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Two-Process Systems

Companion slides forDistributed Computing

Through Combinatorial TopologyMaurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum

Distributed Computing through Combinatorial Topology

Two-Process Systems

Two-process systems can be captured by elementary

graph theorygentle introduction to more general structures needed

later for larger systemsDistributed Computing through

Combinatorial Topology2

Road MapElementary Graph Theory

Tasks

Models of Computation

Approximate Agreement

Task Solvability


3


Tasks



Task Solvability


4

A Vertex


5

A VertexCombinatorial: an element of a set.


6

A VertexCombinatorial: an element of a set.

Geometric: a point in Euclidean Space


7

An Edge

8

An EdgeCombinatorial: a set of two vertexes.

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An EdgeCombinatorial: a set of two vertexes.

Geometric: line segment joining two points

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A Graph


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A GraphCombinatorial: a set of sets of vertices.


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A GraphCombinatorial: a set of sets of vertices.

Geometric: points joined by line segments


13

Graphs

finite set V with a collection G of subsets of V,


14

Graphs

simplices(singular: simplex)


vertices


15

Graphs


(1) If X 2 G, then |X| · 2


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Graphs


(1) If X 2 G, then |X| · 2


vertex: |X| = 1 edge: |X|= 2

17

Graphs

(1) If X 2 G, then |X| · 2(2) for all v 2 V, {v} 2 G


18

Graphs

(1) If X 2 G, then |X| · 2(2) for all v 2 V, {v} 2 G

(3) for all X 2 G, and Y ½ X, Y 2 G


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Dimension

dim(X) = |X|-1.

dimension 0

dimension 1


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Pure Graphs

pure of dim 0

pure of dim 1


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Graph Coloring


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Graph Coloring

Â: G ! C


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Graph Coloring

Â: G ! Cfor each edge (s0, s1) 2 G, Â(s0) Â(s1).


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Graph Coloring

Â: G ! Cfor each edge (s0, s1) 2 G, Â(s0) Â(s1).

usually process nameschromatic graphs


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Graph Labeling

1 0

0

1


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Graph Labeling

1 0

0

1f: G ! L


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Graph Labeling

1 0

0

1f: G ! Lusually values from some domain


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Labeled Chromatic Graph

0 1

1

0

name(s) = Â(s) view(s) = f(s)


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Simplicial Maps

Vertex-to-vertex map …

that also sends edges to edges.


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Rigid Simplicial Maps

A simplicial map can send an edge to a vertex …


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Rigid Simplicial Maps

A simplicial map can send an edge to a vertex …

A simplicial map that sends distinct vertices to distinct vertices is rigid.


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A Path Between two Vertices


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A Path Between two Vertices

A graph is connected if there is a path between every pair of vertices


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Not Connected

A graph is connected if there is a path between every pair of vertices


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Theorem


Theorem

Á

The image of a connected graph under a simplicial map is connected.

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Carrier Maps


©: G ! 2H

For graphs G, H, a carrier map

Carries each simplex of G to a subgraph of H …

©

37

Carrier Maps


©: G ! 2H

For graphs G, H, a carrier map

Carries each simplex of G to a subgraph of H …

©

satisfying monotonicity:for all ¾,¿2G, if ¾µ¿, then ©(¾)µ©(¿).

38

Monotonicity

Strict Carrier Maps


For all ¾,¿2G, if ¾µ¿, then ©(¾)µ©(¿).

39

Monotonicity

Strict Carrier Maps



Equivalent to …©(¾Å¿) µ ©(¾) Å ©(¿)

40

Monotonicity

Strict Carrier Maps



Equivalent to …©(¾Å¿) µ ©(¾) Å ©(¿)

Definition© is strict if ©(¾Å¿) = ©(¾) Å ©(¿)

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Tasks



Task Solvability


42

Two Processes


Hello! I’m

AliceHello! I’m

Bob

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Informal Task Definition


Processes start with input values …

They communicate …

They halt with output values …

legal for those inputs.

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Formal Task Definition


Input graph Iall possible assignments of input values

45




Output graph Oall possible assignments of output values

46




Output graph Oall possible assignments of output values

Carrier map ¢: I ! 2O

all possible assignments of output values for each input

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Task Input Graph: Consensus


1

10

0

I

48

Task Input Graph


0 1

10

I

49

Task Input Graph


1 1

00

Pure

Colored by process names

Labeled by input values

50

Task Output Graph


1 1

00

O

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Task Carrier Map


1 1

00

1 1

00

¢: I ! 2OI O

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Task Carrier Map


1 1

00

1 1

00

¢: I ! 2OI O

If Bob runs alone with input 1 …

then he decides output 1.

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Task Carrier Map


1 1

00

1 1

00

¢: I ! 2OI O

If Bob and Alice both have input 1 …

then they both decide output 1.

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Task Carrier Map


1 1

00

1 1

00

¢: I ! 2OI O

If Bob has 1 and Alice 0 …

then they must agree, on either one.

55

56

Example: Coordinated Attack

Alice and Bob winIf they both attack

together

Alice Bob


Enemy

Indifferent

Attack at dawn!

Attack at noon!

Input Graph


I

57

Output Graph


dawn!

noon!

failed!

O

58

Carrier Map


¢

I

O

59

Carrier Map


¢

dawn!

failed!

dawn! I

O

60

Carrier Map


¢

noon!

failed!

noon!I

O

61

Carrier Map


¢

dawn!

dawn! I

O

62

63

Example: Coordinated AttackAlice Bob


Enemy

64



Enemy

Alice and Bob realize that they do not need to agree

on an exact time …

65



Enemy

Alice and Bob realize that they do not need to agree

on an exact time …they will win if attack times

are sufficiently close.

