Distributed computation of homology in wireless sensor networks

Domen Šoberl (2013) Distributed computation of homology in wireless sensor networks. MSc thesis.

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Abstract

Wireless sensor networks are networks of small and simple autonomous devices or nodes which perform measurements in the environment where they are located. The data obtained is processed and transmitted over a wireless communication channel to other neighboring nodes. Individual measurements are performed in a limited geometric range around sensors therefore random deployment of nodes may cause some areas to remain uncovered. Since wireless sensor network is capable of distributed processing it is able to autonomously check the coverage of its domain even if no information about geometric position and orientation of its nodes is available. Methods used to check the coverage using only connectivity information of the network have been introduced only a few years ago. The general idea is based on principles of computational topology. Using the connectivity graph of the network a simplicial complex called Rips complex is built which has been shown to contain information about the holes in the measurement domain. To capture this information, homology is used which produces a sequence of homology groups that we can use to derive a computable criterion for coverage verification. First algorithms to compute homology in wireless sensor networks have been centralized. In centralized approach all the connectivity data has to be gathered in one place where the processing is then carried out. The unit on which such algorithm is implemented requires high processing capabilities as computation of homology groups typically applies operations on large matrices. Networks without a central processing unit usually cannot run such algorithms since one node alone lacks the required processing performance. Divide-and-conquer approaches turned out to be the most efficient way to compute homology distributively. The network is divided into smaller parts where each node takes over the computation of its neighborhood. The main drawback of these algorithms are complicated procedures for dividing the network which usually require information about the edge of the domain called the fence. Since topology of the fence is limited to a cycle, such algorithms are not expected to work properly in all network topologies. In this thesis we present an algorithm for distributed computation of homology in wireless sensor networks which does not require a fence and works in any network topology. We take the approach of constructing larger network segments from smaller ones, capturing homology of the union during the process. We use a spanning tree of the network as a predefined path for merging, following the direction from leaves to the root. Initial network segments are small enough not to contain any holes. The criterion for detecting holes in the union of two segments was derived from the Mayer-Vietoris sequence and extended to fit the merging of larger sets of segments. Correctness of the algorithm has been tested and validated by extensive simulations. We had also measured its computational and communicational complexity. According to our results we conclude that the algorithm is suitable for use in real wireless sensor networks.

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