# Random leaders and random spanning trees

## Abstract

The problem of distributively constructing a minimum spanning tree has been thoroughly studied. The root of this spanning tree is often elected as a leader, and then centralized algorithms are run in the distributed system. If, however, we have fault tolerance in mind, selecting a random spanning tree and a random leader are more desirable. If we manage to select a random tree, the probability that a bad channel will disconnect some nodes from the random tree is relatively small. Otherwise, a small number of predetermined edges will greatly effect the system's behavior.

In this paper we present an algorithm for choosing a *random leader* (RL), and *distributed random spanning tree* algorithms (RST), where *random* means, that each spanning tree in the underlying graph has the same probability of being selected. We give optimal algorithms for the complete graph and the ring. We also describe an RST algorithm for the general graph, and discuss the relation between RST and RL algorithms.

## Preview

Unable to display preview. Download preview PDF.

## References

- [AG]Afek, Y. and Gafni, E.,
*Time and Message Bounds for Election in Synchronous and Asynchronous Complete Networks*, PODC 1985, pp 199–207.Google Scholar - [AHU]Aho, V. A., Hopcroft, J.E. and Ullman, J.D.,
*The Design and Analysis of Computer Algorithms*, Addison-Wesley, 1974.Google Scholar - [A]Awerbuch, B.,
*Linear Time Distributed Algorithms for Minimum Spanning Trees, leader election, counting and related problems*, 19th STOC, 1987, pp. 230–240.Google Scholar - [B]Broder, A. Z.,
*Generating Random Spanning Trees*, To appear in 30th FOCS, 1989.Google Scholar - [Bi]Biggs, N.,
*Algebraic Graph Theory*Google Scholar - [BK]Broder, A. Z. and Karlyn, A. R.,
*Bounds on Cover Time*29th FOCS, 1988.Google Scholar - [Bu]Burns, J. E.,
*A Formal Model for Message Passing Systems*, TR-91, Indiana University, 1980.Google Scholar - [E]Even, S.,
*Graph Algorithms*, Maryland, Computer Science Press, 1979.Google Scholar - [FL]Fredrickson, G. N. and Lynch N. A.,
*The Impact of Synchronous Communication On the Problem of Electing a Leader in a Ring*, STOC84, pp. 493–503.Google Scholar - [Ga]Gafni, E.,
*Improvements in the Time Complexity of Two Message-Optimal Election Algorithms*, PODC85, pp 175–184.Google Scholar - [GHS]Gallager, R. G., Humblet, P. A. and Spira, P. M.,
*A Distributed Algorithm for Minimum Weight Spanning Trees*, ACM Trans. Program. Lang. Syst., vol 5. pp. 66–77, January 1983.Google Scholar - [G]Guenoche, A.,
*Random Spanning Trees*, J. Algorithm, vol 4, pp. 214–220, 1983.Google Scholar - [H]Harary, F.,
*Graph Theory*Addison-Wesley, 1972.Google Scholar - [KS]Kemeny, J.G and Snell, J.L.,
*Finite Markov Chains*Lect. Notes in Math, vol 69.Google Scholar - [MVV]Mulmuley, K., Vazirani, U. V. and Vazirani, V. J.,
*Matching is as Easy as Matrix Inversion*, 19th STOC, 1987, pp. 345–354.Google Scholar - [QN]Quinn, M. J. and Narsingh Deo,
*Parallel Graph Algorithms*, Computing Surveys, vol 16. No. 3, September 1984, pp 319–348.Google Scholar - [V]Vitanyi, P.,
*Distributed Election in an Archimedian Ring of Processors*, STOC84, pp 542–547.Google Scholar