Specification, implementation and application of randomized regular registers (brief announcement)
Page 338
Abstract
This paper presents a definition of a randomized regular register, shows that the definition is implemented by the probabilistic quorum algorithm of [3], and shows how to program with such registers using the framework of [4]. Consequently, existing iterative algorithms for a large class of problems (including solving systems of linear equations, finding shortest paths, etc.) will converge with high probability if executed in a system in which the shared data is implemented with registers satisfying the new condition. A modified definition is presented and its expected time for convergence is calculated and compared experimentally with that for the original definition.
We define a random regular register to satisfy: every operation invocation has a matching response; every read reads from a write that starts before the read ends; and for every finite execution that ends in a write invocation, the probability that this write W is read from infinitely often is 0, if an infinite number of writes are performed in the extension. We show in [1] that the probabilistic quorum algorithm of [2, 3] implements a random regular register, by showing that the probability that at least one replica from W's quorum survives ℓ subsequent writes is at most k(n-k/n)ℓ, where n is the number of processes and k is the quorum size.
The class of algorithms considered in [4] are those in which a function is applied repeatedly to a vector to produce another vector. In typical applications, each vector component may be computed by a separate process, based on that process' current best estimate of the values of all the vector components — estimates which might be out of date. Üresin and Dubois show that if the function satisfies certain properties and if the outdatedness of the vector component estimates is not too extreme, then this iterative procedure will eventually converge to the fixed point of the function, in increments called pseudocycles. Functions satisfying the desired properties are called asynchronously contracting operators (ACOs). In [1] we prove: If F is an ACO, then in every execution using random regular registers, the computed vector eventually converges to the fixed point of F with probability 1.
Our second definition requires the register to be monotone, meaning that if a read reads from a certain write, then no subsequent read by the same process reads from an earlier write. Additionally, let Y be a random variable whose value is the number of reads by a process after a write W until W or a later write is read from by that process. Then there must exist p, 0 < p ≤ 1, such that for all r ≥ 1, Pr(Y = r) ≤ (1 - p)r - 1 · p.
In [1] we define a round to be a minimal length (contiguous) subsequence of an execution in which each process performs at least one update of its vector components, and we prove the following: In every execution using monotone random regular registers with constant p, the expected number of rounds per pseudocycle is at most 1/p. Thus the expected number of rounds for an ACO requiring M pseudocycles for convergence is at most M/p.
In [1] we prove that a simple modification to the probabilistic quorum algorithm for n processes with quorum size k implements a monotone random regular register with p = 1 - (n-k k)/(n k). This implies that the expected number of rounds per pseudocycle is at most 1/1-(n - k/n)k.
References
[1]
H. Lee and J. L. Welch, "Specification, Implementation and Application of Randomized Regular Registers," TR00-012, Department of Computer Science, Texas A&M University, March 2000.
[2]
D. Malkhi and M. Reiter, "Byzantine Quorum Systems," In Proceedings of the ~9th ACM Symposium on Theory of Computing, pages 569-578, May 1997.
[3]
D. Malkhi, M. Reiter, and R. Wright, "Probabilistic Quorum Systems," In Proceedings of the 16th Annual A CM Symposium on Principles of Distributed Computing, pages 267-273, Aug. 1997.
[4]
A. Uresin and M. Dubois, "Parallel Asynchronous Algorithms for Discrete Data," J. ACM, Vol. 37, No. 3, pages 558-606, July 1990.
Index Terms
- Specification, implementation and application of randomized regular registers (brief announcement)
Recommendations
Brief Announcement: An Exponential Separation Between Randomized and Deterministic Complexity in the LOCAL Model
PODC '16: Proceedings of the 2016 ACM Symposium on Principles of Distributed ComputingOver the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially ...
Brief Announcement: Symmetry Breaking in the CONGEST Model: Time- and Message-Efficient Algorithms for Ruling Sets
PODC '17: Proceedings of the ACM Symposium on Principles of Distributed ComputingWe study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. Our work is motivated by the following central question: can we break the long-...
Brief Announcement: A Randomness-efficient Massively Parallel Algorithm for Connectivity
PODC'21: Proceedings of the 2021 ACM Symposium on Principles of Distributed ComputingWe give a randomness-efficient Massively Parallel Computation (MPC) algorithm for deciding whether an undirected graph is connected. For Connectivity on n-vertex, m-edge graphs whose components have diameter at most D = 2o(log n/ log log n), our ...
Comments
Information & Contributors
Information
Published In
Copyright © 2000 ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]
Sponsors
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 16 July 2000
Check for updates
Qualifiers
- Article
Conference
Acceptance Rates
PODC '00 Paper Acceptance Rate 32 of 117 submissions, 27%;
Overall Acceptance Rate 740 of 2,477 submissions, 30%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 144Total Downloads
- Downloads (Last 12 months)13
- Downloads (Last 6 weeks)7
Reflects downloads up to 26 Jan 2025
Other Metrics
Citations
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in