1. CSCE 668 DISTRIBUTED ALGORITHMS AND SYSTEMS
Set 19: Asynchronous Solvability
CSCE 668 DISTRIBUTED ALGORITHMS AND SYSTEMS
CSCE 668
Fall 2011
Prof. Jennifer Welch

2. 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
Set 19: Asynchronous Solvability
CSCE 668

3. Model Assumptions asynchronous shared memory with read/write registers
heavy use of atomic snapshot objects
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
Set 19: Asynchronous Solvability
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4. 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
Set 19: Asynchronous Solvability
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5. 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.
new
Set 19: Asynchronous Solvability
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6. 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 the minimum value appearing in any segment
Set 19: Asynchronous Solvability
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7. 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.
Set 19: Asynchronous Solvability
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8. 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 pi.
So pi's final scan misses at least f + 1 values, contradicting the code.
Set 19: Asynchronous Solvability
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9. Synchronous vs. Asynchronous?
How does the previous, asynchronous, algorithm compare to the synchronous algorithm for k-set consensus from Chapter 5 homework?
Recall the synchronous algorithm runs in f/k + 1 rounds.
Set 19: Asynchronous Solvability
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10. Set Consensus Lower Bound
Theorem: There is no asynchrounous algorithm for solving k-set consensus in the presence 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 very involved
Set 19: Asynchronous Solvability
CSCE 668

11. 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)
Set 19: Asynchronous Solvability
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12. 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.
new
new
Set 19: Asynchronous Solvability
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13. Approximate Agreement Algorithm
Assume procs know the range from which input values are drawn:
let D be the length of this range
wait-free: 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 log2 D/ rounds
Set 19: Asynchronous Solvability
CSCE 668

14. Approximate Agreement Algorithm
Shared atomic snapshot objects ASO[1], ASO[2],...
Initially local variable v = pi's input
Initially local variable r = 1
while true do
update pi's segment of ASO[r] to be v
let scan be set of values obtained by scanning ASO[r]
v := midpoint(scan)
if r = log2 (D/) + 1 then decide v and terminate
else r++
Set 19: Asynchronous Solvability
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15. Analysis of Approx. Agreement Alg.
Definitions w.r.t. a particular execution:
M = log2 (D/) + 1
U0 = set of input values
Ur = set of all values ever written to ASO[r]
Set 19: Asynchronous Solvability
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16. 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:
min(Ur)
(min(Ur)+u)/2 u
(max(Ur)+u)/2
max(Ur)
elements of Ur+1 are in here
Set 19: Asynchronous Solvability
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17. 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(Ur+1)  range(Ur)
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(Ur+1) ≤ spread(Ur)/2
Set 19: Asynchronous Solvability
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18. 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(UM) ≤ spread(U0)/2M
≤ D/2M
≤ 
Set 19: Asynchronous Solvability
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19. 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
Set 19: Asynchronous Solvability
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20. 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
Set 19: Asynchronous Solvability
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21. Unknown Input Range Algorithm
shared atomic snapshot object A; initially all segments  updatei(A,[x,1,x]), where x is pi's input // [original input, rd#, current estimate] repeat scan A let S be spread of all inputs in non- segments if S = 0 then maxRound := 0 else maxRound := log2(S/) let rmax be largest round in non- segments let values be set of candidates in segments with round number rmax update pi's segment in A with [x,rmax+1,midpt(values)] until rmax ≥ maxRound decide midpoint(values)
Set 19: Asynchronous Solvability
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22. Analysis of Unknown Input Range Algorithm
Definitions w.r.t. a particular execution:
U0 = set of all input values
Ur = set of all values ever written to A with round number r
M = largest r s.t. Ur is not empty
With these changes, correctness proof is similar to that for known input range algorithm.
Set 19: Asynchronous Solvability
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23. 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 pi's input is observed by pj the last time pj computes its maxRound, same argument as before.
Otherwise, when pi wakes up, it ignores its own input and uses values from maxRound or later.
Set 19: Asynchronous Solvability
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24. 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
Set 19: Asynchronous Solvability
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25. Renaming Problem Definition
Termination: Eventually every nonfaulty proc pi decides on a new name yi Uniqueness: If pi and pj are distinct nonfaulty procs, then yi ≠ yj. 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.
Set 19: Asynchronous Solvability
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26. 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.
Set 19: Asynchronous Solvability
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27. 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 2n – 1, for wait-free case if n satisfies a certain number- theoretic property
If n does not satisfy the property, there is a wait-free algorithm with M = 2n – 2. (includes n = 6, 10, 14,...)
Set 19: Asynchronous Solvability
CSCE 668

28. Wait-Free Renaming Algorithm
Shared atomic snapshot object A; initially all segments  s := 1 // suggestion for my new name while true do update pi's segment of A to be [x,s], where x is pi’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
Set 19: Asynchronous Solvability
CSCE 668

29. Analysis of Renaming Algorithm
Uniqueness: Suppose in contradiction pi and pj choose same new name, s.
pi's last
update
before
deciding:
suggests s
pj's last
scan before
deciding s
pi's last
scan before
deciding s
sees s as pi's
suggestion and
doesn't decide s
contradiction!
Set 19: Asynchronous Solvability
CSCE 668

30. Analysis of Renaming Algorithm
New name space is {1, …, 2n – 1}.
Why?
rank of a proc pi'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, so suggested names are {1, …, n – 1}.
Then pi suggests n + n – 1 = 2n – 1.
Set 19: Asynchronous Solvability
CSCE 668

31. 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).
Set 19: Asynchronous Solvability
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32. 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 z1, z2,… 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 pi be proc whose original name has smallest rank (among this set of original names). Let r be this rank.
Set 19: Asynchronous Solvability
CSCE 668

33. Analysis of Renaming Algorithm
Eventually procs other than pi stop suggesting zr 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 pi has a larger rank and thus continually suggests a new name for itself that is larger than zr, once it does the first scan in .
Set 19: Asynchronous Solvability
CSCE 668

34. Analysis of Renaming Algorithm
Eventually pi does suggest zr as its new name:
By choice of zr as r-th smallest free new name, and fact that eventually other trying procs stop suggesting z1 through zr, eventually pi sees zr as free name with r-th smallest rank.
Contradicts assumption that pi is trying (i.e., stuck).
So termination holds.
Set 19: Asynchronous Solvability
CSCE 668

35. 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.
Set 19: Asynchronous Solvability
CSCE 668

36. 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.
Set 19: Asynchronous Solvability
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37. k-Exclusion Algorithm
cf. paper by Afek et al. [5].
Set 19: Asynchronous Solvability
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38. 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 pi and pj 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
Set 19: Asynchronous Solvability
CSCE 668

39. k-Assignment Algorithm Schema
k-assignment entry section
k-exclusion entry section
renaming using m = 2k-1 names
what about repeated invocations?
k-assignment exit section
k-exclusion exit section
Set 19: Asynchronous Solvability
CSCE 668

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