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The Computability of Relaxed Data Structures:Queues and Stacks as Examples

Nir Shavit and Gadi Taubenfeld

Version: August 2015

SIROCCO 2015 Gadi Taubenfeld © 2015

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Semantics of concurrent data structures

Sequential specification -- set of legal runs/sequences.

Consistency condition -- e.g. linearizability, sequential consistency, …

relaxed

This paper

SIROCCO 2015 Gadi Taubenfeld © 2015

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Why to relax ?

Synchronization inherently limits parallelism. Semantically weaker DS reduce the need from synchronization.

Thus, provides potential to achieve better performance

and scalability. There are many published efficient implementations

of relaxed data structures. (See related work.) We are interested in

computability not complexity.

SIROCCO 2015 Gadi Taubenfeld © 2015

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Queue[a,b,c]

enqueue dequeue

peek

a b

c

* -- can insert/remove/return a value at arbitrary position 0 -- operation not supported

SIROCCO 2015 Gadi Taubenfeld © 2015

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Stack[a,b,c]

push pop

top

a b

c

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Queue[a,b,c]

enqu

eue

dequ

eue

peek

Queue[1,1,1] -- traditional FIFO queue Queue[1,1,0] -- does not support peek Queue[1,1,*] -- peek returns a random value

Stack[ 1,0,1] -- atomic read/write register

Queue[*,*,0] = Stack[*,*,0] -- multiset object

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Object Consensus number

queue[1,1,1]

queue[*,*,0], queue[1,1,0] 2stack[*,*,0], stack[1,1,0],

stack[1,1,1] 2

atomic register, stack[1,0,1] 1

Objects are wait-free & linearizable

Consensus Numberknown results

SIROCCO 2015 Gadi Taubenfeld © 2015

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Relaxing the enqueue operation

SIROCCO 2015 Gadi Taubenfeld © 2015

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Relaxing the enqueue operationObject Consensus number

queue[1,1,1]

queue[2,1,1]queue[3,1,1]

…queue[*,1,1]queue[*,2,1]queue[*,3,1]

…queue[*,*,1]queue[*,*,2]

queue[*,*,0] 2

SIROCCO 2015 Gadi Taubenfeld © 2015

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Relaxing the enqueue operationObject Consensus number

queue[1,1,1]

queue[2,1,1]queue[3,1,1]

…queue[*,1,1] 2queue[*,2,1]queue[*,3,1]

…queue[*,*,1]queue[*,*,2]

queue[*,*,0] 2

SIROCCO 2015 Gadi Taubenfeld © 2015

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Relaxing the enqueue operationObject Consensus number

queue[1,1,1]

queue[2,1,1]queue[3,1,1]

…queue[*,1,1] 2queue[*,2,1] 2queue[*,3,1] 2

… 2queue[*,*,1] 2queue[*,*,2] 2

2queue[*,*,0] 2

SIROCCO 2015 Gadi Taubenfeld © 2015

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Relaxing the enqueue operationObject Consensus number

queue[1,1,1]

queue[2,1,1]

queue[3,1,1]

…

queue[*,1,1] 2queue[*,2,1] 2queue[*,3,1] 2

… 2queue[*,*,1] 2queue[*,*,2] 2

2queue[*,*,0] 2

SIROCCO 2015 Gadi Taubenfeld © 2015

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Relaxing the enqueue operationObject Consensus number

queue[1,1,1], queue[1,0,1]

queue[2,1,1], queue[2,0,1]

queue[3,1,1], queue[3,0,1]

… …

queue[*,1,1] 2queue[*,2,1] 2queue[*,3,1] 2

… 2queue[*,*,1] 2queue[*,*,2] 2

2queue[*,*,0] 2SIROCCO 2015 Gadi Taubenfeld ©

2015

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Relaxing the peek operation

SIROCCO 2015 Gadi Taubenfeld © 2015

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Relaxing the peek operationObject Consensus number

queue[1,1,1]

queue[1,1,2]queue[1,1,3]

…queue[1,1,*]queue[1,2,*]queue[1,3,*]

