1 consensus hierarchy part 2. 2 fifo (queue) fifo object headtail

1

Consensus HierarchyPart 2

2

FIFO (Queue)

FIFO Object

),( xQenq

Q

headtail

x

a

aQdeq )( x

b

ac b

c

bc

3

Special case: empty queue

)(Qdeq

tail head

4

Theorem: A FIFO object has consensusnumber 2

Proof:1. We can solve wait-free consensus using FIFO (and read/write) objects for 2 processes

2. We cannot solve wait-free consensus using FIFO (and read/write) variables for 3 or more processes

5

A Wait-Free Consensus algorithmfor 2 processors using a FIFO object(and read/write objects)

Initially:

tail head

0

Proof-Part 1

6

Shared Memory

Queue

Other Variables

0tail

head

7

0x 1x0p 1p

Local variables

}1,0{, ii yx

:initial values for the consensus problem

0y 1y

ix

iy:resulting values for the consensus problem

8

Shared Memory

Queue

Other Variables

0tail

head

1x

0xPrefer[0]Prefer[1]

Initial values

9

Code for processor ip }1,0{i

If then else

0)( Qdeq

ii xy ]1[ ipreferyi

//am I the first?

//yes, choose my value

//no, choose the other processor’s value

Note: the algorithm uses a FIFO object and read/write objects

ixiprefer ][

10

Example execution:

Shared Memory

Queue0tail

head

01 x

10 xPrefer[0]Prefer[1]

10 x

01 x

0p

1p

0y

1y

11

Suppose that accesses first the queue

Shared Memory

Queuetail

head

01 x

10 xPrefer[0]Prefer[1]

10 x

01 x

0p

1p

0y

01 y

1p

)(0 Qdeq

decides on its own value

12

Suppose that accesses second

Shared Memory

Queuetail

head

01 x

10 xPrefer[0]Prefer[1]

10 x

01 x

0p

1p

00 y

01 y

2p

decides on the other processor’s value

)(Qdeq

Consensus Reached

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There is no wait-free consensusalgorithm using only FIFO(and read-write) objects for 3n

Consider three processors(the same proof generalizes to more)

Proof-Part 2

We will prove:

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There is a bivalent initial configuration(we proved it before)

We will show that every bivalent configurationhas a processor which is not critical

Therefore, we can construct an infiniteexecution with bivalent configurations whereconsensus is never reached

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Assume for contradiction that all processors are critical

C

0C

1C

1nC

bivalent0p

1p

2p

univalent

univalent

univalent

Possibleexecutions

16

bivalent

valent

It cannot be that all have the same valence

v

)1 or 0( v

valentv

valentv

valentv

Contradiction

C

0C

1C

2C

0p

1p

2p

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There must exist two processors with different valences

valent-0

valent-1C

0C

1C

2C

bivalent0p

1p

2p

univalent

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Shared Memory

1QQueue

2QQueue

zQQueue

Read/WriteObjects

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Shared Memory

1QQueue

2QQueue

zQQueue

Read/WriteObjects

valent-0

valent-1

C

0C

1C

2C

bivalent0p

1p

2p

Case: the processors access different objects

0p

1p

Note: if an object was read/write the analysis is the similar

univalent

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valent-0

valent-1

C

0C

1C

2C

bivalent0p

1p

2p

op Q

1

op Q2

op Q2

op Q1

C

C

1p

0p

valent-0

valent-1

CCp

2

Impossible since

Two possible executions

univalent

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Shared Memory

1QQueue

QQueue

zQQueue

Read/WriteObjects

valent-0

valent-1

C

0C

1C

2C

bivalent0p

1p

2p

Case: the processors access same object

0p

1p

Note: if the object was read/write the analysisis the same as in the case with read/write objects

univalent

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valent-0

valent-1

C

0C

1C

2C

bivalent0p

1p

2punivalent

deq(

Q)

deq(Q)

Subcase: deq/deq

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valent-0

valent-1

C

0C

1C

2C

bivalent0p

1p

2punivalent

deq(

Q)

deq(Q)

deq(Q)

deq(Q)

C

C

1p

0p

valent-0

valent-1

CCp

2

Impossible since

Queue Q before operations: bc

a b

a d

a b

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valent-0

valent-1

C

0C

1C

2C

bivalent0p

1p

2punivalent

deq(

Q)

enq(Q,x)

Subcase: deq/enq

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valent-0

valent-1

C

0C

1C

2C

bivalent0p

1p

2punivalent

deq(

Q)

enq(Q,x)

enq(Q,x)

deq(Q)

C

C

1p

0p

valent-0

valent-1

CCp

2

Impossible since

Suppose Q was not empty a bc

a bx

a bx

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valent-0

valent-1

C

0C

1C

2C

bivalent0p

1p

2punivalent

deq(

Q)

enq(Q,x)

enq(Q,x)C 1pvalent-0

1

2

CCp

Impossible since

Suppose Q was empty

x

x

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valent-0

valent-1

C

0C

1C

2C

bivalent0p

1p

2punivalent

enq(

Q,a)

enq(Q,b)

