CSCE 668 DISTRIBUTED ALGORITHMS AND SYSTEMS

1. Set 18: Wait-Free Simulations Beyond Registers CSCE 668DISTRIBUTED ALGORITHMS AND SYSTEMS Fall 2011 Prof. Jennifer Welch

2. Data Types Beyond Registers • Registers support the operations read and write • We've seen wait-free simulations of one kind of register out of another kind • different numbers of values, readers, writers • What about (wait-free) simulating a significantly different kind of data type out of registers? • More generally, what about (wait-free) simulating an object of type X out of objects of type Y ? Set 18: Wait-Free Simulations Beyond Registers

3. Key Insight • Ability of objects of type Y to be used to simulate an object of type X is related to the ability of those data types to solve consensus! • We are focusing on systems that are • asynchronous • shared memory • wait-free Set 18: Wait-Free Simulations Beyond Registers

4. FIFO Queue Example • Sequential specification of a FIFO queue: • operation with invocation enq(x) and response ack • operation with invocation deq and response return(x) • a sequence of operations is allowable iff each deq returns the oldest enqueued value that has not yet been dequeued (returns  if queue is empty) Set 18: Wait-Free Simulations Beyond Registers

5. one shared FIFO queue two shared registers write my input into my register use shared queue to arbitrate between the 2 procs: first one to dequeue the initial 0 wins, decision value is its input loser obtains decision value from other proc's register Consensus Algorithm for n = 2 Using FIFO Queue Initially Q = [0] and Prefer[i] =  Prefer[i] := pi's input val := deq(Q) if val = 0 then decide on pi's input else temp := Prefer[1 - i] decide temp Set 18: Wait-Free Simulations Beyond Registers

6. Implications of Consensus Algorithm Using FIFO Queue • Suppose we want to wait-free simulate a FIFO queue using read/write registers. • Is this possible? • No! If it were possible, we could solve consensus: • simulate a FIFO queue using registers • use simulated queue and previous algorithm to solve consensus Set 18: Wait-Free Simulations Beyond Registers

7. Extend Algorithm to More Procs? • Can we use FIFO queues to solve consensus with more than 2 procs? • The ability to atomically dequeue a value was key to the 2-proc alg: • one proc. learns it is the winner • the other learns it is the loser, therefore the id of the winner is obvious • Not clear how to handle 3 procs. • Suppose we have a different data type: Set 18: Wait-Free Simulations Beyond Registers

8. Compare & Swap Specification compare&swap(X : shared memory address, old: value, new: value) previous := X // previous is a local var. if previous = old then X := new return previous occurs atomically X old new Set 18: Wait-Free Simulations Beyond Registers

9. one shared C&S object if First =  then replace with my input simultaneously indicate the winner and the value to be decided by all the losers Consensus Algorithm Using Compare-and-Swap Initially First =  val := compare&swap(First, , my input) if val =  then decide on my input else decide val Set 18: Wait-Free Simulations Beyond Registers

10. Impossibility of 3-Proc Consensus with FIFO Queue Theorem (15.3): Wait-free consensus is impossible using FIFO queues and registers if n > 2. Proof: Same structure as for registers. Key difference is when considering situation when • C is bivalent • p0(C) is 0-valent and p1(C) is 1-valent. Set 18: Wait-Free Simulations Beyond Registers

11. 0/1 0 1 0 1 Impossibility of 3-Proc Consensus with FIFO Queues • p0 and p1 must be accessing the same FIFO queue. Case 1: Both steps are deq's. C p0 deq's p1 deq's p0 deq's p1 deq's look same to p2 contradiction! Set 18: Wait-Free Simulations Beyond Registers

12. 0/1 0 1 ? Impossibility Proof Case 2:p0 deq's and p1enq's. Case 2.1: The queue is not empty in C C p0 deq's p1 enq's p1 enq's p0 deq's contradiction! Set 18: Wait-Free Simulations Beyond Registers

13. 0/1 0 1 1 Impossibility Proof Case 2:p0 deq's and p1enq's. Case 2.2: The queue is empty in C queue is empty C p0 deq's  p1 enq's queue is still empty p0 deq's look the same to p2 queue is empty again contradiction! Set 18: Wait-Free Simulations Beyond Registers

14. 1 0/1 0 1 0 Impossibility Proof Case 3:Both p0 and p1 enq (on same queue). C p0 enq's A p1 enq's B p1 enq's B p0 enq's A why do  and  exist? : p0 takes steps until deq'ing B : p0 takes steps until deq'ing A : p1 takes steps until deq'ing B : p1 takes steps until deq'ing A look the same to p2 contradiction! Set 18: Wait-Free Simulations Beyond Registers

15. 0/1 0 1 0 1 Impossibility Proof Case 3 cont'd: Suppose  does not exist: C p0 enq's A p1 enq's B p1 enq's B p0 enq's A p0 takes steps until deciding but never deq's A; decides 0 p0 takes same number of steps as on the left; never deq's B; also decides 0 contradiction! Set 18: Wait-Free Simulations Beyond Registers

