Optimal Patterns for Four-Connectivity and Full Coverage in Wireless Sensor Networks

Abstract

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 r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ) to their sensing range (denoted by r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> ). We propose a new pattern, the Diamond pattern, which can be viewed as a series of evolving patterns. When r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> /r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> ¿ ¿(3), the Diamond pattern coincides with the well-known triangle lattice pattern; when r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> /r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> ¿ ¿(2), it degenerates to a Square pattern (i.e., a square grid). We prove that our proposed pattern is asymptotically optimal when r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> /r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> > ¿(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.