributed Computing through MITRO207, P4, 2017 Administrivia Language: (fr)anglais? Lectures: Fridays (28.04, 20.05-23.06, 30.06), Thursday (29.06), 8:30-11:45, B555-557 Web page: http://perso.telecom-paristech.fr/~kuznetso/