Common2 Extended to Stacks

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