0 1

0 1/5 2/5 3/5 4/5 1

Coordinated Attack Graphs


¢

I

O

66

0 1

0 1/5 2/5 3/5 4/5 1



¢

I

O

67

0 1

0 1/5 2/5 3/5 4/5 1



¢

I

O

68

0 1

0 1/5 2/5 3/5 4/5 1



¢

I

O

69


Tasks



Task Solvability


70

Protocols



71

Alice’s Protocol

72

shared mem array 0..1 of Valueview: Value := my input value;for i: int := 0 to L do mem[A] := view; view := view + mem[A] + mem[B];return δ(view)

Finite program

Bob’s protocol is symmetric


shared mem array 0..1 of Valueview: Value := my input value;for i: int := 0 to L do mem[A] := view; view := view + mem[B];return δ(view)

Alice’s Protocol

73Distributed Computing through Combinatorial Topology

shared two-element memory


Alice’s Protocol


Start with input value


Alice’s Protocol


Run for L layers


Alice’s Protocol


Alice writes her value, read Bob’s value, and concatenate it to my view


Alice’s Protocol


Alice writes her value, read Bob’s value, and concatenate it to my view

(full-information protocol)


Alice’s Protocol


finally, apply task-specific decision map to view

Formal Protocol Definition



79




Protocol graph Pall possible process views after execution

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Protocol graph Pall possible process views after execution

Carrier map ¥: I ! 2P

all possible assignments of views

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One-Round Protocol Graph


10

0? 0101 ?1

¥

I

P

82



0? 0101 ?1

Colored by process names

Labeled with final views

P

83



0? 0101 ?1

Alice finishes before Bob starts, doesn’t see his value

P

84



0? 0101 ?1

Alice and Bob run together,she sees his value.

P

85



0? 0101 ?1

Alice finishes, then Bob starts

P

86



0? 0101 ?1

Alice and Bob run together

P

87



0? 0101 ?1

Bob can’t tell whether Alice saw him

P

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Execution Carrier Map


10

0? 0101 ?1

¥

I

P

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Execution Carrier Map


10

0? 0101 ?1

¥

I

P

¥: I ! 2P

strict carrier map¥(¾) Å ¥(¿) = ¥(¾ Å ¿)90

Output graph0 2/31/3 1

0? 0101 ?1 Protocol graph

δ

The Decision Map


δ

91

All Together


0 1

0 2/31/3 1

0? 0101 ?1¢

δ

I

P

O

¥

92

Definition


Decision map δ is carried by carrier map ¢ if

for each input vertex s,

for each input edge ¾,

δ(¥(s)) µ ¢(s)

δ(¥(¾)) µ ¢(¾).

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Meaning

94

δ(¥(s)) µ ¢(s)

process strarts in state s

Meaning

95

δ(¥(s)) µ ¢(s)

runs the protocol to completion

Meaning

96

δ(¥(s)) µ ¢(s)

makes a decision …

Meaning

97

δ(¥(s)) µ ¢(s)

decision is permitted by task carrier map

Definition

Solving a Task


The protocol (I,P,¥) solves the task (I, O, ¢)

98

Definition

Solving a Task



if there is …

99

Definition

Solving a Task



if there is …

a simplicial decision mapδ:P ! O

100

Definition

Solving a Task



if there is …

a simplicial decision mapδ:P ! O

101

such that δ is carried by ¢.(δ agrees with ¢)

Layered Read-Write Model

102

Layered Read-Write Protocol


103

shared mem array 0..1,0..L of Valueview: Value := my input value;for i: int := 0 to L do mem[i][A] := view; view := view + mem[A] + mem[B];return δ(view)


As before, run for L layers



104




Each layer uses a distinct, “clean” memory

105

Layered R-W Protocol Graph


10

0? 0101 ?1

¥

I

P

106

Layered R-W Protocol Graph


¥

P is always a subdivision of I

I P

107


Tasks



Task Solvability


108

Alice’s 1/3-Agreement Protocol


mem[A] := 0other := mem[B]if other == ? then decide 0else decide 1/3

109



mem[A] := 0if mem[B] == ? then decide 0else decide 1/3

Alice writes her value to memory

110




If she doesn’t see Bob’s value, decide her own.

111




If she see’s Bob’s value, jump to the middle

112

0 2/31/3 1

0 2/31/3 1

0 2/31/3 1


113

One-Layer 1/3-Agreement Protocol


0 1

0 2/31/3 1

0? 0101 ?1

δ

I

P

O

¥

114

No 1-Layer 1/5-Agreement Protocol


0 1

1/5 3/52/5 4/5

0? 0101 ?1

δ

P

O

¥

10

(no map possible)

I

115

10

0 1/5 2/5 3/5 4/5 1

¥

0? 0101 ?1

2-Layer 1/5-Agreement


¥

I

O

layer 1

layer 2

δ

116

Fact


In the layered read-write model,

The 1/K-Agreement Task

Has a dlog3 Ke–layer protocol

117


Tasks



Task Solvability


118

Fact


The protocol graph for any L-layer protocol with input graph I is a subdivision of I, where each edge is subdivided 3L times.

119

Main Theorem


The two-process task (I, O, ¢) is solvable in the layered read-write model if and only if there exists a connected carrier map ©: I ! 2O carried by ¢.

120

Corollary


The consensus task has no layered read-write protocol

121

Corollary


Any ²–agreement task has a layered read-write protocol

122

123

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