1. CPSC 668Set 19: Asynchronous Solvability1 CPSC 668 Distributed Algorithms and Systems Fall 2006 Prof. Jennifer Welch

2. CPSC 668Set 19: Asynchronous Solvability2 Problems Solvable in Failure- Prone Asynchronous Systems Although consensus is not solvable in failure- prone asynchronous systems (neither message passing nor read/write shared memory), there are some interesting problems that are solvable: –set consensus –approximate agreement –renaming –k-exclusion weakenings of consensus - "opposite" of consensus - fault-tolerant variant of mutex

3. CPSC 668Set 19: Asynchronous Solvability3 Model Assumptions asynchronous shared memory with read/write registers at most f crash failures of procs. results can be translated to message passing if f < n/2 (cf. Chapter 10) may be a few asides into message passing

4. CPSC 668Set 19: Asynchronous Solvability4 Set Consensus Motivation By judiciously weakening the definition of the consensus problem, we can overcome the asynchronous impossibility We've already seen a weakening of consensus: –weaker termination condition for randomized algorithms How about weakening the agreement condition? One weakening is to allow more than one decision value: –allow a set of decisions

5. CPSC 668Set 19: Asynchronous Solvability5 Set Consensus Definition Termination: Eventually, each nonfaulty processor decides. k-Agreement: The number of different values decided on by nonfaulty processors is at most k. Validity: Every nonfaulty processor decides on a value that is the input of some processor.

6. CPSC 668Set 19: Asynchronous Solvability6 Set Consensus Algorithm Uses a shared atomic snapshot object X –can be implemented with read/write registers update your segment of X with your input repeatedly scan X until there are at least n - f nonempty segments decide on minimum value appearing in any segment

7. CPSC 668Set 19: Asynchronous Solvability7 Correctness of Set Consensus Algorithm Termination: at most f crashes. Validity: every decision is some proc's input Why does k-agreement hold? –We'll show it does as long as k > f. –Sanity check: When k = 1, we have standard consensus. As long as there is less than 1 failure, we can solve the problem.

8. CPSC 668Set 19: Asynchronous Solvability8 k-Set Agreement Condition Let S be set of min values in final scan of each nf proc; these are the nf decisions Suppose in contradiction |S| > f + 1. Let v be largest value in S, the decision of p i. So p i 's final scan misses at least f + 1 values, contradicting the code.

9. CPSC 668Set 19: Asynchronous Solvability9 Set Consensus Lower Bound Theorem: There is no algorithm for solving k- set consensus in the presents of f failures, if f ≥ k. Straightforward extensions of consensus impossibility result fail; even proving the existence of an initial bivalent configuration is quite involved. Original proof of set-consensus impossibility used concepts from algebraic topology Textbook's proof uses more elementary machinery, but still rather involved

10. CPSC 668Set 19: Asynchronous Solvability10 Approximate Agreement Motivation An alternative way to weaken the agreement condition for consensus: Require that the decisions be close to each other, but not necessarily equal Seems appropriate for continuous- valued problems (as opposed to discrete)

11. CPSC 668Set 19: Asynchronous Solvability11 Approximate Agreement Definition Termination: Eventually, each nonfaulty processor decides.  -Agreement: All nonfaulty decisions are within  of each other. Validity: Every nonfaulty decision is within the range of the input values.

12. CPSC 668Set 19: Asynchronous Solvability12 Approximate Agreement Algorithm Assume procs know the range from which input values are drawn: –let D be the length of this range up to n - 1 procs can fail algorithm is structured as a series of "asynchronous rounds": –exchange values via a snapshot object, one per round –compute midpoint for next round continue until spread of values is within , which requires about log 2 D/  rounds

13. CPSC 668Set 19: Asynchronous Solvability13 Approximate Agreement Algorithm Initially local variable v = p i 's input Initially local variable r = 1 1.update p i 's segment of ASO[r] to be v 2.let scan be set of values obtained by scanning ASO[r] 3.v := midpoint(scan) 4.if r =  log 2 (D/  )  + 1 then decide v and terminate 5.else r++

14. CPSC 668Set 19: Asynchronous Solvability14 Analysis of Algorithm Definitions w.r.t. a particular execution: M =  log 2 (D/  )  + 1 U 0 = set of input values U r = set of all values ever written to ASO[r]

