Range-Free Ranking in Sensors Networks and Its Applications to Localization Zvi Lotker ∗ , Marc Martinez de Albeniz ∗ , and St´ ephane P´ er´ ennes ⋆ CNRS-INRIA-Sophia Antipolis-I3S, 06902, France zvilo@eng.tau.ac.il, marcalbeniz@hotmail.com, stephane.perennes@inria.fr Abstract. We address the question of finding sensors’ coordinates, or at least an approximation of them, when the sensors’ abilities are very weak. In a d dimen- sional space, we define an extremely relaxed notion of coordinates along dimen- sion i. The ranki of a sensor s is the number of sensors with ith-coordinate less than the i-coordinate of s. In this paper we provide a theoretical foundation for sensor ranking, when one assumes that a few anchor sensors know their locations and that the others determine their rank only by exchanging information. We show that the rank problem can be solved in linear time in R and that it is NP-Hard in R 2 . We also study the usual localization problem and show that in general one cannot solve it; unless one knows a priori information on the sensors distribution. 1 Introduction One of the fundamental problems in sensor networks is the localization problem, i.e., each sensor wishes to know its exact location. In the last two years several variations of this problem had been studied [6,13,11,7,16]. For an application to localization we refer the reader to [17] Since sensors are supposed to be cheap the idea of using a GPS devices in every sensor is not practical. One possibility to reduce the cost is to divide the sensors into two sets. One set will serve as base stations and will be equipped with a GPS. The sensors in the second set will use the base stations in order to compute their location. Many localization algorithms for sensor networks have been proposed. There are mainly two groups of algorithms: range based and range free. In the range based case it is assumed that the sensors have an estimation of the point-to-point distance or angle. In the range free case no such assumption is made. Because of the hardware limitations of sensors networks devices, solutions in range-free localization are being pursued as a cost-effective alternative to more expensive range-based approaches. In this paper we study the range free localization and ranking problems. The main questions that arise are: Uniqueness do we have enough information so that all the solutions that are consistent with the sensors observations are “the same”? Practical feasibility: Can we find at least one solution in reasonable time? Distributed efficiency: can we provide a fast distributed algorithm that solve the problem ? Best solution: Can we find the solution which is the most consistent with the observations? ⋆ Project Mascotte CNRS-INRIA-Sophia Antipolis-I3S, 06902, France, this work was supported by the European project Aracne. I. Nikolaidis et al. (Eds.): ADHOC-NOW 2004, LNCS 3158, pp. 158–171, 2004. c  Springer-Verlag Berlin Heidelberg 2004