In this paper, we study the optimal deployment pattern problem in wireless sensor networks (WSNs). We propose a new set of patterns, particularly when sensors' communication range ($r_{\rm c}$) is relatively small compared to their sensing range ($r_{\rm s}$), and prove their optimality. In this study, we discover an interesting phenomenon—pattern mutation. To the best of our knowledge, this is the first time that mutation in pattern deployments has been discovered. This phenomenon, which contradicts the conjecture presented in a previous work that there exists a universal elemental pattern among optimal pattern deployment, significantly furthers our understanding of optimal patterns in WSNs.

In this paper, we study the optimal deployment pattern problem in wireless sensor networks (WSNs). We propose a new set of patterns, particularly when sensors' communication range (rc) is relatively small compared with their sensing range (rs), and prove their optimality among regular patterns. In this study, we discover a surprising and interesting phenomenon-pattern mutation. This phenomenon contradicts the conjecture presented in a previous work that there exists a universal elemental pattern among optimal pattern evolution and that pattern evolution is continuous. For example, we find mutation happens among the patterns for full-coverage and 3-connectivity when rc/rs= 1.0459, among the patterns for full-coverage and 4-connectivity when rc/rs= 1.3903, and among the patterns for full-coverage and 5-connectivity when rc/rs= 1.0406. To the best of our knowledge, this is the first time that mutation in pattern evolution has been discovered. Also, our work further completes the exploration of optimal patterns in WSNs.

In this paper, we study optimal deployment in terms of the number of sensors required to achieve four-connectivity and full coverage under different ratios of sensors' communication range (denoted by rc) to their sensing range (denoted by rs). We propose a new pattern, the Diamond pattern, which can be viewed as a series of evolving patterns. When rc/rs¿ ¿(3), the Diamond pattern coincides with the well-known triangle lattice pattern; when rc/rs¿ ¿(2), it degenerates to a Square pattern (i.e., a square grid). We prove that our proposed pattern is asymptotically optimal when rc/rs> ¿(2) to achieve four-connectivity and full coverage. We also discover another new deployment pattern called the Double-strip pattern. This pattern provides a new aspect to research on optimal deployment patterns. Our work is the first to propose an asymptotically optimal deployment pattern to achieve four-connectivity and full coverage for WSNs. Our work also provides insights on how optimal patterns evolve and how to search for them.

In this paper, we study deployment patterns to achieve full coverage andk-connectivity(k≤ 6) under different ratios of the sensor communication range (denoted byRc) to the sensing range (denoted byRs) for homogeneous wireless sensor networks (WSNs). In particular, we propose new patterns for 3- and 5-connectivity. We also discover that there exists a hexagon-based universally elemental pattern that can generate all known optimal patterns. The previously proposed Voronoi-based approach cannot be applied to prove the optimality of the new patterns due to their special features. We propose a new deployment-polygon-based methodology. We prove the optimality of deployment patterns to achieve 3-connectivity, 4-connectivity, and 5-connectivity for certain ranges ofRc/Rs, respectively, and prove the optimality of deployment patterns to achieve 6-connectivity under all ranges ofRc/Rs.

We study the issue of optimal deployment to achieve four connectivity and full coverage for wireless sensor networks (WSNs) under different ratios of sensors' communication range (denoted by rc) to their sensing range (denoted by rs). We propose a "Diamond" pattern, which can be viewed as a series of different evolving patterns. When rc/rsges radic3, the Diamond pattern coincides with the well-known triangle lattice pattern; when rc/rsges radic2, it degenerates to a "Square" pattern. We prove the Diamond pattern to be asymptotically optimal when rc/rsges radic2- Our work is the first to propose an asymptotically optimal deployment pattern to achieve four connectivity and full coverage for WSNs. We hope our work will provide some insights on how optimal patterns evolve and how to search for them.

In this paper, we propose deployment patterns to achieve full coverage and three-connectivity, and full coverage and five-connectivity under different ratios of sensor communication range (denoted by Rc) over sensing range (denoted by Rs) for wireless sensor networks (WSNs). We also discover that there exists a hexagon-based universally elemental pattern which can generate all known optimal patterns. The previously proposed Voronoi-based approach can not be applied to prove the optimality of the new patterns due to their special features. We propose a new deployment-polygon based methodology, and prove their optimality among regular patterns when Rc/Rs ≥ 1. We conjecture that our patterns are globally optimal to achieve full coverage and three-connectivity, and full coverage and five-connectivity, under all ranges of Rc/Rs. With these new results, the set of optimal patterns to achieve full coverage and k-connectivity (k≤6) is complete, for the first time.

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 con-sidered. 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.