Self-Stabilizing Symmetry Breaking in Constant Space
Abstract
We investigate the problem of self-stabilizing round-robin token management on a bidirectional ring of identical processors. Each processor is an asynchronous probabilistic finite state (i.e., constant space) machine which sends and receives constant-size messages and whose state transition is triggered by the receipt of a message. We also show that this problem is equivalent to symmetry breaking (i.e., leader election).
Wejustify and suggest a two-layer (hardware and software) solution to the token management problem: The subproblem of reducing an arbitrary but nonzero number of tokens (in an otherwise arbitrary initial system state) to exactly one token (and a legal system state) is solved in hardware and takes only small polynomial time. The detection of a complete lack of tokens (communication deadlock) is done by a software clock. In high-speed networks the hardware layer can be implemented using fast universal switches (i.e., finite state machines) independent of the size of the network. We note that randomization is essential, since Dijkstra showed that for arbitrary rings the subproblem does not have a deterministic solution (regardless of the computational power of the identical processors). The use of the software layer (deadlock detection) in our solution is minimized.
MSC codes
Keywords
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Karl Abrahamson, Andrew Adler, Rachel Gelbart, Lisa Higham, David Kirkpatrick, The bit complexity of randomized leader election on a ring, SIAM J. Comput., 18 (1989), 12–29
2.
K. Abrahamson, A. Adler, L. Higham, and D. Kirkpatrick, Probabilistic solitude verification on a ring, in Proceedings of the 5th ACM Symposium on Principles of Distributed Computing, 1986.
3.
Karl Abrahamson, Andrew Adler, Lisa Higham, David Kirkpatrick, Probabilistic leader election on rings of known size, Lecture Notes in Comput. Sci., Vol. 519, Springer, Berlin, 1991, 481–495
4.
Y. Afek and G. M. Brown, Self‐stabilization over unreliable communication media, Distrib. Comput., 7 (1993), pp. 27–34.
5.
A. Arora and M. Gouda, Closure and convergence: A foundation for fault‐tolerant computing, in Proceedings of the 22nd International Conference on Fault‐Tolerant Computing Systems, IEEE, 1992.
6.
Yossi Matias, Yehuda Afek, Simple and efficient election algorithms for anonymous networks, Lecture Notes in Comput. Sci., Vol. 392, Springer, Berlin, 1989, 183–194
7.
Yehuda Afek, Shay Kutten, Moti Yung, Memory‐efficient self‐stabilizing protocols for general networks (extended abstract), Lecture Notes in Comput. Sci., Vol. 486, Springer, Berlin, 1991, 15–28
8.
D. Angluin, Local and global properties in networks of processors, in Proceedings of the 12th ACM Symposium on Theory of Computing, 1980.
9.
B. Awerbuch and R. Ostrovsky, Memory efficient and self stabilizing network reset, in Proceedings of the 14th ACM Symposium on Principles of Distributed Computing, 1994.
10.
B. Awerbuch, B. Patt‐Shamir, and G. Varghese, Self‐stabilization by local checking and correction, in Proceedings of the 33nd IEEE Symposium on Foundations of Computer Science, 1991, pp. 268–277.
11.
Geoffrey Brown, Mohamed Gouda, Chuan‐Lin Wu, Token systems that self‐stabilize, IEEE Trans. Comput., 38 (1989), 845–852
12.
J. E. Burns and J. Pachl, Uniform self‐stabilizing rings, ACM TOPLAS, 11 (1989), pp. 330–344.
13.
W. Bux, F. H. Closs, K. Kümmerle, H. J. Keller, and H. R. Müller, Architecture and design of a reliable token‐ring network, IEEE J. Selected Areas Comm., 1 (1983), pp. 756–765.
14.
I. Cidon and Y. Ofek, MetaRing ‐ A full‐duplex ring with fairness and spatial reuse, IEEE Trans. Comm., 41 (1993), pp. 110–120.
15.
Edsger Dijkstra, Selected writings on computing: a personal perspective, Texts and Monographs in Computer Science, Springer‐Verlag, 1982xvii+362, Including a paper co‐authored by C. S. Scholten
16.
E. W. Dijkstra, Self‐stabilizing systems in spite of distributed control, Comm. ACM, 17 (1974), pp. 643–644.
17.
S. Dolev, A. Israeli, and S. Moran, Uniform dynamic self‐stabilizing leader election, in Proceedings of the 5th International Workshop on Distributed Algorithm, Vol. 579, Springer‐Verlag, 1991, pp. 167–180.
