# An optimal bit complexity randomized distributed MIS algorithm

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## Abstract

We present a randomized distributed maximal independent set (MIS) algorithm for arbitrary graphs of size *n* that halts in time *O*(log *n*) with probability 1 − *o*(*n* ^{−1}), and only needs messages containing 1 bit. Thus, its bit complexity par channel is *O*(log *n*). We assume that the graph is anonymous: unique identities are not available to distinguish between the processes; we only assume that each vertex distinguishes between its neighbours by locally known channel names. Furthermore we do not assume that the size (or an upper bound on the size) of the graph is known. This algorithm is optimal (modulo a multiplicative constant) for the bit complexity and improves the best previous randomized distributed MIS algorithms (deduced from the randomized PRAM algorithm due to Luby (SIAM J. Comput. 15:1036–1053, 1986)) for general graphs which is *O*(log^{2} * n*) per channel (it halts in time *O*(log *n*) and the size of each message is log *n*). This result is based on a powerful and general technique for converting unrealistic exchanges of messages containing real numbers drawn at random on each vertex of a network into exchanges of bits. Then we consider a natural question: what is the impact of a vertex inclusion in the MIS on distant vertices? We prove that this impact vanishes rapidly as the distance grows for bounded-degree vertices and we provide a counter-example that shows this result does not hold in general. We prove also that these results remain valid for Luby’s algorithm presented by Lynch (Distributed algorithms. Morgan Kaufman 1996) and by Wattenhofer (http://dcg.ethz.ch/lectures/fs08/distcomp/lecture/chapter4.pdf, 2007). This question remains open for the variant given by Peleg (Distributed computing—a locality-sensitive approach 2000).

## Keywords

Arbitrary Graph Active Neighbour Asymptotic Independence Distant Vertex Neighbour Degree## Preview

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## References

- 1.Alon N., Babai L., Itai A.: A fast and simple randomized parallel algorithm for the maximal independent set. J. Algorithms
**7**(4), 567–583 (1986)CrossRefzbMATHMathSciNetGoogle Scholar - 2.Awerbuch, B., Goldberg, A.V., Luby, M., Plotkin, S.A.: Network decomposition and locality in distributed computation. In: Proceedings of the 30th ACM Symposium on FOCS, pp. 364–369. ACM Press (1989)Google Scholar
- 3.Bodlaender H.L., Moran S., Warmuth M.K.: The distributed bit complexity of the ring: from the anonymous case to the non-anonymous case. Inf. comput.
**114**(2), 34–50 (1994)CrossRefMathSciNetGoogle Scholar - 4.Chalopin J., Métivier Y.: An efficient message passing election algorithm based on mazurkiewicz’s algorithm. Fundam. Inform.
**80**(1-3), 221–246 (2007)zbMATHGoogle Scholar - 5.Dinitz, Y., Moran, S., Rajsbaum, S.: Bit complexity of breaking and achieving symmetry in chains and rings. J. ACM.
**55**(1) (2008)Google Scholar - 6.Ghosh S.: Distributed Systems—An Algorithmic Approach. CRC Press, Boca Raton (2006)CrossRefGoogle Scholar
- 7.Habib, M., McDiarmid, C., Ramirez-Alfonsin, J., Reed, B. (eds.): Probabilistic Methods for Algorithmic Discrete Mathematics, of Algorithms and Combinatorics, vol. 16. Springer-Verlag, Berlin (1998)Google Scholar
- 8.Karp, R.M., Widgerson, A.: A fast parallel algorithm for the maximal independent set problem. In: Proceedings of the 16th ACM Symposium on Theory of Computing (STOC), pp. 266–272. ACM Press (1984)Google Scholar
- 9.Kothapalli, K., Onus, M., Scheideler, C., Schindelhauer, C.: Distributed coloring in Õ\({\left(\sqrt{\log n}\right)}\) bit rounds. In: 20th International Parallel and Distributed Processing Symposium (IPDPS 2006), Proceedings April 2006, Rhodes Island, Greece. IEEE pp. 25–29 (2006)Google Scholar
- 10.Kuhn, F., Moscibroda, T., Nieberg, T., Wattenhofer, R.: Fast deterministic distributed maximal independent set computation on growth-bounded graphs. In: DISC, pp. 273–287 (2005)Google Scholar
- 11.Kuhn, F., Moscibroda, T., Wattenhofer, R.: What cannot be computed locally! In: Proceedings of the 24 Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 300–309 (2004)Google Scholar
- 12.Kuhn, F., Wattenhofer, R.: On the complexity of distributed graph coloring. In: Proceedings of the 25 Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 7–15. ACM Press (2006)Google Scholar
- 13.Kushilevitz E., Nisan N.: Communication Complexity. Cambridge University Press, Cambridge, MA (1999)Google Scholar
- 14.Linial N.: Locality in distributed graph algorithms. SIAM J. Comput.
**21**, 193–201 (1992)CrossRefzbMATHMathSciNetGoogle Scholar - 15.Luby M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput.
**15**, 1036–1053 (1986)CrossRefzbMATHMathSciNetGoogle Scholar - 16.Lynch, N.A.: Distributed Algorithms. Morgan Kaufman (1996)Google Scholar
- 17.Moscibroda T., Wattenhofer R.: Coloring unstructured radio networks. Distrib. Comput.
**21**(4), 271–284 (2008)CrossRefGoogle Scholar - 18.Naor M., Stockmeyer L.J.: What can be computed locally?. SIAM J. Comput.
**24**(6), 1259–1277 (1995)CrossRefzbMATHMathSciNetGoogle Scholar - 19.Peleg, D.: Distributed computing—A Locality-sensitive approach. SIAM Monographs on Discrete Mathematics and Applications (2000)Google Scholar
- 20.Santoro N.: Design and Analysis of Distributed Algorithms. Wiley, London (2007)zbMATHGoogle Scholar
- 21.Tel G.: Introduction to Distributed Algorithms. Cambridge University Press, Cambridge, MA (2000)zbMATHGoogle Scholar
- 22.Wattenhofer, R.: http://dcg.ethz.ch/lectures/fs08/distcomp/lecture/chapter4.pdf. (2007)
- 23.Yao, A.C.: Some complexity questions related to distributed computing. In: Proceedings of the 11th ACM Symposium on Theory of Computing (STOC), pp. 209–213. ACM Press (1979)Google Scholar