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forDistributed ComputingThrough Combinatorial TopologyMaurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum

1

Slide2

Previously

4-Mar-15

2

Used Sperner’s Lemma to show k-set agreement impossible when protocol complex is a manifold.

But in many models, protocol complexes are

not

manifolds …

Slide3

Road Map

4-Mar-15

3

Consensus Impossibility

General theorem

Application to read-write models

k-set agreement Impossibility

General theorem

Application to read-write models

Slide4

Road Map

4-Mar-15

4

Consensus Impossibility

General theorem

Application to read-write models

k-set agreement Impossibility

General theorem

Application to read-write models

Slide5

A Path

4-Mar-15

5

vertex

vertex

vertex

vertex

vertex

edge

edge

edge

edge

simplicial

complex

Slide6

Path Connected

4-Mar-15

6

Any two vertexes can be linked by a path

Slide7

Theorem

4-Mar-15

7

If, for protocol

(

I

,

P

,

¥

)

…

Slide8

If, for protocol (I,P,¥) …

Theorem

4-Mar-15

8

For every

n

-simplex

¾

n

,

¥

(

¾

n

)

is path-connected …

Slide9

If, for protocol (I,P,¥) …

Theorem

4-Mar-15

9

(If no process fails)

For every

n

-simplex

¾

n

,

¥

(

¾

n

)

is path-connected …

Slide10

If, for protocol (I,P,¥) …

Then …

Theorem

4-Mar-15

10

For every (n-1)-simplex ¾n-1, ¥ (¾n-1) is non-empty …

For every n-simplex ¾n, ¥(¾n) is path-connected …

(If one process fails)

Slide11

If, for protocol (I,P,¥) …

Then …

Theorem

4-Mar-15

11

For every (n-1)-simplex ¾n-1, ¥ (¾n-1) is non-empty …

For every n-simplex ¾n, ¥ (¾n) is path-connected …

¥

(

¢

)

cannot solve consensus.

Slide12

Model Independence

4-Mar-15

12

Holds for message-passing or shared memory ….

Synchronous, asynchronous, or in-between …

Any adversarial scheduler …

As long as one failure is possible.

Slide13

Protocol Complex Notation

All 3 participate

¥

(PQR)

2 participate

¥

(PQ), ¥(QR), ¥(PR),

1 participates

¥

(

P),

¥

(Q),

¥

(R),

Slide14

Proof

4-Mar-15

14

¥

(PQR)

Slide15

Proof

4-Mar-15

15

¥

(PQ)

¥

(PQR)

Slide16

Proof

4-Mar-15

16

¥

(P)

¥

(PQR)

¥

(PQ)

Slide17

Proof

4-Mar-15

17

¥

(P)

¥

(R)

¥

(PQR)

¥

(PQ)

¥

(QR)

Slide18

Complex is Path-Connected

4-Mar-15

18

¥

(PQR)

¥

(PQ)

¥

(QR)

¥

(P)

¥

(R)

Slide19

¥

(PQR)

One-dimensional Sperner’s Lemma

4-Mar-15

19

Some edge has two “colors”

An execution that decides two distinct values

QED

¥

(PQ)

¥

(QR)

¥

(P)

¥

(R)

Slide20

Road Map

4-Mar-15

20

Consensus Impossibility

General theorem

Application to read-write models

k-set agreement Impossibility

General theorem

Application to read-write models

Slide21

Application

4-Mar-15

21

We now show that consensus is impossible in wait-free read-write memory

For every (n-1)-simplex ¾n-1, ¥ (¾n-1) is non-empty.

