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Common2 Extended to Stacks

Common2 Extended to Stacks. Adam Morrison joint with. Yehuda Afek. Eli Gafni. Model. Shared memory model. Wait-free linearizable algorithms. Computability. Stack. ?. Computability. Snapshot implementation from read-write registers.

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Common2 Extended to Stacks

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  1. Common2 Extendedto Stacks Adam Morrison joint with Yehuda Afek Eli Gafni

  2. Model • Shared memory model. • Wait-free linearizable algorithms.

  3. Computability

  4. Stack ? Computability Snapshot implementationfrom read-write registers. X implementationfrom stacks and read-write registers?

  5. Wait-free hierarchy [Herlihy 91] Object’s consensus number: Maximum number of processes that can implement consensus using copies of the object and R/W registers. … Consensusnumber 2 1

  6. Wait-free hierarchy [Herlihy 91] … Consensusnumber 2 Stack Register 1

  7. Wait-free hierarchy [Herlihy 91] F&A(x): old := v v += x return old swap(x): old := v v := x return old … F&A Consensusnumber Swap 2 Queue Stack Register 1

  8. Wait-free hierarchy [Herlihy 91] Common2 … More than 2 processes? F&A Consensusnumber Swap 2 Queue Stack Register 1

  9. This talk Common2 … F&A Consensusnumber Swap 2 Queue Stack Register 1

  10. Stack algorithm Ø Ø Ø Ø Ø Ø Ø cells r 0

  11. Swap Swap Stack algorithm Pop() { Push(x) { for t:=F&A(r,0)-1 t := F&A(r,1) downto 0 { cells[t] := x x := swap(cells[t], Ø) } if (x != Ø( return x } return EMPTY { Ø Wait-free Ø Ø Ø Ø Ø x Ø cells r Push(x) 1 2 0 Pop Push(y)

  12. Linearizability proof

  13. Linearizability proof • Concurrent matching Push/Pop pairs can be ignored. Obtain an execution where Pop starts after corresponding Push has finished. Push(x) Push(y) Pop: y Push(z) Pop: z

  14. Linearizability proof Linearizability: assign each operation a linearization point at some event during its execution. Handling ties: build explicit sequential order to breaks ties.

  15. Proof technique Linearizing procedure that processes the execution, using an auxiliary array, and outputs a linearization. aux

  16. Auxiliary array Lin=Push(y) Pop:y Push(x) x aux

  17. Linearizing Push() Push is linearized when it writes: If to top of array, linearized now. Else, with the Push above it. Push(z) z Lin=Push(y) Pop:y Push(x) Lin=Push(y) Pop:yPush(w)Push(x)Push(z) x Push(w) w aux

  18. EMPTY Linearizing Pop() Pops are linearized as soon as their return value is at top of the auxiliary array. Lin=Push(y) Pop:y Push(w) Push(x) Push(z) Pop:z Pop:x Pop:x Pop:z Pop:x Pop:z z z x x w w cells aux

  19. Linearization proc correctness • Linearization is a valid stack execution. • Linearization respects real-time order. 

  20. Linearizing Push() Push is linearized when it writes: If to top of array, linearized now. Else, with the Push above it.  Push(z) z F&A write Push(x) x  F&A write Push(w) w Push(w) aux

  21. Linearization proc correctness • Linearization is a valid stack execution. • Linearization respects real-time order. • Every operation is linearized.  

  22. Every Pop() is linearized Pops are linearized as soon as their return value is at top of the auxiliary array. Pop:x Pop:x Pop:z z z w w x x cells aux Pop:x – first to finish without being linearized

  23. Linearization proc correctness • Linearization is a valid stack execution. • Linearization respects real-time order. • Every operation is linearized.   

  24. Conclusion Common2 … F&A Consensusnumber Swap ? 2 Queue Stack Register 1

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