We propose an algorithm that builds and maintains clusters over a network subject to mobility. This algorithm is fully decentralized and makes all the different clusters grow concurrently. The algorithm uses circulating tokens that collect data and move according to a random walk traversal scheme. Their task consists in (i) creating a cluster with the nodes it discovers and (ii) managing the cluster expansion; all decisions affecting the cluster are taken only by a node that owns the token. The size of each cluster is maintained higher than $m$ nodes ($m$ is a parameter of the algorithm). The obtained clustering is locally optimal in the sense that, with only a local view of each clusters, it computes the largest possible number of clusters (\emph{ie} the sizes of the clusters are as close to $m$ as possible). This algorithm is designed as a decentralized control algorithm for large scale networks and is mobility-adaptive: after a series of topological changes, the algorithm converges to a clustering. This recomputation only affects nodes in clusters in which topological changes happened, and in adjacent clusters.
Scalability in distributed system has become a major challenge nowadays, in structuring and managing communications. We propose a solution to manage large-scale networks based on the division of the system into subsystems, called clusters. We focus in this paper on algorithms that build clusters and maintain them after topological reconfiguration. The algorithms we propose are decentralized: all nodes execute the same code. This allows all clusters to be built concurrently, which is desirable for efficiency.
Large-scale networks are often subject to mobility: their components can connect or disconnect. This phenomenon has to be taken into account. The algorithm being decentralized also allows the algorithm to have no distinguished node, the failure of which would lead to a major re-clustering. The connection or disconnection of a node has only a limited impact (that we can state) on the algorithm.
Random walks are naturally adaptive to dynamic networks such as ad-hoc sensors network [BBCD02,DSW06] because they make use only of local up-to-date information. Moreover they can easily manage connections and disconnections occurring in the network.
Our solution takes these different constraints into account. It is based on the circulation of several tokens. Each token creates a cluster and coordinates its growing in a decentralized way.
A random walk based algorithm is a token circulation algorithm in which a token randomly moves among the nodes in the network. A random walk can be used as base of a distributed token circulation algorithm. This token collects and disseminates information in the network. At each step of the execution of the algorithm, the random walk (the token) is on a node i of the network. The node that owns the token chooses one of its neigbour j with a probability 1/degree(i). It is important to remark that this definition ensures that all nodes, with high probability, eventually own the token, and that the token, with high probability, eventually hits all nodes [Lov93].
In [BBF04,BBFR06], we introduced and used the combination of a circulating word, i.e. the token has a content to collect and broadcast data (this concept is formally defined in Section 2.3) and a random walk as moving scheme of the token. Using this combination, we proposed solutions to build adaptive spanning trees for systems like ad-hoc sensor networks. These solutions are tolerant to transient failures in the network.
In these works, we also proposed a fully decentralized solution to the communication deadlock problem, introducing a new control mechanism called reloading wave. These works have been used to propose a solution to the resource allocation problem in ad-hoc networks [BBFN10]. Such a combination has also been used in [BBFR06] to build and maintain spanning structures to solve on-the-fly resources research in peer-to-peer grids.
Although the token perpetually circulates in the network in order to update the underlying structure, we bound the size of the circulating word to 2n -1 in the case of bidirectional communication links and to n 2 /4 in the case of unidirectional communication links, by retaining only the most recent data necessary to build the tree (with n the size of the network, [Ber06]).
We use the content of the circulating word to structure the network into different clusters. Their construction and their maintenance are achieved in a decentralized way. Using the properties of random walks and of circulating words, the clusters are able to adapt to topological reconfigurations. Thus, this solution can be used to design distributed control algorithm on large scale dynamic networks.
Unlike solutions described in [Bas99,JN06], our solution does not use any local leader on a cluster. The advantage of such solutions is that if a node “moves” in the network, this never entails a total reconstruction of the clusters. After a topological change, the system eventually converges to a correct global state without having to rebuild all clusters. This kind of approach on a 1-hop solution is described in [TIMF05] in which re-clustering mechanism are used. Our solution is totally decentralized as opposed to [BBCD02], in which a spanning structure of the whole network is built in a first step, to be divided using a global mechanism. Our solution is realized in a fully concurrent way. As stated in [ABCP96], it considerably accelerates the construction of the different clusters. Thus our solution satisfies the property highlighted in [TV08].
Moreover, we guarantee that after a topological change, only a bounded portion of the system is affected. Nodes that are in clusters that are not adjacent to the one in which it occurs have no extra work, and are not even aware of this event.
In the first section, we present some preliminary notions about random walk based distributed algorithms, and we present with more details the clustering problem we solve. The second section gives the fundamental distributed clus
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