18.
Greg Frederickson, Nancy Lynch, Electing a leader in a synchronous ring, J. Assoc. Comput. Mach., 34 (1987), 98–115
19.
Ted Herman, Probabilistic self‐stabilization, Inform. Process. Lett., 35 (1990), 63–67
20.
A. Hopper and R. M. Needham, The Cambridge fast ring networking system, IEEE Trans. Comput., 37 (1988), pp. 1214–1223.
21.
A. Israeli and M. Jalfon, Token management schemes and random walks yield self stabilizing mutual exclusion, in Proceedings of the 9th ACM Symposium on Principles of Distributed Computing, 1990.
22.
Amos Israeli, Marc Jalfon, Self‐stabilizing ring orientation, Lecture Notes in Comput. Sci., Vol. 486, Springer, Berlin, 1991, 1–14
23.
Alon Itai, Michael Rodeh, Symmetry breaking in distributed networks, Inform. and Comput., 88 (1990), 60–87
24.
Alon Itai, On the computational power needed to elect a leader (extended abstract), Lecture Notes in Comput. Sci., Vol. 486, Springer, Berlin, 1991, 29–40
25.
, 35th Annual Symposium on Foundations of Computer Science, Proceedings of the IEEE Symposium held in Santa Fe, NM, November 20–22, 1994, IEEE Computer Society Press, 1994, 0–0, xiii+837
26.
Gene Itkis, Chengdian Lin, Janos Simon, Deterministic, constant space, self‐stabilizing leader election on uniform rings, Lecture Notes in Comput. Sci., Vol. 972, Springer, Berlin, 1995, 288–302
27.
S. Katz and K. J. Perry, Self‐stabilizing extensions for message‐passing systems, Distrib. Comput., 7 (1993), pp. 17–26.
28.
Leslie Lamport, Nancy Lynch, Distributed computing: models and methods, Elsevier, Amsterdam, 1990, 1157–1199
29.
A. A. Lazar, A. T. Temple, and R. Gidron, MAGNET II: A metropolitan area network based on asynchronous time sharing, IEEE J. Selected Areas Comm., 8 (1990), pp. 1582–1594.
30.
J. Misra, Detecting termination of distributed computations using markers, in Proceedings of the 2nd ACM Symposium on Principles of Distributed Computing, 1983.
31.
Alain Mayer, Rafail Ostrovsky, Moti Yung, Self‐stabilizing algorithms for synchronous unidirectional rings, ACM, New York, 1996, 564–573
32.
Y. Ofek and M. Yung, Principles for high‐speed network control, in Proceedings of the 9th ACM Symposium on Principles of Distributed Computing, 1990.
33.
H. Ohnishi, N. Morita, and S. Suzuki, ATM ring protocol and performance, in Proceedings of the IEEE Conference on Communications, 1989.
34.
J. Pachl, E. Korach, and D. Rotem, A technique for proving lower bounds for distributed maximum‐finding algorithms, in Proceedings of the 14th ACM Symposium on Theory of Computing, 1982.
35.
G. Parlati and M. Yung, Non‐exploratory self‐stabilization for constant‐space symmetry breaking, in Algorithms ‐ ESA’94, Lecture Notes in Comput. Sci. 855, J. van Leeuwen, ed., Springer‐Verlag, Berlin, 1994, pp. 183–201.
36.
M. O. Rabin and D. Lehmann, On the advantage of free choice: A symmetric solution to the dining philosophers problem, in Proceedings of the ACM Symposium on Principles of Programming Languages, 1981.
37.
F. E. Ross, FDDI ‐ A tutorial, IEEE Communication Magazine, 24 (1986), pp. 10–17.
38.
M. Schneider, Self‐stabilization, ACM Comput. Surveys, 25 (1993), pp. 45–67.
39.
B. Schieber and M. Snir, Calling names on nameless networks, in Procedings of the 8th ACM Symposium on Principles of Distributed Computing, 1989.
Information & Authors
Information
Published In
SIAM Journal on Computing
Pages: 1571 - 1595
ISSN (online): 1095-7111
Copyright
Copyright © 2002 Society for Industrial and Applied Mathematics.
History
Published online: 17 February 2012
MSC codes
Keywords
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Fast and lean self-stabilizing asynchronous protocolsProceedings 35th Annual Symposium on Foundations of Computer Science | 1 Jan 1994
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].