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1 Applications of Probabilistic Quorums to Iterative Algorithms HyunYoung Lee, University of Denver Jennifer L. Welch, Texas A&M University presented at ICDCS 2001

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2 Outline The Probabilistic Quorum Algorithm (PQA) Abstracting PQA into Random Register (RR) Using RRs in Iterative Convergent Algorithms Monotone RRs and their Performance Simulation Results Conclusions

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3 Outline The Probabilistic Quorum Algorithm (PQA) Abstracting PQA into Random Register (RR) Using RRs in Iterative Convergent Algorithms Monotone RRs and their Performance Simulation Results Conclusions

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4 Distributed Shared Memory Provides illusion of shared variables for inter- process communication on top of a message- passing distributed system Benefits of shared memory paradigm: –familiar from uniprocessor case –supports good software development practice Examples: Treadmarks [Amza+], DASH [Gharachorloo+],...

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5 Distributed Shared Memory app proc 1 client 1 server 1 server n client r app proc r network send recv send recv write(X,3)ack(X) read(Y)return(Y,5) Implements shared variables X, Y, Z,...

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6 Replicated Data with Quorums Keep a copy of shared variable at n replica servers that communicate by messages. A quorum is a subset of replica servers. To write: client updates copies in a quorum with new value plus timestamp. To read: client receives copies from a quorum and returns value with latest timestamp.

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7 Quorum Intersection To ensure each read obtains latest value written, every read quorum must intersect every write quorum. 12,7:00 4,9:00 10,8:00 a write quorum a read quorum

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8 Performance Measures for Quorum Systems Availability: minimum number of servers that must fail to disable every quorum [Peleg & Wool]. –Optimal (largest) availability is (n). Achieved when every set of size n/2 +1 is a quorum. Load: probability of accessing the busiest server, in the best case [Naor & Wool]. –Optimal (smallest) load is (1/ n). Tradeoff Theorem [Naor & Wool]: For any quorum system, if load is optimal (1/ n), then availability is at most ( n).

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9 Breaking the Tradeoff with Probabilistic Quorums [Malkhi, Reiter & Wright] Relax requirement that every read quorum overlap every write quorum. Instead, choose each quorum uniformly at random from the set of all k-sized subsets of the n replica servers, for k < n/2. Theorem: If k = ( n), then –availability is n - k = (n) –load is (1/ n) To handle server failures: keep trying until enough responses to form a quorum are received.

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10 Probabilistic Quorums [MRW] a write quorum Drawback: A read quorum might not overlap the most recent write quorum, causing a read to return an out-of-date value. 12,7:00 4,9:00 12,7:00 10,8:00 A read quorum Theorem: Probability of not overlapping is < e -h 2, when k = h n.

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11 Programming with PQs What are the semantics of the shared variable (register) implemented by the PQA? What kind of applications can tolerate reads returning, with low probability, out-of-date values?

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12 Outline The Probabilistic Quorum Algorithm (PQA) Abstracting PQA into Random Register (RR) Using RRs in Iterative Convergent Algorithms Monotone RRs and their Performance Simulation Results Conclusions

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13 Definition of Random Register One writer and multiple readers. [R1] Every read or write invocation has a response. [R2] Every read R reads 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. R(c) W 1 (c)W 2 (b)W 3 (a)W 4 (c)

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14 Definition of Random Register [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). Related Work: –Most work on randomized shared objects concerns termination, not correct responses. –[Afek+] and [Jayanti+] assumed a fixed subset of shared objects that can return incorrect values.

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15 PQA implements an RR Theorem 1: PQA implements an RR. Proof: [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.

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16 PQA Implements an RR [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 Q 1 … j Q h ) = k h i=1 Pr( j Q i ) = k((n-k)/n) h 0 as h .

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17 Outline The Probabilistic Quorum Algorithm (PQA) Abstracting PQA into Random Register (RR) Using RRs in Iterative Convergent Algorithms Monotone RRs and their Performance Simulation Results Conclusions

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18 Iterative Convergent Algorithms [Uresin & Dubois] Repeatedly apply a function to a vector to produce another vector until reaching a fixed point. Responsibility for vector components is distributed across several processes. Vector component updates are based on possibly out-of-date views of the vector components.

