Common2 Extended Common2 Extended to Stacks to Stacks Adam Morrison Adam Morrison joint with joint with Eli Gafni Eli Gafni Yehuda Afek Yehuda Afek
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Common2 ExtendedCommon2 Extendedto Stacksto Stacks
Adam MorrisonAdam Morrisonjoint withjoint with
Eli GafniEli GafniYehuda AfekYehuda Afek
ModelModel
• Shared memoryShared memory model. model.
• Wait-free linearizable Wait-free linearizable algorithms. algorithms.
ComputabilityComputability
ComputabilityComputability
Snapshot Snapshot implementationimplementation fromfrom read-write registersread-write registers..
StackStack??
X X implementationimplementation fromfrom stacks stacks and and read-write registersread-write registers??
Wait-free hierarchyWait-free hierarchyCo
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[[Herlihy 91Herlihy 91]]
Object’s Object’s consensus numberconsensus number::Maximum number of Maximum number of processes that can processes that can implement consensus usingimplement consensus usingcopies of the object andcopies of the object and R/W registers.R/W registers.
Wait-free hierarchyWait-free hierarchyCo
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Stack
Register
[[Herlihy 91Herlihy 91]]
Wait-free hierarchyWait-free hierarchyCo
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Stack
F&A SwapQueue
Register
[[Herlihy 91Herlihy 91]]
swap(x):swap(x):
old := old := vv
vv := x := x
returnreturn old old
F&A(x):F&A(x):
old := old := vv
vv += x += x
returnreturn old old
Queue
Wait-free hierarchyWait-free hierarchyCo
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F&A Swap
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[[Herlihy 91Herlihy 91]]
More than 2 processes?More than 2 processes?
Common2Common2
Queue
This talkThis talkCo
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Stack
F&A Swap
Register
Common2Common2
Stack algorithmStack algorithm
0
rr
cellscells
ØØ
ØØ
ØØ
ØØ
ØØ
ØØ
ØØ
Pop() {Pop() { Push(x) {Push(x) {
for t:=F&A(r,0)-1for t:=F&A(r,0)-1 t := F&A(r,1) t := F&A(r,1)
downto 0 {downto 0 { cells]t[ := x cells]t[ := x
x := swap(cells]t[, Ø)x := swap(cells]t[, Ø) }}
if (x != Øif (x != Ø(( return x return x
}}
return EMPTYreturn EMPTY
{{
Stack algorithmStack algorithm
0
rr
cellscells
Push(x)Push(x)1
x
Push(y)Push(y)
2
PopPop
SwapSwap
SwapSwap
ØØ
ØØ
ØØ
ØØ
ØØ
ØØ
ØØ
Wait-free
Wait-free
Linearizability proofLinearizability proof
Linearizability proofLinearizability proof• Concurrent matching Push/Pop pairs can Concurrent matching Push/Pop pairs can
be ignored.be ignored.
Obtain an execution where Pop starts after Obtain an execution where Pop starts after corresponding Push has finished.corresponding Push has finished.
Push(x)Push(x) Push(y)Push(y) Pop: yPop: y
Push(z)Push(z)
Pop: zPop: z
Linearizability proofLinearizability proofLinearizability: Linearizability: assign each assign each
operation a linearization point at operation a linearization point at some some eventevent during its execution. during its execution.
Handling tiesHandling ties: build explicit : build explicit sequential order to breaks ties.sequential order to breaks ties.
Linearizing procedure thatLinearizing procedure thatprocesses the execution,processes the execution,using an using an auxiliary arrayauxiliary array,,and outputs aand outputs alinearization.linearization.
Proof techniqueProof technique
auxaux
Auxiliary arrayAuxiliary array
auxaux
xLin=Push(y) Pop:y Push(x)Lin=Push(y) Pop:y Push(x)
Push is linearized when it writes:Push is linearized when it writes:If to top of array, linearized now.If to top of array, linearized now.Else, with the PushElse, with the Pushabove it.above it.
Linearizing PushLinearizing Push()()
auxaux
x
zPush(z)Push(z)
Lin=Push(y) Pop:y Push(x)Lin=Push(y) Pop:y Push(x)
Push(w)Push(w) w
Lin=Push(y) Pop:yLin=Push(y) Pop:y Push(w)Push(w) Push(x)Push(x) Push(z)Push(z)
Pops are linearized as soon as their returnPops are linearized as soon as their returnvalue is at top value is at top of the auxiliary arrayof the auxiliary array..
Linearizing PopLinearizing Pop()()
cellscells
xx
zz
w
Lin=Push(y) Pop:y Push(w) Push(x) Push(z)Lin=Push(y) Pop:y Push(w) Push(x) Push(z)
Pop:xPop:x Pop:zPop:z
Pop:xPop:x
xx
zz
ww
Pop:xPop:x
auxaux
Pop:zPop:z
Pop:zPop:z
EMPTYEMPTY
Linearization proc correctnessLinearization proc correctness•Linearization is a valid stack Linearization is a valid stack
execution. execution. •Linearization respects real-time Linearization respects real-time
order.order.
Push is linearized when it writes:Push is linearized when it writes:If to top of array, linearized now.If to top of array, linearized now.Else, with the PushElse, with the Pushabove it.above it.
Linearizing PushLinearizing Push()()
auxaux
x
zPush(z)Push(z)
Push(w)Push(w) w
Push(w)Push(w)
Push(x)Push(x)
F&AF&A
F&AF&A writewrite
writewrite
Linearization proc correctnessLinearization proc correctness•Linearization is a valid stack Linearization is a valid stack
execution. execution. •Linearization respects real-time Linearization respects real-time
order.order.•Every operation is linearized.Every operation is linearized.
Pops are linearized as soon as their returnPops are linearized as soon as their returnvalue is at top of the auxiliary array.value is at top of the auxiliary array.
Every Pop() is linearizedEvery Pop() is linearized
cellscells
xx
zzPop:xPop:x
xx
zz
Pop:xPop:x
auxaux
Pop:zPop:z
ww ww
Pop:x – first to finish without being linearizedPop:x – first to finish without being linearized
Linearization proc correctnessLinearization proc correctness•Linearization is a valid stack Linearization is a valid stack
execution. execution. •Linearization respects real-time Linearization respects real-time
order.order.•Every operation is linearized.Every operation is linearized.
Queue
ConclusionConclusionCo
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Stack
F&A Swap
Register
Common2Common2
??
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