For every

n

-simplex

¾

n

,

¥

(

¾

n

)

is path-connected …

Slide22

22

Reachable Complexes

A moves

Protocol complex

Reachable part of protocol complex

B moves

Reachable part of protocol complex

Slide23

23

Reachable Complexes

Initially,

entire protocol complex

reachable

At the end, single simplex reachable

Slide24

24

Eventual Property

Satisfied by

simplexes

Slide25

25

Eventual Property

Suppose it does not hold at start

Slide26

26

Critical States

critical

Slide27

27

Critical States Exist

critical

Non-satisfying

Slide28

28

Eventual Property

Critical states

Lowest non-satisfying states in tree

Slide29

Path-Connectivity is an Eventual Property

4-Mar-15

29

Individual simplex is path-connected

Slide30

Path-Connectivity has a Critical State

4-Mar-15

30

C

each path connected

next steps

not path connected

Slide31

Critical State in Layered IS

4-Mar-15

31

C

U

V

W

each path connected

participants next layer

not path connected

Slide32

Notation

4-Mar-15

32

C " U

Configuration reached by running processes in

U

in next layer

Slide33

Critical State in Layered IS

4-Mar-15

33

C

U

V

W

¥

(

C

"

U

)

¥

(C " V)

¥(C " W)

each path connected

participants next layer

not path connected

Slide34

Proof Strategy

4-Mar-15

34

C

U

V

W

¥

(

C

"

U

)

¥

(C " V)

¥(C " W)

each path connected

not path connected

path connected

Math!

Contradiction!

It was path-connected all along!

Slide35

One-Dimensional Nerve Lemma

4-Mar-15

35

Reason about path-connectivity of a graph …

From path-connectivity of components …

And how they fit together.

Slide36

Graph K

4-Mar-15

36

Slide37

Covering

K

i

4-Mar-15

37

K

0

K

1

K

2

K

=

[

i

K

i

Slide38

Nerve Graph Vertices

4-Mar-15

38

K

0

K

1

K

2

Each

K

i

is a vertex

Slide39

Nerve Graph Edges

4-Mar-15

39

K

0

K

1

K

2

Edge between

K

i

and

K

j

if

K

i

Å

K

j



;

Slide40

One-Dimensional Nerve Lemma

4-Mar-15

40

If each Ki is path-connected …

and the nerve graph N i(Ki ) is path-connected …

then Ki is path-connected …

Slide41

Critical State in Layered IS

4-Mar-15

41

C

U

V

W

¥

(

C

"

U

)

¥

(

C " V)

¥(C " W)

each path connected

participants next layer

not path connected

Slide42

Configurations

4-Mar-15

42

The

¥(C " U) form a cover for reachable complex

C

U

V

W

¥

(

C

"

U

)

¥

(

C " V)

¥

(

C

"

W

)

Slide43

Want to show

Configurations

4-Mar-15

43

each ¥(C " U) is path-connected

each ¥(C " U) Å ¥(C " V) is non-empty

C

U

V

W

¥

(

C

"

U

)

¥

(

C " V)

¥(C " W)

Slide44

PQRwritewritewritesnapsnapsnap

Computing Intersections

44

Distributed Computing Through Combinatorial Topology

C " U

PQRwritewritesnapsnapwritesnap

V = {P,Q}

U = {P,Q,R}

(C " V) " U n V

C

"

V

Memory state same

Only V can distinguish

Slide45

Intersections (Case 1)

4-Mar-15

45

¥(C " U) Å ¥(C " V) is the complex reachable from C " U in executions where no process in V takes another step

If

V

µ

U,

then

Slide46

Notation

4-Mar-15

46

(¥ # U)(C)

Complex reachable from

C

in executions where processes in

U

halt and the rest finish.

Slide47

Notation

4-Mar-15

47

(¥ # U)(C)

Complex reachable from C in executions where processes in U halt and the rest finish.

¥(C " U) Å ¥(C " V) = (¥ # V)(C " U)

If

V

µ

U,

then

Slide48

Notation

4-Mar-15

48

(¥ # U)(C)

Complex reachable from C in executions where processes in U halt and the rest finish.

