Robust Localization from Incomplete Local Information

We consider the problem of localizing wireless devices in an ad-hoc network embedded in a $d$-dimensional Euclidean space. Obtaining a good estimation of where wireless devices are located is crucial in wireless network applications including environment monitoring, geographic routing and topology control. When the positions of the devices are unknown and only local distance information is given, we need to infer the positions from these local distance measurements. This problem is particularly challenging when we only have access to measurements that have limited accuracy and are incomplete. We consider the extreme case of this limitation on the available information, namely only the connectivity information is available, i.e., we only know whether a pair of nodes is within a fixed detection range of each other or not, and no information is known about how far apart they are. Further, to account for detection failures, we assume that even if a pair of devices is within the detection range, it fails to detect the presence of one another with some probability and this probability of failure depends on how far apart those devices are. Given this limited information, %the above set of noisy and incomplete measurements, we investigate the performance of a centralized positioning algorithm {\sc MDS-MAP} introduced by Shang et al. \cite{SRZ03}, and a distributed positioning algorithm {\hopterrain} introduced by Savarese et al. \cite{SLR02}. In particular, for a network consisting of $n$ devices positioned randomly, we provide a bound on the resulting error for both algorithms. We show that the error is bounded, decreasing at a rate that is proportional to $R_{\rm Critical}/R$, where $R_{\rm Critical}$ is the critical detection range when the resulting random network starts to be connected, and $R$ is the detection range of each device.