…queue[1,*,*]queue[2,*,*]

queue[*,*,0] 2

SIROCCO 2015 Gadi Taubenfeld © 2015

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Relaxing the peek operationObject Consensus number

queue[1,1,1]

queue[1,1,2] 2queue[1,1,3]

…queue[1,1,*]queue[1,2,*]queue[1,3,*]

…queue[1,*,*]queue[2,*,*]

queue[*,*,0] 2

SIROCCO 2015 Gadi Taubenfeld © 2015

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Relaxing the peek operationObject Consensus number

queue[1,1,1]

queue[1,1,2] 2queue[1,1,3] 2

… 2queue[1,1,*] 2queue[1,2,*] 2queue[1,3,*] 2

… 2queue[1,*,*] 2queue[2,*,*] 2

2queue[*,*,0] 2

SIROCCO 2015 Gadi Taubenfeld © 2015

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Relaxing the dequeue operation

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Relaxing the dequeue operationObject Consensus number

queue[1,1,1], queue[1,0,1]

queue[1,1,2] 2queue[1,2,2]

…queue[1,*,2]queue[1,*,3]queue[1,*,4]

…queue[1,*,*]queue[2,*,*]

queue[*,*,0] 2SIROCCO 2015 Gadi Taubenfeld ©

2015

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Relaxing the dequeue operationObject Consensus number

queue[1,1,1], queue[1,0,1]

queue[1,1,2] 2queue[1,2,2] 2

… 2queue[1,*,2] 2queue[1,*,3] 2queue[1,*,4] 2

… 2queue[1,*,*] 2queue[2,*,*] 2

2queue[*,*,0] 2

SIROCCO 2015 Gadi Taubenfeld © 2015

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Not supporting the dequeue operationObject Consensus number

queue[1,1,1], queue[1,0,1]

queue[1,1,2] 2queue[1,0,2]queue[2,0,2]queue[3,0,2]

…queue[*,0,2]queue[0,0,2] 1

SIROCCO 2015 Gadi Taubenfeld © 2015

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Not supporting the dequeue operationObject Consensus number

queue[1,1,1], queue[1,0,1]

queue[1,1,2] 2queue[1,0,2] 1queue[2,0,2]queue[3,0,2]

…queue[*,0,2]queue[0,0,2] 1

SIROCCO 2015 Gadi Taubenfeld © 2015

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Not supporting the dequeue operationObject Consensus number

queue[1,1,1], queue[1,0,1]

queue[1,1,2] 2queue[1,0,2] 1queue[2,0,2] 1queue[3,0,2] 1

… 1queue[*,0,2] 1queue[0,0,2] 1

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Atomic registers vs. relaxed queues

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Atomic registers vs. relaxed queuesObject Consensus number

queue[1,1,1], queue[1,0,1]

queue[1,1,2] 2queue[1,0,2] 1queue[2,0,2] 1queue[3,0,2] 1

… 1queue[*,0,2] 1

Atomic registers 1

yes

???

Can atomic registers implement … ?

SIROCCO 2015 Gadi Taubenfeld © 2015

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Atomic registers vs. relaxed queuesObject Consensus number

queue[1,1,1], queue[1,0,1]

queue[1,1,2] 2queue[1,0,2] 1queue[2,0,2] 1queue[3,0,2] 1

… 1queue[*,0,2] 1

Atomic registers 1

yes

No!!!

Theorem: A queue[a,0,c] has no wait-free implementation from atomic registers, for every two positive integers a and c.

SIROCCO 2015 Gadi Taubenfeld © 2015

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Discussion Each one of the infinitely many relaxed objects

has one of the following consensus numbers: 1, 2, . Why only three?

Queue is more sensitive than stack to changes in its semantics.

Consider other types of relaxed data structures. What is the internal structure among relaxed

objects with the same consensus number? queue[1,1,0] + registers queue[1,1,2] ?

SIROCCO 2015 Gadi Taubenfeld © 2015

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Thank you for listening

SIROCCO 2015 Gadi Taubenfeld © 2015