15. CPSC 668Set 19: Asynchronous Solvability15 Helpful Lemma Lemma (16.8): Consider any round r < M. Let u be the first value written to ASO[r]. Then the values written to ASO[r+1] are in this range: umin(U r )max(U r )(min(U r )+u)/2(max(U r )+u)/2 elements of U r+1 are in here

16. CPSC 668Set 19: Asynchronous Solvability16 Implications of Lemma The range of values written to the ASO object for round r + 1 is contained within the range of values written to the ASO object for round r. –range(U r+1 )  range(U r ) The spread (max - min) of values written to the ASO object for round r + 1 is at most half the spread of values written to the ASO object for round r. –spread(U r+1 ) ≤ spread(U r )/2

17. CPSC 668Set 19: Asynchronous Solvability17 Correctness of Algorithm Termination: Each proc executes M asynchronous rounds. Validity: The range at each round is contained in the range at the previous round.  -Agreement: spread(U M ) ≤ spread(U 0 )/2 M ≤ D/2 M ≤ 

18. CPSC 668Set 19: Asynchronous Solvability18 Handling Unknown Input Range Range might not be known. Actual range in an execution might be much smaller than maximum possible range. First idea: have a preprocessing phase in which procs try to determine input range –but asynchrony and possible failures makes this approach problematic

19. CPSC 668Set 19: Asynchronous Solvability19 Handling Unknown Input Range Use just one atomic snapshot object Dynamically recalculate how many rounds are needed as more inputs are revealed Skip over rounds to try to catch up to maximum observed round Only consider values associated with maximum observed round Still use midpoint

20. CPSC 668Set 19: Asynchronous Solvability20 Unknown Input Range Algorithm shared atomic snapshot object A; initially all segments  update i (A,[x,1,x]), where x is p i 's input repeat scan A let S be spread of all inputs in non-  segments if S = 0 then maxRound := 0 else maxRound := log 2 (S/  ) let r max be largest round in non-  segments let values be set of candidates in segments with round number r max update p i 's segment in A with [x,r max +1,midpt(values)] until r max ≥ maxRound decide midpoint(values)

21. CPSC 668Set 19: Asynchronous Solvability21 Analysis of Unknown Input Range Algorithm Definitions w.r.t. a particular execution: U 0 = set of all input values U r = set of all values ever written to A with round number r M = largest r s.t. U r is not empty With these changes, correctness proof is similar to that for known input range algorithm.

22. CPSC 668Set 19: Asynchronous Solvability22 Key Differences in Proof Why does termination hold? –a proc's local maxRound variable can only increase if another proc wakes up and increases the spread of the observable inputs. This can happen at most n - 1 times. Why does  -agreement hold? –If p i 's input is observed by p j the last time p j computes its maxRound, same argument as before. –Otherwise, when p i wakes up, it ignores its own input and uses values from maxRound or later.

23. CPSC 668Set 19: Asynchronous Solvability23 Renaming Procs start with unique names from a large domain Procs should pick new names that are still distinct but that are from a smaller domain Motivation: Suppose original names are serial numbers (many digits), but we'd like the procs to do some kind of time slicing based on their ids

24. CPSC 668Set 19: Asynchronous Solvability24 Renaming Problem Definition Termination: Eventually every nonfaulty proc p i decides on a new name y i Uniqueness: If p i and p j are distinct nonfaulty procs, then y i ≠ y j. We are interested in anonymous algorithms: procs don't have access to their indices, just to their original names. Code depends only on your original name.

25. CPSC 668Set 19: Asynchronous Solvability25 Performance of Renaming Algorithm New names should be drawn from {1,2,…,M}. We would like M to be as small as possible. Uniqueness implies M must be at least n. Due to the possibility of failures, M will actually be larger than n.

