Tight Space Bounds for ℓ-Exclusion
The simplest deadlock-free algorithm for mutual exclusion requires only one single-writer non-atomic bit per process [4,6,13]. This algorithm is known to be space optimal [5,6]. For over 20 years now it has remained an intriguing open problem whether a similar type of algorithm, which uses only one single-writer bit per process, exists also for ℓ-exclusion for some ℓ ≥ 2.
We resolve this longstanding open problem. For any ℓ and n, we provide a tight space bound on the number of single-writer bits required to solve ℓ-exclusion for n processes. It follows from our results that it is not possible to solve ℓ-exclusion with one single-writer bit per process, for any ℓ ≥ 2.
In an attempt to understand the inherent difference between the space complexity of mutual exclusion and that of ℓ-exclusion for ℓ ≥ 2, we define a weaker version of ℓ-exclusion in which the liveness property is relaxed, and show that, similarly to mutual exclusion, this weaker version can be solve using one single-writer non-atomic bit per process.
KeywordsMutual Exclusion ℓ-exclusion space complexity tight bounds
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- 2.Afek, Y., Stupp, G., Touitou, D.: Long-lived adaptive collect with applications. In: Proc. 40th IEEE Symp. on Foundations of Computer Science, pp. 262–272 (October 1999)Google Scholar
- 3.Anderson, J.H., Moir, M.: Using local-spin k-exclusion algorithms to improve wait-free object implementations. Distributed Computing 11 (1997)Google Scholar
- 5.Burns, J.E., Lynch, A.N.: Mutual exclusion using indivisible reads and writes. In: 18th Annual Allerton Conference on Communication, Control and Computing, pp. 833–842 (1980)Google Scholar
- 8.Dolev, D., Gafni, E., Shavit, N.: Toward a non-atomic era: ℓ-exclusion as a test case. In: Proc. 20th ACM Symp. on Theory of Computing, pp. 78–92 (1988)Google Scholar
- 9.Fischer, M.J., Lynch, N.A., Burns, J.E., Borodin, A.: Resource allocation with immunity to limited process failure. In: Proc. 20th IEEE Symp. on Foundations of Computer Science, pp. 234–254 (October 1979)Google Scholar
- 15.Peterson, G.L.: New bounds on mutual exclusion problems. Technical Report TR68, University of Rochester (February 1980) (Corrected, November 1994)Google Scholar
- 16.Peterson, G.L.: Observations on ℓ-exclusion. In: 28th Annual Allerton Conference on Communication, Control and Computing, pp. 568–577 (October 1990)Google Scholar
- 17.Taubenfeld, G.: Synchronization Algorithms and Concurrent Programming, 423 pages. Pearson / Prentice-Hall (2006) ISBN 0-131-97259-6Google Scholar