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CSCE 668 DISTRIBUTED ALGORITHMS AND SYSTEMS

CSCE 668 DISTRIBUTED ALGORITHMS AND SYSTEMS. Fall 2011 Prof. Jennifer Welch. 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

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CSCE 668 DISTRIBUTED ALGORITHMS AND SYSTEMS

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  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

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