¥(C " U) Å ¥(C " V) = (¥ # V)(C " U)

If

V

µ

U,

then

Slide49

PQRwritesnapwritesnap

Computing Intersections

49

Distributed Computing Through Combinatorial Topology

C " U

PQRwritesnapwritesnap

V = {Q}

U = {P}

(C " U) " V n U

C

"

V

Memory state same

Only U [ V can distinguish

(C

"

V)

"

U

n

V

Slide50

Intersections (Case 2)

4-Mar-15

50

¥(C " U) Å ¥(C " V) = (¥ # U [ V)(C " U [ V)

If U, V incomparable, then

Run processes in

U [ V

Then crash processes in U [ V

Slide51

Lemma

4-Mar-15

51

¥(C " U) Å ¥(C " V) = (¥ # W)(C " U [ V)

Where

U if U µ V

V if V µ U

U [ V otherwise

W =

Slide52

Lemma

4-Mar-15

52

The nerve graph N(¥(C " U)) is path-connected

Proof

Consider vertex v = ¥(C " ¦)

Show every vertex has an edge to

v

Slide53

Proof (con’t)

4-Mar-15

53

4-Mar-15

53

Consider vertex

v =

¥

(C

"

¦

)

Slide54

Proof (con’t)

4-Mar-15

54

4-Mar-15

54

Consider vertex v = ¥(C " ¦)

for every

U

½

¦

consider possible edge …

Slide55

Proof (con’t)

Consider vertex v = ¥(C " ¦)

¥(C " ¦) Å ¥(C " U) = (¥ # U)(C " ¦)

for every

U

½

¦

consider possible edge …

Slide56

Proof (con’t)

4-Mar-15

56

4-Mar-15

56

Consider vertex v = ¥(C " ¦)

¥(C " ¦) Å ¥(C " U) = (¥ # U)(C " ¦)

Run everyone in next layer

for every

U

½

¦

consider possible edge …

Slide57

Proof (con’t)

4-Mar-15

57

4-Mar-15

57

Consider vertex v = ¥(C " ¦)

¥(C " ¦) Å ¥(C " U) = (¥ # U)(C " ¦)

Run everyone in next layer

crash everyone in

U

for every

U

½

¦

consider possible edge …

Slide58

Proof (con’t)

4-Mar-15

58

4-Mar-15

58

Consider vertex v = ¥(C " ¦)

¥(C " ¦) Å ¥(C " U) = (¥ # U)(C " ¦)

Run everyone in next layer

crash everyone in

U

for every

U ½ ¦ consider possible edge …

Because

U

½

¦

, complex non-empty, hence edge exists

Slide59

Theorem

4-Mar-15

59

For every input simplex ¾, the layered IS protocol complex ¥(¾) is path-connected

Proof

Induction on n

Case n=0: ¥(¾) is a single vertex

Case

induction step …

Slide60

Proof

4-Mar-15

60

C

U

V

W

¥

(

C

"

U

)

¥

(C " V)

¥(C " W)

assume ¥(C)not path-connected

each ¥(C " U) is path connected

their nerve graph path-connected

covering of

¥

(C)

Slide61

covering of ¥(C)

Proof

4-Mar-15

61

¥

(

C

" U)

¥(C " V)

¥(C " W)

each ¥(C " U) is path connected

their nerve graph path-connected

¥(C) is path connected by Nerve Lemma

contradiction!

Slide62

Road Map

4-Mar-15

62

Consensus Impossibility

General theorem

Application to read-write models

k-set agreement Impossibility

General theorem

Application to read-write models

Slide63

So Far …

4-Mar-15

63

Expressed solvability of consensus as a topological property of protocol complex …

And applied the result to wait-free read-write memory.

Next: do the same for

k

-set agreement!

Slide64

0-sphere

1-disc

Rethinking Path Connectivity

Let’s call this complex

0-connected

Slide65

1-Connectivity

1-sphere

2-disc

Slide66

?

