It is well-known that placing disks in the triangular lattice pattern is optimal for achieving full coverage on a plane. With the emergence of wireless sensor networks, however, it is now no longer enough to consider coverage alone when deploying a wireless sensor network; connectivity must also be considered. While moderate loss in coverage can be tolerated by applications of wireless sensor networks, loss in connectivity can be fatal. Moreover, since sensors are subject to unanticipated failures after deployment, it is not enough to have a wireless sensor network just connected, it should be k-connected (for k > 1). In this paper, we propose an optimal deployment pattern to achieve both full coverage and 2-connectivity, and prove its optimality for all values of rc/rs, where rc is the communication radius, and rs is the sensing radius. We also prove the optimality of a previously proposed deployment pattern for achieving both full coverage and 1-connectivity, when rc/rs < √ 3. Finally, we compare the efficiency of some popular regular deployment patterns such as the square grid and triangular lattice, in terms of the number of sensors needed to provide coverage and connectivity.

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Aims and Scope With this book series, we aim to publish monographs of high quality in all areas of mathematics. Both research monographs and books of an expository nature are welcome. This series is the continuation of the "Mathematics Studies", previously published by Elsevier. All books published after November 2010 are promoted, distributed and sold by Springer, both as e-books and in print. The books are also part of SpringerLink and included in the relevant Springer subject collections. All book proposals submitted to this series will be reviewed by the Series Editor. After the manuscript has been completed, it will be entirely reviewed by one of our editors or reviewers. Only after this review will the book be published.

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