16. Impossibility Proof Case 3 cont'd: Prove existence of  similarly. Thus there is no wait-free algorithm for consensus with 3 procs using FIFO queues and registers. Set 18: Wait-Free Simulations Beyond Registers

17. Implications • Suppose we want to wait-free simulate a compare&swap object using FIFO queues (and registers). • Is this possible? • Not if n > 2! If it were possible, we could solve consensus using FIFO queues (and registers): • simulate a compare&swap object using FIFO queues (and registers) • use simulated compare&swap object and c&s algorithm to solve consensus Set 18: Wait-Free Simulations Beyond Registers

18. Generalize these Arguments • Previous results concerning FIFO queues and compare&swap suggest a criterion for determining if wait-free simulations exist: • based on ability of the data types to solve consensus for a certain number of procs. Set 18: Wait-Free Simulations Beyond Registers

19. Consensus Number Data type X has consensus numbern if n is the largest number of procs. for which consensus can be solved using only objects of type X and read/write registers. Set 18: Wait-Free Simulations Beyond Registers

20. … X X X Y … reg reg reg Using Consensus Numbers Theorem (15.5): If data type X has consensus number m and data type Y has consensus number n with n > m, then there is no wait-free simulation of an object of type Y using objects of type X and read/write registers in a system with more than m procs. Set 18: Wait-Free Simulations Beyond Registers

21. Using Consensus Numbers Proof: Suppose in contradiction there is a wait-free simulation S of Y using X and registers in a system with k procs, where m < k ≤ n. • Construct consensus algorithm for k > m procs using objects of type X (and registers): • Use S to simulate some objects of type Y using objects of type X (and registers) • Use the (simulated) type Y objects (and registers) in the k-proc consensus algorithm that exists since CN(Y) = n. contradicts CN(X) < k Set 18: Wait-Free Simulations Beyond Registers

22. Corollaries • There is no wait-free simulation of any object with consensus number > 1 using just read/write registers. • There is no wait-free simulation of any object with consensus number > 2 using just FIFO queues and read/write registers. Set 18: Wait-Free Simulations Beyond Registers

23. Universality • Let's now consider positive results relating to consensus number. • A data type is universal if objects of that type (together with read/write registers) can wait-free simulate any data type. • Theorem: If data type X has consensus number n, then it is universal in a system with at most n procs. Set 18: Wait-Free Simulations Beyond Registers

24. Proving Universality Result • Describe an algorithm that simulates any data type • uses compare&swap (instead of any object with consensus number n) • simulation is only non-blocking, weaker than wait-free • Modify to use any object with consensus number n • Modify to be wait-free • Modify to bound shared memory used Set 18: Wait-Free Simulations Beyond Registers

25. Non-Blocking • Non-blocking vs. wait-free is analogous to no-deadlock vs. no-lockout for mutual exclusion. • Non-blocking simulation:at any point in an execution, if at least one operation is pending (response is not yet ready to be done), then there is a finite sequence of steps by a single proc that completes one of the pending operations. • Does not ensure that every pending operation is eventually completed. Set 18: Wait-Free Simulations Beyond Registers

26. Universal Construction • Keep history of operations that have been applied to the simulated object as a shared linked list. • To apply an operation on the simulated object, the invoking proc. must insert an appropriate "node" into the linked list: • it is convenient to put the newest node at the head of the list • A compare&swap object is used to keep track of the head of the list Set 18: Wait-Free Simulations Beyond Registers

27. Details on Linked List Each linked list node has • operation invocation • new state of the simulated object • operation response • pointer to previous node (previous op) anchor Head Set 18: Wait-Free Simulations Beyond Registers

28. Simulation Initially Head points to anchor node • represents initial state of simulated object When inv is invoked: allocate a new linked list node in shared memory, pointed to by local var point point.inv := inv repeat h := Head // h is a local var point.state, point.response := apply(inv,h.state) point.before := h until compare&swap(Head,h,point) = h do the output indicated by point.response depends on simulated data type if Head still points to same node h points to, then make Head point to new node. Set 18: Wait-Free Simulations Beyond Registers

29. Simulation Figure pi invocation point state response h before … if compare&swap indicates that Head has moved on, then try again to insert the new node, at the new location Head Set 18: Wait-Free Simulations Beyond Registers

30. Strengthenings of Algorithm • To replace compare&swap object with any object with consensus number n (the number of procs): • define a consensus object (data type version of consensus problem) • get around the difficulty that a consensus object can only be used once by adding a consensus object to each linked list node that points to next node in the list Set 18: Wait-Free Simulations Beyond Registers

31. Strengthenings of Algorithm • To get a wait-free implementation, use idea of helping: procs help each other to finish pending operations (not just their own) • To reduce the size of the linked list (so it doesn't grow without bound), need to keep track of which list nodes can be recycled. Set 18: Wait-Free Simulations Beyond Registers

32. Effect of Randomization • Suppose we relax the liveness condition for linearizable shared memory: • operations must terminate with high probability • Now a randomized consensus algorithm can be used to simulate any data type out of any other data type, including read/write registers • I.e., hierarchy collapses. Set 18: Wait-Free Simulations Beyond Registers