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19 Iterative Convergent Algorithms [UD] Requirements: [A1]: All views come from the past. [A2]: Every component is updated infinitely often. [A3]: Each view is used only finitely often timevector components Arrows indicate views used in last update. Red views are updated ones. 0

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20 Iterative Convergent Algorithms [UD] [A1], [A2], [A3] are equivalent to the existence of a partition of the update sequence into pseudocycles (p.c.’s): –at least one update per component, and –every view used was created in current or previous p.c. p.c. i -1p.c. i X

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21 Asynchronously Contracting Operators 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 D 0, D 1,… such that –each D K is Cartesian product of m sets (independence) –D 0 D 1 … D M = D M+1 = …= { fixed point } –If x D K, then F(x) D K+1 for all K. Why? At end of K-th p.c., computed vector is in D K. DMDM D M-1... D1D1 D0D0 m-vector fixed point

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22 Example: All Pairs Shortest Path G is weighted directed graph with n nodes. Compute n x n vector x; process i updates i-th row of x, 1 i n. Initially x is adjacency matrix for G. F(x) computes y, where y ij = min 1 k n { x ik + x kj }. Shown to be an ACO by [UD]. Claim: Worst-case number of pseudocycles for F to converge is log 2 diameter(G) .

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23 ACOs Correct with RRs 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].

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24 Implications RRs can be used to implement any ACO, which includes algorithms for –APSP –transitive closure –constraint satisfaction –solving system of linear equations If PQA is used for the RRs, improved load and availability are provided. Convergence is guaranteed with probability 1. But how long does it take to converge?

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25 Measuring Time with Rounds A round finishes when every process has –read all the vector components –applied the function –updated its own vector components at least once. How many (expected) rounds per p.c.? We don’t know with current RR definition, so modify definition...

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26 Outline The Probabilistic Quorum Algorithm (PQA) Abstracting PQA into Random Register (RR) Using RRs in Iterative Convergent Algorithms Monotone RRs and their Performance Simulation Results Conclusions

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27 Monotone RR 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. R(c)R(c)R'(b) W '(b)W(c)W(c) X

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28 Monotone RR Definition (cont’d) [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-1 q. So q is the probability of a “successful” read (w.r.t. W).

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29 Monotone RR Algorithm Same as previous probabilistic quorum algorithm, except: Read client keeps track of value with latest timestamp that it has seen so far. This value is returned if its timestamp is later than all those obtained from current quorum. Theorem 3: Attains q = 1 - W ’s or later value is read if a subsequent read quorum overlaps W ’s quorum. nknk ( ) () n - k k

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30 Monotone RR Rounds per P.C. 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].

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31 Messages vs. Rounds for ACOs 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?

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32 Message Complexity for ACOs Messages per round in synchronous case: Each of the m vector components is read by each of the p processes and written by one. Each operation generates two messages to each of the k quorum members. 2m(p+1)k. MPQA: When k = n, expected # messages per p.c. is c2m(p+1) n, 1 < c < 2.

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33 Comparing Message Complexity Recall when k = n, expected # messages per p.c. for MPQA is c2m(p+1) n, 1 < c < 2. High availability (n): –Strict: k = n/2 + 1, so # messages per p.c. is 2m(p+1)( n/2 +1). Worse. Low load (1/ n): –Strict: k = n (e.g., rows and columns of grid), so # messages per p.c. is 2m(p+1) n. Asymptotically same.

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34 Outline The Probabilistic Quorum Algorithm (PQA) Abstracting PQA into Random Register (RR) Using RRs in Iterative Convergent Algorithms Monotone RRs and their Performance Simulation Results Conclusions

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35 Simulation Purpose Simulated non-monotone and monotone RR implementations using PQs with APSP application to study: difference between synchronous and asynchronous cases expected convergence time in non-monotone case (no analysis) actual expected convergence time in monotone case compared to computed upper bound

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36 Simulation Details Input graph: log 2 33 = 6 pseudocycles to converge. Measured rounds till convergence (when simulated results equaled precomputed actual answer). Each plotted point is average of 7 runs

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37 Simulation Results Computed upper bound is not tight. Synch & asynch are very similar. Monotone is better than non-monotone.

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38 Outline The Probabilistic Quorum Algorithm (PQA) Abstracting PQA into Random Register (RR) Using RRs in Iterative Convergent Algorithms Monotone RRs and their Performance Simulation Results Conclusions

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39 Summary Proposed two specifications of randomized shared variables that can return wrong answers, monotone and non-monotone random read-write registers. Both specs can be implemented with PQA of [MRW]. Our specs can be used to implement a significant class of iterative convergent algorithms, characterized by [UD]; algorithms converge with probability 1. Computed bounds on convergence time and message complexity for ACOs in monotone case. Simulation results indicate monotone is faster than non- monotone, asynch and synch are similar, and computed upper bound is not tight.

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40 Future Work Are our specs of more general interest? Other good algs that implement them? Different specs better? Useful applications for other shared data structures (e.g., stack) with errors? How to specify and implement them? How to tolerate client failures? Approximate agreement as an application?

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