More is More: The Benefits of Denser Sensor Deployment

Positioning disk-shaped sensors to optimize certain coverage parameters is a fundamental problem in ad-hoc sensor networks. The hexagon grid lattice is known to be optimally efficient, but the 20.9% of the area covered by two sensors may be considered a waste. Furthermore, any movement of a sensor from its designated grid position or sensor failure, due to placement error or obstacle avoidance, leaves some region uncovered, as would the failure of any one sensor. We explore how shrinking the grid can help to remedy these shortcomings. First, shrinking to obtain a denser hexagonal lattice allows all sensors to move about their intended positions independently while nonetheless guaranteeing full coverage. Second, sufficiently increasing the lattice density will naturally yield k-coverage for k > 1. Moreover, we show that a density increase tantamount to fc copies of the lattice can yield k' -coverage, for k j > k (e.g. k = 11, k j = 12), through the exploitation of the double-coverage regions. Our examples' savings provably converge in the limit to the ap 20.9% maximum. We also provide analogous results for the square lattice and its ap 57% inefficiency, including k = 3, k j = 4, k = 5,k j = 7, indicating that for multi-coverage, the square lattice can actually be more efficient than the hexagon lattice. All these efficiency gains can be used to provide 1-coverage or fc-coverage even in the face of probabilistic sensor failure. We conclude by construing the shrinking factor as a budget to be divided among these three benefits.