Subcase: enq/enq

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valent-0

valent-1

C

0C

1C

2C

bivalent0p

1p

2punivalent

enq(

Q,a)

enq(Q,b)

enq(Q,b)

enq(Q,a)

C

C

1p

0p

valent-0

valent-1

Suppose Q was not empty yx

yx

yx

a

a

b

b

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valent-1

C 1C

2C

bivalent

0p

1p

2punivalent

enq(

Q,a

)

enq(Q,b) enq(Q,a)C 0p

valent-1

valent-0

0C enq(Q,b)C 1p

yxab b

0p

z w t w

1p0D

yxab z w t w

0p 1p1D

a

valent-0

Dequeue a Dequeue b

Dequeue b Dequeue a

Impossible since

10

2

DDp

30

C

bivalent

0p

enq(

Q,a

)

valent-0

0C enq(Q,b)C 1p

yxab b

0p

a

0 decidesvalent-0

Suppose does not dequeue a

0

A decision will be reached since the consensus algorithm is wait-free

0p

Explanation

0p

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valent-1

C 1Cbivalent

0p

1p

enq(

Q,a

)

enq(Q,b) enq(Q,a)C 0p

valent-0

0C enq(Q,b)C 1p

yxab

0p

yxab

0p

0

0

valent-0

valent-1

The same value will be decided by ,since sees the same shared memory values in both executions

0p

decides 00p

decides 00p

contradiction

0

ba

ba

0p

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In all cases we obtained contradiction;Therefore, there exists a processorwhich is not critical

C

0C

1C

1nC

bivalent0p

1p

1np

univalent

univalent

univalent

kC bivalent(not critical)

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Therefore, we can construct an execution

0C 1C 2C

Initialconfiguration

0ip

1ip

2ip Never

ends

bivalent bivalent bivalent

Consensus can never be reached

End of Theorem Proof

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Compare&Swap

Compare&Swap(X,A,B) { Temp X;

If X==A then X B;Return Temp; }

Shared Memory

X

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Theorem: The consensus number ofthe Compare&Swap object is

Proof:

Given processes, for any , we can solve wait-free consensus using a Compare&Swap object(and read/write objects)

n n

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A Wait-Free Consensus algorithm for n processors using a compare&swap object

Shared Memory

1nx

0x

1np

0p

1ny

0y

First

Local Memory

(compare&swap object)

Initial value

Final value

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Code for processor ip }1,0{i

If then else

),,(& ii xFirstSwapComparev

ii xy

ii vy

//am I the first?

//yes, choose my value

Note: the algorithm uses a compare&swap and read/write objects

iv

//no, choose the value of the first process which is stored in First

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Shared Memory00 x

0p0y

First

Local Memory

Example execution:

11 x1p

1y

02 x2p

2y

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00 x0p

0y

Local Memory

Suppose executes first

11 x1p

1y

02 x2p

2y

2p

Compare&Swap(First, ,0)

Shared Memory

First

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00 x0p

0y

Local Memory

Suppose executes first

11 x1p

1y

02 x2p

02 y

2p

0

Shared Memory

First

Realizes it is first, decides on its own value

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00 x0p

0y

Local Memory

Suppose executes second

11 x1p

1y

02 x2p

1p

Compare&Swap(First, ,1)

Shared Memory

First0

02 y

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00 x0p

0y

Local Memory

11 x1p

01 y

02 x2p

Shared Memory

First0

0

Realizes is not first, decides on value of First

02 y

Suppose executes second 1p

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00 x0p

00 y

Local Memory

11 x1p

01 y

02 x2p

Shared Memory

First0

0

Realizes is not first, decides on value of First

02 y

Similarly for 0p Consensus has beenreached

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The algorithm is wait-free, since after the completion of the Compare&Swap operation, every processor decides (without consideringwhat the other processors do)

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Consensus HierarchyPart 3

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Consensus Number

Consensus Number of an object type:

The maximum number of processes for which the object can be used tosolve the wait-free consensus problem(together with read/write objects)

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Object Type Consensus Number

Read/Write 1

FIFO 2

Compare&Swap (infinity)

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Simulation:Object Type B

Object Type A

Object Type A

Read/Write Object

Object of type A simulates object of Type B

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Theorem:Objects of type A with consensus numbercannot simulate another object of type B withconsensus number

n

nm

Proof: Since otherwise, object A wouldhave consensus numberm

End of Proof

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Universality

can simulate in a wait-free mannerany other arbitrary object

Universal object:

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Objects with consensus number ncan simulate in a wait-free mannerany other arbitrary object of up toprocessors

n

We can show:

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new-state

Compare&Swap

inv

responsebefore

new-stateinv

responsebefore

initial

head

anchor

Non-Blocking Simulation

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new-state

Consensus Object

inv

responseafter

new-stateinv

responseafter

initial

Head

anchor

Non-Blocking Simulation

seq seq 1

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new-stateinv

responseafter

Announce

seq

Wait-free Simulation