This Complex is not 1-Connected

Slide67

2-Connectivity

3-disk

2-sphere

Slide68

n-connectivity

4-Mar-15

68

C is n-connected, if, for m · n, every continuous map of the m-sphere

can be extended to a continuous map of the (m+1)-disk

(

-1

)-connected is non-empty

Slide69

Road Map

4-Mar-15

69

Consensus Impossibility

General theorem

Application to read-write models

k-set agreement Impossibility

General theorem

Application to read-write models

Slide70

Connectivity and k-Set Agreement

4-Mar-15

70

Theorem

(I,O,¢) an (n+1)-process k-set agreement task…

(I,P,¥) a protocol …

such that ¥(¾) is (k-1)-connected for all ¾ in I …

then

(

I

,

P

,

¥

)

cannot solve

k

-set

agreement.

Slide71

Lemma

4-Mar-15

71

carrier map ©: A  2B

such that for all ® 2 A,

Then © has a simplicial approximationÁ: DivN A ! B.

©

(

®

)

is

((dim

®

) – 1)

-connected.

Slide72

Lemma Proof Sketch

4-Mar-15

72

carrier map ©: A ! 2B

has continuous approximation f: |A| ! |B|

f(|¾|) µ |©(¾)|

Inductive construction …

Slide73

Lemma Proof Sketch

4-Mar-15

73

continuous approximation f: |A| ! |B|

Base

define on vertices …

f

0

Slide74

Lemma Proof Sketch

4-Mar-15

74

Step

connectivity allows “filling in”

f

1

f

2

Slide75

Lemma Proof Sketch

4-Mar-15

75

continuous approximation

f: |A| ! |B|

take simplicial approximation

Á

:

Div

A

!

B

of

f

: |

A

|

!

|

B

|

Slide76

Theorem Proof Sketch

4-Mar-15

76

let ¾ 2 I have k+1 distinct input values

let ¢k be simplex labeled with k+1 values

¢k its (k-1)-skeleton

c: ¥(¾) ! ¢k well-defined simplicial map

By lemma, ¥ has simplicial approximation

Á

:

Div

¾

!

¥

(

¾

)

of

f

: |

A

|

!

|

B

|

Slide77

Theorem Proof Sketch

4-Mar-15

77

c: ¥(¾) ! ¢k well-defined simplicial map

By lemma, ¥ has simplicial approximation

Á: Div ¾ ! ¥(¾) of f: |A| ! |B|

composition Div ¾ ! ¥(¾) ! ¢k

defines a Sperner coloring of Div ¾

some ¿ in Div ¾ maps to all of ¢k

Contradiction!

Slide78

k-Connectivity is an Eventual Property

4-Mar-15

78

Individual simplex is

k

-connected

Slide79

k-Connectivity has a Critical State

4-Mar-15

79

C

each

k

-connected

next steps

not

k

-connected

Slide80

Critical State in Layered IS

4-Mar-15

80

C

U

V

W

¥

(

C

"

U

)

¥

(C " V)

¥(C " W)

each k-connected

participants next layer

not

k

-connected

Slide81

Proof Strategy

4-Mar-15

81

C

U

V

W

¥

(

C

"

U

)

¥

(C " V)

¥(C " W)

each k-connected

not k- connected

k

-connected

Math!

Contradiction!

It was

k

-

connected all along!

Slide82

Nerve Lemma

4-Mar-15

82

Reason about connectivity of a complex…

From connectivity of components …

And how they fit together.

Slide83

Covering

4-Mar-15

83

Complex

Slide84

Covering

4-Mar-15

84

Complex

Covering

Slide85

4-Mar-15

85

Vertex

Simplex, if

ÅC

i

non-empty

Slide86

4-Mar-15

86

Nerve Example: Sphere

Slide87

4-Mar-15

87

Reasoning About Connectivity

Covering

Nerve

Slide88

Nerve Lemma

4-Mar-15

88

k

-connected

(

k-|U|+

1)

-connected

If …

Slide89

…Then

4-Mar-15

89

if and only if ..

