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Coverage and Connectivity

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Wireless Sensor Networks

Abstract

A major advantage of wireless sensor networks (WSNs) over wired networks is the potential for ad hoc deployment of the network. If the monitoring of a dangerous environment is required, then one may not be able to deploy a wired network.

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Author information

Authors and Affiliations

  1. Louisiana Tech University, Ruston, LA, USA

    Rastko R. Selmic

  2. Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY, USA

    Vir V. Phoha

  3. Department of Computer Science, Texas Tech University, Lubbock, TX, USA

    Abdul Serwadda

Authors
  1. Rastko R. Selmic
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  2. Vir V. Phoha
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  3. Abdul Serwadda
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Corresponding author

Correspondence to Rastko R. Selmic .

Questions and Exercises

Questions and Exercises

  1. 1.

    What is the degree of a vertex in a graph G? Provide an example.

  2. 2.

    Explain when is a graph connected? Draw examples of connected and disconnected graphs.

  3. 3.

    Demonstrate by example that if a graph G is disconnected, then its complement \(\bar{G}\) is connected. Show another example that the opposite is not true—if the graph G is connected, then its complement \(\bar{G}\) is not necessary disconnected. Finally, prove this statement rigorously.

  4. 4.

    Study the Chvátal’s art gallery theorem and find out what is an upper bound on the minimal number of guards that are required to cover the gallery? Please describe “upper bound on the minimal number” in case of this problem.

  5. 5.

    If the gallery floor plan can be divided into square rooms, what would be the upper bound on the minimal number of guards required to cover the gallery in this case?

  6. 6.

    When one can say that complete sensing coverage of a convex region implies connectivity of the network?

  7. 7.

    Given a graph \(V = \{ 1,2,3,4,5,6,7\}\), \(E = \{ (1,2),(1,3),(1,5),(2,3),(3,5),(4,7),(6,7)\}\), using a graph Laplacian check if the graph is connected. Find the number of spanning trees using Laplacian eigenvalues.

  8. 8.

    Given a graph \(V = \{ 1,2,3,4,5,6,7,8,9\}\), \(E = \{ (1,2),(1,3),(1,5),(2,3),(3,5),(3,6),(4,5),(5,8),(5,9),(6,7),(6,8)\}\), estimate the lower and upper bounds on the graph diameter.

  9. 9.

    For a graph with vertices \(V = \{ 1,2,3,4,5\}\), draw a complex and a simplicial complex.

  10. 10.

    For a set of points in two-dimensional space with the following coordinates \(X = \{ ( - 1, - 2),(1,0),(2,3),(3, - 4),( - 3, - 5)\}\), draw a Voronoi diagram and derive alpha shape complex.

  11. 11.

    Describe a k-coverage in sensor networks and its relationship with the Rips complex. What does it mean by 2-coverage specifically and in terms of the Rips complex?

  12. 12.

    Use linear quadratic control problem setup similar to the one given in Eq. (5.31) and formulate a sensor network coverage control problem where it is required for mobile nodes to be attracted to the target node, to be repelled from the static sensor nodes, and to be repelled from each other (mobile nodes).

  13. 13.

    How can you extend the previous problem formulation and include constraints that would ensure collision avoidance between mobile sensor nodes?

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Selmic, R.R., Phoha, V.V., Serwadda, A. (2016). Coverage and Connectivity. In: Wireless Sensor Networks. Springer, Cham. https://doi.org/10.1007/978-3-319-46769-6_5

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  • DOI: https://doi.org/10.1007/978-3-319-46769-6_5

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  • Online ISBN: 978-3-319-46769-6

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