Tight bounds on the round complexity of distributed 1-solvable tasks
A distributed task T is 1-solvable if there exists a protocol that solves it in the presence of (at most) one crash failure. A precise characterization of the 1-solvable tasks was given in [BMZ]. In this paper we determine the number of rounds of communication that are required, in the worst case, by a protocol which 1-solves a given 1-solvable task T for n processors. We define the radius R (T) of T, and show that if R (T) is finite, then this number is Θ(logn R (T)); more precisely, we give a lower bound of log(n−1)R(T), and an upper bound of 2+[log(n−1)R(T)]. The upper bound implies, for example, that each of the following tasks: renaming, order preserving renaming ([ABDKPR]) and binary monotone consensus [BMZ] can be solved in the presence of one fault in 3 rounds of communications. All previous protocols that 1-solved these tasks required Ω(n) rounds. The result is also generalized to tasks whose radii are not bounded, e.g., the approximate consensus and its variants [DLPSW, BMZ].
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- [ABDKPR]C. Attiya, A. Bar-Noy, D. Dolev, D. Koller, D. Peleg, R. Reischuk, Achievable cases in an asynchronous environment, Proc. of the 28th FOCS, October 1987, pp. 337–346.Google Scholar
- [BMZ]O. Biran, S. Moran and S. Zaks, A combinatorial characterization of the distributed tasks which are solvable in the presence of one faulty processor, Proc. of the 7th PODC, 1988, pp. 263–273.Google Scholar
- [DDS]D. Dolev, C. Dwork and L. Stockmeyer, On the minimal synchronism needed for distributed consensus, Journal of the ACM, Vol. 34 no. 1, pp. 77–97.Google Scholar
- [DLPSW]D. Dolev, N. A. Lynch, S. Pinter, E. Stark and W. Weihl, Reaching approximate agreement in the presence of faults, Journal of the ACM, Vol. 33 no. 3 (1986), pp. 499–516.Google Scholar
- [Fe]A. D. Fekete, Asynchronous Approximate Agreement, Proc. of the 6th PODC, 1987, pp. 64–76.Google Scholar
- [FL]G. N. Frederickson and N. A. Lynch, Electing a leader in a synchronous ring Journal of the ACM, Vol. 34 No. 1 (1987), pp. 98–115.Google Scholar
- [FLP]M. J. Fischer, N. A. Lynch and M. S. Paterson, Impossibility of distributed consensus with one faulty process, Journal of the ACM, Vol. 32 No. 2 (1985), pp. 373–382.Google Scholar
- [KMZ]E. Korach, S. Moran and S. Zaks, Tight lower and upper bounds for some distributed algorithms for a complete network of processors, Proc. of the 3rd PODC, pp. 199–207.Google Scholar
- [MW]S. Moran and Y. Wolfstahl, Extended impossibility results for asynchronous complete networks, Information Processing Letters, 26, 1987, pp. 145–151.Google Scholar
- [NT]G. Neiger and S. Toueg, Automatically increasing the Fault-tolerance of distributed systems, Proc. of the 7th PODC, pp. 248–262.Google Scholar
- [TKM]G. Taubenfeld, S. Katz and S. Moran, Initial failures in distributed computations, to appear in Journal of Parallel and Distributed Computing.Google Scholar