Consensus Hierarchy Part 2

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Consensus HierarchyPart 2
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FIFO (Queue)
head
tail
FIFO Object
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Special case empty queue
tail
head
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A Wait-Free Consensus algorithm for 2 processors
using a FIFO object
tail
head
Initially
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Shared Memory
Queue
tail
head
Other Variables
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Local variables
initial values for the consensus problem
resulting values for the consensus problem
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Shared Memory
Queue
tail
head
Other Variables
Initial values
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Code for processor
//am I the first?
If then else
//yes, choose my value
//no, choose the other processors value
Note the algorithm uses a FIFO object
and read/write objects
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Example execution
Shared Memory
Queue
tail
head
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Suppose that accesses first the queue
Shared Memory
decides on its own value
Queue
tail
head
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Suppose that accesses second
Shared Memory
Queue
Consensus Reached
tail
head
decides on the other processors value
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Theorem
There is no wait-free consensus algorithm using
only FIFO objects and read-write objects for
Proof of Theorem
Consider three processors (the same proof
generalizes to more)
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There is a bivalent initial configuration (we
proved it before)
We will show that every bivalent
configuration has a processor which is not
critical
Therefore, we can construct an infinite execution
with bivalent configurations where consensus is
never reached
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Assume for contradiction that all processors are
critical
univalent
bivalent
Possible executions
univalent
univalent
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It cannot be that all have the same valence
Contradiction
valent
bivalent
valent
valent
valent
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There must exist two processors with different
valences
bivalent
univalent
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Shared Memory
Queue
Queue
Queue
Read/Write Objects
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Shared Memory
Case the processors access different
objects
Queue
Queue
bivalent
Queue
Read/Write Objects
univalent
Note if an object was read/write the analysis is
the similar
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Two possible executions
op Q2
op Q1
bivalent
op Q2
op Q1
Impossible since
univalent
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Shared Memory
Case the processors access same object
Queue
Queue
bivalent
Queue
Read/Write Objects
univalent
Note if the object was read/write the
analysis is the same as in the case with
read/write objects
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Subcase deq/deq
deq(Q)
bivalent
deq(Q)
univalent
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Queue Q before operations
deq(Q)
deq(Q)
bivalent
deq(Q)
deq(Q)
Impossible since
univalent
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Subcase deq/enq
deq(Q)
bivalent
enq(Q,x)
univalent
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Suppose Q was not empty
enq(Q,x)
deq(Q)
bivalent
enq(Q,x)
deq(Q)
Impossible since
univalent
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Suppose Q was empty
enq(Q,x)
deq(Q)
bivalent
enq(Q,x)
Impossible since
univalent
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Subcase enq/enq
enq(Q,a)
bivalent
enq(Q,b)
univalent
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Suppose Q was not empty
enq(Q,b)
enq(Q,a)
bivalent
enq(Q,b)
enq(Q,a)
univalent
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enq(Q,b)
Dequeue a
Dequeue b
enq(Q,a)
bivalent
enq(Q,b)
enq(Q,a)
Dequeue b
Dequeue a
univalent
Impossible since
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Explanation
decides
enq(Q,b)
enq(Q,a)
Suppose does not dequeue a
bivalent
A decision will be reached since the consensus
algorithm is wait-free
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decides
enq(Q,b)
enq(Q,a)
contradiction
bivalent
decides
enq(Q,b)
enq(Q,a)
The same value will be decided by , since
sees the same shared memory values in both
executions
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In all cases we obtained contradiction
Therefore, there exists a processor which is not
critical
univalent
bivalent
univalent
bivalent
(not critical)
univalent
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Therefore, we can construct an execution
bivalent
bivalent
bivalent
Never ends
Initial configuration
Consensus can never be reached
End of Theorem Proof
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CompareSwap
Shared Memory
CompareSwap(X,A,B)
Temp X If XA then X B Return Temp
X
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A Wait-Free Consensus algorithm for n processors
using a compareswap object
Local Memory
Shared Memory
First
Initial value
compareswap object
Final value
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Code for processor
If then else
//am I the first?
//yes, choose my value
//no, choose the value of the first which
is stored in First
Note the algorithm uses a compareswap
and read/write objects
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Example execution
Local Memory
Shared Memory
First
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Suppose executes first
Local Memory
Shared Memory
First
CompareSwap(First, ,0)
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Suppose executes first
Local Memory
Shared Memory
First
Realizes it is first, decides on its own value
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Suppose executes second
Local Memory
Shared Memory
CompareSwap(First, ,1)
First
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Suppose executes second
Local Memory
Shared Memory
First
Realizes is not first, decides on value of First
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Similarly for
Consensus has been reached
Local Memory
Shared Memory
First
Realizes is not first, decides on value of First
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The algorithm is wait-free, since after the
completion of the CompareSwap operation, every
processor decides (without considering whether
the other processors decide)