26. CPSC 668Set 19: Asynchronous Solvability26 Renaming Results Algorithm for wait-free case (f = n - 1) with M = n + f = 2n - 1. Algorithm for general f with M = n + f. Lower bound that M must be at least n + 1, for wait-free case. –Proof similar to impossibility of wait-free consensus Stronger lower bound that M must be at least n + f, if f is the number of failures –Proof uses algebraic topology and is related to lower bound for set consensus

27. CPSC 668Set 19: Asynchronous Solvability27 Wait-Free Renaming Algorithm Shared atomic snapshot object A; initially all segments  s := 1 // suggestion for my new name while true do update p i 's segment of A to be [x,s], where x is p i 's original name scan A if s is also someone else's suggestion then let r be rank of x among original names of non-  segments let s be r-th smallest positive integer not currently suggested by another proc else decide on s for new name and terminate

28. CPSC 668Set 19: Asynchronous Solvability28 Analysis of Renaming Algorithm Uniqueness: Suppose in contradiction p i and p j choose same new name, s. p i 's last scan before deciding s p j 's last scan before deciding s p i 's last update before deciding: suggests s sees s as p i 's suggestion and doesn't decide s

29. CPSC 668Set 19: Asynchronous Solvability29 Analysis of Renaming Algorithm New name space is {1,…,2n - 1}. Why? rank of a proc p i 's original name is at most n (the largest one) worst case is when each of the n - 1 other procs has suggested a different new name for itself, say {1,…,n - 1}. Then p i suggests n + n - 1 = 2n - 1.

30. CPSC 668Set 19: Asynchronous Solvability30 Analysis of Renaming Algorithm Termination: Suppose in contradiction some set T of nonfaulty procs never decide in some execution. Consider the suffix  of the execution in which –each proc in T has already done at least one update and –only procs in T take steps (others have either already crashed or decided).

31. CPSC 668Set 19: Asynchronous Solvability31 Analysis of Renaming Algorithm Let F be the set of new names that are free (not suggested at the beginning of  by any proc not in T) -- the trying procs need to choose new names from this set. Let z 1, z 2,… be the names in F in order. By the definition of , no proc wakes up during  and reveals an additional original name, so all procs in T are working with the same set of original names during . Let p i be proc whose original name has smallest rank (among this set of original names). Let r be this rank.

32. CPSC 668Set 19: Asynchronous Solvability32 Analysis of Renaming Algorithm Eventually procs other than p i stop suggesting z r as a new name: –After  starts, every scan indicates a set of free names that is no larger than F. –Every trying proc other than p i has a larger rank and thus continually suggests a new name for itself that is larger than z r, once it does the first scan in .

33. CPSC 668Set 19: Asynchronous Solvability33 Analysis of Renaming Algorithm Eventually p i does suggest z r as its new name: –By choice of z r as r-th smallest free new name, and fact that eventually other trying procs stop suggesting z 1 through z r, eventually p i sees z r as free name with r-th smallest rank. Contradicts assumption that p i is trying (i.e., stuck). So termination holds.

34. CPSC 668Set 19: Asynchronous Solvability34 General Renaming Suppose we know that at most f procs will fail, where f is not necessarily n - 1. We can use the wait-free algorithm, but it is wasteful in the size of the new name space, 2n - 1, if f < n - 1. We can do better (if f < n - 1) with a slightly different algorithm: –keep track in the snapshot object of whether you have decided –an undecided proc suggests a new name only if its original name is among the f + 1 lowest names of procs that have not yet decided.

35. CPSC 668Set 19: Asynchronous Solvability35 k-Exclusion Problem A fault-tolerant version of mutual exclusion. Processors can fail by crashing, even in the critical section (stay there forever). Allow up to k processors to be in the critical section simultaneously. If < k processors fail, then any nonfaulty processor that wishes to enter the critical section eventually does so.

36. CPSC 668Set 19: Asynchronous Solvability36 k-Exclusion Algorithm cf. paper by Afek et al. [5].

37. CPSC 668Set 19: Asynchronous Solvability37 k-Assignment Problem A specialization of k-Exclusion to include: Uniqueness: Each proc in the critical section has a variable called slot, which is an integer between 1 and m. If p i and p j are in the C.S. concurrently, then they have different slots. Models situation when there is a pool of identical resources, each of which must be used exclusively: –k is number of procs that can be in the pool concurrently –m is the number of resources –To handle failures, m should be larger than k

38. CPSC 668Set 19: Asynchronous Solvability38 k-Assignment Algorithm Schema k-exclusion entry section renaming using m = 2k-1 names k-assignment entry section k-exclusion exit section k-assignment exit section

39. CPSC 668Set 19: Asynchronous Solvability39 k-Assignment Algorithm Schema k-exclusion entry section request-name for long-lived renaming using m = 2k-1 names k-assignment entry section k-exclusion entry section release-name for long-lived renaming using m = 2k-1 names k-assignment exit section