Applications of Probabilistic Quorums to Iterative Algorithms. HyunYoung Lee, University of Denver Jennifer L. Welch, Texas A&M University presented at ICDCS 2001. Outline. The Probabilistic Quorum Algorithm (PQA) Abstracting PQA into Random Register (RR)
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Applications of Probabilistic Quorums to Iterative Algorithms
HyunYoung Lee, University of Denver
Jennifer L. Welch, Texas A&M University
presented at ICDCS 2001
X, Y, Z, ...
a write quorum
a read quorum
Achieved when every set of size n/2 +1 is a quorum.
Tradeoff Theorem [Naor & Wool]: For any quorum system, if load is optimal (1/n), then availability is at most (n).
Theorem: If k = (n), then
A read quorum
Drawback: A read quorum might not overlap the most recent write quorum, causing a read to return an out-of-date value.
a write quorum
Theorem: Probability of not overlapping is < e-h2, when k = hn.
One writer and multiple readers.
[R1] Every read or write invocation has a response.
[R2] Every read Rreads from some write W:
(1) W begins before R ends.
(2) R’s value is same as W ’s value.
(3) W is latest such write.
[R3] For every finite execution ending with a write W, probability that W is read from infinitely often is 0 (over all extensions with an infinite number of writes).
Theorem 1: PQA implements an RR.
[R1]: Each invocation gets a response since no lost messages and only crash failures of servers.
[R2]: Each read reads a value written by a previous or overlapping write, since no data corruption.
[R3]: Show probability that at least one replica in a write quorum is never overwritten is 0:
Pr( 1 replica survives h writes )
k Pr( replica j survives h writes )
= k Pr( j Q1 … j Qh )
= k hi=1 Pr( j Qi )
0 as h .
[A1]: All views come from the past.
[A2]: Every component is updated infinitely often.
[A3]: Each view is used only finitely often.
Red views are updated
Arrows indicate views
used in last update.
[A1], [A2], [A3] are equivalent to the existence of a partition of the update sequence into pseudocycles (p.c.’s):
p.c. i -1
Theorem [UD]: Sufficient condition on F for convergence to fixed point, if update sequence satisfies [A1]-[A3]: There exists integer M and sequence of sets D0, D1,… such that
Shown to be an ACO by [UD].
Claim: Worst-case number of pseudocycles for F to converge is log2 diameter(G).
Theorem 2: If F is an ACO, then every iterative execution using RRs for the vector components converges with probability 1.
Proof: Show the sequence of updates in the execution satisfies [A1], [A2] and [A3] with probability 1.
[A1]: All views are from the past by [R2].
[A2]: Application ensures every component is updated i.o.
[A3] holds with probability 1: Each view is used finitely often with probability 1 by [R3].
at least once.
How many (expected) rounds per p.c.?
We don’t know with current RR definition, so modify definition...
[R1] - [R3] plus
[R4]: If read R by process i reads from W and a later read R' by i reads from W' , then W' does not precede W.
[R5]: There exists q s.t. for all r,
Pr[r reads are needed until W or a later write is read from] (1 - q)r-1q.
So q is the probability of a “successful” read (w.r.t. W).
Same as previous probabilistic quorum algorithm, except:
Theorem 3: Attains q = 1 -
W ’s or later value is read if a subsequent read quorum overlaps W ’s quorum.
n - k
Theorem 4: Expected number of rounds per pseudocycle, when implementing an ACO with monotone RRs, is at most 1/q.
Proof: For p.c. h to end, each process i must read from a write first write in p.c. h-1.
Once this read occurs for i, every later read by i is at least as recent, since monotone.
Expected # rounds for first read is 1/q by [R5].
Corollary: For monotone PQA, expected # rounds per p.c. is (1 - ((n-k)/n)k)-1.
Expression is between 1 and 2 when k = n.
Strict quorum system has 1 round per p.c.
Monotone PQA has > 1 expected round per p.c. but may have fewer messages per p.c.
Which has better message complexity?
Messages per round in synchronous case:
MPQA: When k = n, expected # messages per p.c. is c2m(p+1)n, 1 < c < 2.
Recall when k = n, expected # messages per p.c. for MPQA is c2m(p+1)n, 1 < c < 2.
Simulated non-monotone and monotone RR implementations using PQs with APSP application to study:
log2 33 = 6 pseudocycles to converge.
Measured rounds till convergence (when simulated results equaled precomputed actual answer).
Each plotted point is average of 7 runs.
bound is not tight.
Synch & asynch are
Monotone is better