Probabilistic leader election on rings of known size
We are interested in the average case complexity of leader election (and related problems) on asynchronous processor rings, whose size, n, is known to all constituent processors.
Duris and Galil  prove an Ω(n log n) lower bound for the average (over all assignments of identifiers) of the number of messages required by any deterministic leader election algorithm, when n is a power of 2 and the processor identifier space is sufficiently (exponentially) large. More recently Bodlaender
Both Duris and Galil's and Bodlaender's lower bounds extend naturally (but significantly) to the expected message complexity of Las Vegas algorithms (randomized algorithms that terminate, always correctly, with probability 1) [7, 11, 5]. These bounds, in turn, can be shown to apply to the evaluation of several natural functions on rings, including AND, OR and XOR .
We show that the lower bound technique introduced by Duris and Galil can be modified to provide a direct proof of an Ω(n log n) lower bound for the expected message complexity of Las Vegas leader election on anonymous (identifier free) rings, that is substantially simpler than the original. This simplicity not only serves to highlight the important structure of technique but also facilitates its extension to both arbitrary ring size and to Monte Carlo algorithms (randomized algorithms that err with probability at most ε). Specifically, we prove that the expected message complexity of any probabilistic algorithm that selects a leader with probability at least 1 − ε on an anonymous ring of known size n, is Ω (n min (log n, log log (1/ε))). A number of common function evaluation problems (including AND, OR, PARITY, and SUM) on rings of known size, are shown to inherit this complexity bound; furthermore these bounds are tight to within a constant factor.
KeywordsFixed Size Monte Carlo Algorithm Ring Size Leader Election Schedule Function
Unable to display preview. Download preview PDF.
- Karl Abrahamson, Andrew Adler, Rachel Gelbart, Lisa Higham, and David Kirkpatrick. The bit complexity of randomized leader election on a ring. SIAM Journal on Computing, 18(1):12–29, 1989.Google Scholar
- Karl Abrahamson, Andrew Adler, Lisa Higham, and David Kirkpatrick. Randomized function evaluation on a ring. Distributed Computing, 3(3):107–117, 1989.Google Scholar
- Karl Abrahamson, Andrew Adler, Lisa Higham, and David Kirkpatrick. Optimal algorithms for probabilistic solitude detection on anonymous rings. Technical Report TR 90–3, University of British Columbia, 1990.Google Scholar
- Karl Abrahamson, Andrew Adler, Lisa Higham, and David Kirkpatrick. Tight lower bounds for probabilistic solitude verification on anonymous rings. Technical Report TR 90–4, University of British Columbia, 1990.Google Scholar
- Hagit Attiya and Mark Snir. Better computing on the anonymous ring. In Proc. Aegean Workshop on Computing, pages 329–338, 1988.Google Scholar
- Hans L. Bodlaender. Distributed Algorithms, Structure and Complexity. PhD thesis, University of Utrecht, 1986.Google Scholar
- Hans L. Bodlaender. New lower bound techniques for distributed leader finding and other problems on rings of processors. Technical Report RUU-CS-88-18, Rijksuniversiteit Utrecht, 1988.Google Scholar
- J. Burns. A formal model for message passing systems. Technical Report TR-91, Indiana University, 1980.Google Scholar
- Danny Dolev, Maria Klawe, and Michael Rodeh. An O(n log n) unidirectional distributed algorithm for extrema finding in a circle. J. Algorithms, 3(3):245–260, 1982.Google Scholar
- Pavol Duris and Zvi Galil. Two lower bounds in asynchronous distributed computation (preliminary version). In Proc. 28nd Annual Symp. on Foundations of Comput. Sci., pages 326–330, 1987.Google Scholar
- Lisa Higham. Randomized Distributed Computing on Rings. PhD thesis, University of British Columbia, Vancouver, Canada, 1988.Google Scholar
- Alon Itai and Michael Rodeh. Symmetry breaking in distributed networks. In Proc. 22nd Annual Symp. on Foundations of Comput. Sci., pages 150–158, 1981.Google Scholar
- Jan Pachl. A lower bound for prbabilistic distributed algorithms. Technical Report CS-85-25, University of Waterloo, Waterloo, Ontario, 1985.Google Scholar
- Jan Pachl, E. Korach, and D. Rotem. Lower bounds for distributed maximum finding. J. Assoc. Comput. Mach., 31(4):905–918, 1984.Google Scholar
- Gary Peterson. An O(n log n) algorithm for the circular extrema problem. ACM Trans. on Prog. Lang. and Systems, 4(4):758–752, 1982.Google Scholar