Abstract
We consider a network of processors in the absence of unique identities, and study the k-Grouping problem of partitioning the processors into groups of size k and assigning a distinct identity to each group. The case k=1 corresponds to the well known problems of leader election and enumeration for which the conditions for solvability are already known. The grouping problem for k≥2 requires to break the symmetry between the processors partially, as opposed to problems like leader election or enumeration where the symmetry must be broken completely (i.e. a node has to be distinguishable from all other nodes). We determine what properties are necessary for solving these problems, characterize the classes of networks where it is possible to solve these problems, and provide a solution protocol for solving them.
For the case k=2 we also consider a stronger version of the problem, called Pairing where each processor must also determine which other processor is in its group. Our results show that the solvable class of networks in this case varies greatly, depending on the type of prior knowledge about the network that is available to the processors. In each case, we characterize the classes of networks where Pairing is solvable and determine the necessary and sufficient conditions for solving the problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Angluin, D.: Local and global properties in networks of processors. In: Proc. 12th ACM Symp. on Theory of Computing (STOC 1980), pp. 82–93 (1980)
Boldi, P., Codenotti, B., Gemmell, P., Shammah, S., Simon, J., Vigna, S.: Symmetry breaking in anonymous networks: Characterizations. In: Proc. of 4th Israeli Symposium on Theory of Computing and Systems (ISTCS 1996), pp. 16–26 (1996)
Boldi, P., Vigna, S.: An effective characterization of computability in anonymous networks. In: Welch, J.L. (ed.) DISC 2001. LNCS, vol. 2180, pp. 33–47. Springer, Heidelberg (2001)
Boldi, P., Vigna, S.: Fibrations of graphs. Discrete Math. 243, 21–66 (2002)
Cardon, A., Crochemore, M.: Partitioning a Graph in O(|A| log2 |V|). Theoretical Computer Science 19, 85–98 (1982)
Chalopin, J., Métivier, Y.: A bridge between the asynchronous message passing model and local computations in graphs. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 212–223. Springer, Heidelberg (2005)
Corneil, D.G., Gotlieb, C.C.: An Efficient Algorithm for Graph Isomorphism. Journal of the ACM 17(1), 51–74 (1970)
Flocchini, P., Roncato, A., Santoro, N.: Computing on anonymous networks with Sense of Direction. Theoretical Computer Science 301, 355–379 (2003)
Godard, E., Métivier, Y., Muscholl, A.: Characterization of classes of graphs recognizable by local computations. Theory of Computing Systems 37(2), 249–293 (2004)
Johnson, R.E., Schneider, F.B.: Symmetry and similarity in distributed systems. In: Proc. 4th Annual ACM Symp. on Principles of Distributed Computing (PODC 1985), pp. 13–22 (1985)
Kranakis, E.: Symmetry and computability in anonymous networks: A brief survey. In: Proc. 3rd Int. Conf. on Structural Information and Communication Complexity (SIROCCO 1997), pp. 1–16 (1997)
Leighton, F.T.: Finite common coverings of graphs. J. Combin. Theory, Ser. B 33, 231–238 (1982)
Mazurkiewicz, A.: Distributed enumeration. Inf. Processing Letters 61(5), 233–239 (1997)
Mullender, S.J., Vitanyi, P.M.B.: Distributed match-making. Algorithmica 3, 367–391 (1998)
Sakamoto, N.: Comparison of initial conditions for distributed algorithms on anonymous networks. In: Proc. 18th ACM Symposium on Principles of Distributed Computing (PODC 1999), pp. 173–179 (1999)
Yamashita, M., Kameda, T.: Computing on anonymous networks: Parts I and II. IEEE Trans. Parallel and Distributed Systems 7(1), 69–96 (1996)
Yamashita, M., Kameda, T.: Computing functions on asynchronous anonymous networks. Mathematical Systems Theory 29, 331–356 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chalopin, J., Das, S., Santoro, N. (2006). Groupings and Pairings in Anonymous Networks. In: Dolev, S. (eds) Distributed Computing. DISC 2006. Lecture Notes in Computer Science, vol 4167. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11864219_8
Download citation
DOI: https://doi.org/10.1007/11864219_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44624-8
Online ISBN: 978-3-540-44627-9
eBook Packages: Computer ScienceComputer Science (R0)