Analytical characterization of computationally efficient localization techniques

Trilateration-based localization techniques have been widely used in sensor networks due to their computational efficiency and distributedness. However in sparse networks or in the boundary area of networks, trilateration-based techniques often fail to localize all localizable nodes. Bilateration-based techniques emerge as a generalization of trilateration techniques to a broader class of networks. Compared with trilateration-based techniques, the main benefit of bilateration-based schemes is that they can localize a higher percentage of nodes while still maintaining the low computational complexity and distributedness properties. One potential drawback of bilateration-based schemes is that the number of estimated possible positions (hence the memory required to store these positions) may grow exponentially with the number of nodes in the network. Despite the empirical observations reported in the literature that such exponential growth is a rare event, there is a lack of rigorous analysis quantifying the complexity of bilateration-based schemes. In this paper, we tackle the challenge by first characterizing a broad subclass of the set of critical sub-networks within which the number of possible estimated positions grows exponentially with the size of these sub-networks. Then using mathematical techniques from percolation theory, we prove that, in random geometric networks, with very high probability the size of these critical sub-networks, which constitute the worst case for bilateration-based localization, is bounded. Therefore the complexity of bilateration-based localization technique does not grow exponentially with the size of the entire network. The significance of this result is to analytically demonstrate that bilateration-based techniques not only localize a higher fraction of nodes than their trilateration counterpart, but also they can be implemented in a very efficient (low computational cost) manner.