N (C0, …Cm) is k-connected.

C

is

k

-connected …

Slide90

4-Mar-15

90

Nerve Example: Sphere

1-connected …

Slide91

4-Mar-15

91

Reasoning About Connectivity

0-connected

Can apply Nerve Lemma!

Slide92

4-Mar-15

92

Reasoning About Connectivity

1-connected …

Slide93

4-Mar-15

93

Reasoning About Connectivity

1-connected !

1-connected !

QED

implies

Slide94

Nerve Complex Lemma

4-Mar-15

94

The nerve complexN(¥(C " U) | ; µ U µ ¦)is n-connected

Proof

Consider vertex v = ¥(C " ¦)

Show the nerve complex is a

cone

with apex

v

Slide95

Nerve Complex Lemma

4-Mar-15

95

i ¥(C " Ui)  ;

each set {¥(C " Ui) | i = 0, …, m}

Is a simplex if and only if

each

¥

(C

"

U

i

)

is a vertex

Slide96

Reasoning About Intersections

4-Mar-15

96

Lemma

Let U0, … , Um sets of process names …

Indexed so

|

U

0

|

¸

…

¸

|

U

m

|

Slide97

Intersection Lemma

4-Mar-15

97

Å ¥(C " Ui) = (¥ # W) (C " [ Ui)

Where

i=1m Ui if i=1 m Ui µ U0

W =



i=0m Ui otherwise

Proof is inductive version of earlier lemma

Slide98

Corollary

4-Mar-15

98

If i Ui = ¦ but each Ui  ¦,

then



i

¥

(

C

"

U

i

)

=

;

.

Slide99

Nerve Complex

So i ¥(C " Ui)  ;

Let ¾ = {¥(C " Ui) } be a simplex

Let vertex v = ¥(C " ¦)

must show that ¾ [ {v} is a simplex …

and



i

U

i



¦

Slide100

Intersection Lemma Proof

to show that ¾ [ {v} is a simplex, show that …

¥(C " ¦) Å (i ¥(C " Ui) )

is non-empty.

Slide101

Intersection Lemma Proof

¥(C " ¦) Å (I ¥(C " Ui)) = (¥ # i Ui ) (C " ¦)

by corollary,

Ui  ¦

don’t halt everyone

complex non-empty

it’s a simplex!

QED

Slide102

Lemma

4-Mar-15

102

i 2 I ¥(C " Ui) is (n-|I|+1)-connected

argue by induction on n

trivial for

n

= 0

…

Slide103

Proof

4-Mar-15

103

i2I ¥(C " Ui) = (¥ # W)(C " X)

a protocol complex for n-|W|+1 processes …

for |W| > 0, W µ X µ [i Ui.

either empty, or n-connected by induction hypothesis.

therefore

(

n

-|I|+1)

-connected

Slide104

Theorem

4-Mar-15

104

For every input simplex ¾, the layered IS protocol complex ¥(¾) is k-connected

Proof

Induction on n

Case n=0: ¥(¾) is a single vertex

Case

induction step …

Slide105

Proof

4-Mar-15

105

C

U

V

W

¥

(

C

"

U

)

¥

(C " V)

¥(C " W)

assume ¥(C)not k-connected

their nerve complex is k-connected

covering of ¥(C)



i

2

I

¥

(

C

"

U

i

)

is

(

n

-|I|+1)

-connected

Slide106

covering of ¥(C)

Proof

4-Mar-15

106

¥

(

C

" U)

¥(C " V)

¥(C " W)

their nerve graph k-connected

¥(C) is k-connected by Nerve Lemma

contradiction!

each



i

2

I

¥

(

C

"

U

i

)

is

(

n

-|I|+1)

-connected

Slide107

Conclusions

4-Mar-15

107

Model-independent topological properties that prevent …

consensus

Model-specific application to wait-free read-write memory

path-connectivity

k-set agreement

k

-connectivity

Slide108

108

         

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