[Lecture Notes in Computer Science] Stabilization, Safety, and Security of Distributed Systems...

Brief Announcement: Robust Self-stabilizing

Construction of Bounded Size Weight-BasedClusters

Colette Johnen1 and Fouzi Mekhaldi2

1 LaBRI, Univ. Bordeaux, CNRS, F-33405 Talence Cedex, France2 LRI, Univ. Paris-Sud XI, CNRS, F-91405 Orsay Cedex, France

Introduction. The clustering problem consists of partitioning network nodesinto groups called clusters. Each cluster has a single clusterhead that acts aslocal coordinator of cluster.

A technique for designing solutions that tolerate transient faults is self-stabilization. Self-stabilizing protocols are attractive because they need not beinitialized: they converge from any configuration to a legitimate one. Also, theyare adaptive to topological changes. If the current configuration is inconsistentwith the network topology, the self-stabilizing protocol eventually converges to alegitimate configuration. Nevertheless, self-stabilizing protocols do not guaranteeany property during the convergence period. In addition, the convergence timemay be proportional to the size of the network; particularly, in weight-basedclustering protocols. In order to overcome these drawbacks, we are interestedto the robust stabilization. Robust stabilization guarantees that from an illegiti-mate configuration, the system reaches quickly a safe configuration, in which thesafety property is satisfied. The safety property has to be defined such that thesystem performs correctly its task in a safe configuration. During the convergenceto a legitimate configuration, the safety property stays always verified.

Related works. In [1], a robust self-stabilizing protocol building a minimumconnected dominating set is proposed. In a safe configuration, the built set is adominating set. A robust self-stabilizing weight-based clustering protocol for adhoc networks is proposed in [2]. In [2], a configuration is safe if and only if thenetwork is partitioned into clusters.

To our knowledge, the only protocols building bounded size clusters are [3,4,5].In [5], the size of obtained clusters is bounded by a lower and an upper bound.This solution cannot be applied to one-hop clusters, because the degree of nodesmay be less than the lower bound. However, [3,5] are not self-stabilizing, and [4]is self-stabilizing but it is not robust.

Contributions. We propose the first robust self-stabilizing protocol that con-structs 1-hop clusters whose size is bounded, and the clusterhead selection isweight-based. The detailed version can be found on [6].

Our protocol is weight based: the clusterhead selection criteria is based onthe weight of nodes. Each node has an input variable, its weight, named w,representing its capacity to be clusterhead. The higher the weight of a node,the more suitable this node is for the role of clusterhead. The weight value canincrease or decrease reflecting the changes in the node’s status.

R. Guerraoui and F. Petit (Eds.): SSS 2009, LNCS 5873, pp. 787–788, 2009.c© Springer-Verlag Berlin Heidelberg 2009

788 C. Johnen and F. Mekhaldi

The proposed clustering protocol provides bounded size clusters; at mostSizeBound ordinary nodes can be in a cluster. This limitation on the num-ber of nodes that a clusterhead handles, ensures the load balancing over thenetwork: no clusterhead is overloaded at any time.

As clusters are bounded, several clusterheads may be neighbors. To limitthe number of clusterheads locally, a node v may become clusterhead only if itdoes not have a suitable clusterhead in the neighborhood. Furthermore, v staysclusterhead only if it cannot join a neighbor cluster : neighbor clusters are full(they contain SizeBound members), or v is the leader having the highest weight.

The obtained clusters have to satisfy the well-balanced clustering properties:

• Affiliation condition: each ordinary node affiliates with a neighboring cluster-head, such that the weight of its clusterhead is greater than its weight.• Size condition: each cluster contains at most SizeBound ordinary nodes.• Clusterhead neighboring condition: if a clusterhead v has a neighboring clus-terhead u such that wu > wv, then the size of u’s cluster is SizeBound.

Convergence and time complexity. Our protocol is silent; no node executesan action once a legitimate configuration is reached. Starting from an arbitraryconfiguration, the protocol reaches a safe configuration in 4 rounds.

Once a safe configuration is reached, each node belongs to a cluster havingan effectual leader, and each cluster contains at most SizeBound members, butclusters may not satisfy the well-balanced clustering properties. During the con-struction of the final clusters, that satisfy the well-balanced clustering properties,the safety property is preserved under any computation, and also under the fol-lowing input changes: (1) change on node’s weight, (2) crash of an ordinary node,(3) failure of a link between: (a) a clusterhead and a nearly ordinary node, (b)two clusterheads, (c) two nearly ordinary nodes, or (d) two ordinary nodes, (4)joining of a sub-network that verifies the safety property.

The time of convergence to a legitimate configuration is at most 7∗N2 + 5,

where N is the number of nodes in the network.

References

1. Kamei, S., Kakugawa, H.: A self-stabilizing approximation for the minimum con-nected dominating set with safe convergence. In: Baker, T.P., Bui, A., Tixeuil, S.(eds.) OPODIS 2008. LNCS, vol. 5401, pp. 496–511. Springer, Heidelberg (2008)

2. Johnen, C., Nguyen, L.H.: Robust self-stabilizing weight-based clustering algorithm.Theoretical Computer Science 410(6-7), 581–594 (2009)

3. Chatterjee, M., Das, S.K., Turgut, D.: WCA: A weighted clustering algorithm formobile ad hoc networks. Journal of Cluster Computing 5(2), 193–204 (2002)

4. Johnen, C., Nguyen, L.H.: Self-stabilizing construction of bounded size clusters. In:ISPA 2008, pp. 43–50 (2008)

5. Tomoyuki, O., Shinji, I., Yoshiaki, K., Kenji, I., Kaori, M.: An adaptive maintenanceof hierarchical structure in ad hoc networks and its evaluation. In: ICDCS 2002, pp.7–13 (2002)

6. Johnen, C., Mekhaldi, F.: Robust self-stabilizing construction of bounded sizeweight-based clusters. Technical Report No 1518, LRI (2009),http://www.lri.fr/~bibli/Rapports-internes/2009/RR1518.pdf