## 1 Introduction  
Obtaining accurate location information is a precondition for many applications that are based on localization technology. Among the existing positioning technologies, GPS satellite positioning systems rely on their global coverage and high accuracy, and have been widely used [1; 2] all over the world. However, these satellite positioning systems also have some significant shortcomings, including high energy consumption, a high cost, fixed auxiliary facilities, and the only use of open space. Since the 1990s, Wireless Sensor Networks (WSNs) have attracted the attention of many relevant researchers. As an extension of a satellite positioning system, WSNs bring localization methods a low cost, low power consumption, and no additional auxiliary facilities. In some applications, WSN-based localization systems still work well while satellite positioning systems may be out of service [3; 4; 5]. For example, WSN localization approaches can provide us with a relatively accurate location in areas where satellite signals are blocked, such as primitive forests and buildings. Some localization methods of WSNs are based on the characteristics of network connectivity and multi-hop information. With the help of known nodes, which are called anchors, the unknown nodes can estimate their locations. Such localization technologies are also called multi-hop range-free localization methods [6; 7]. Localization approaches in WSNs can be divided into two types: range-based algorithms [8; 9] and range-free algorithms [10; 11; 12]. The range-based (e.g., RSSI-based, Received Signal Strength Indication) localization algorithms estimate the locations of sensor nodes by measuring their distances or anglesfrom the anchor nodes. The range-based localization algorithms can deal with anisotropic networks and ensure a relatively accurate estimation of sensor locations; however, these algorithms are easily affected by environmental factors, such as noise and obstacles. In addition, their complexity and cost also increase with the accuracy requirement. For a large-scale sensor network with many sensor nodes, it is not reasonable to equip them all with ranging devices. In contrast, range-free localization algorithms use only the connectivity information among nodes, i.e., they do not require expensive ranging devices. In the case of large-scale WSN localization applications, the range-free algorithms would be preferred where sensor nodes are provided with a small amount of resources. Since the multi-hop range-free positioning technique only needs the hop information between nodes, it can position unknown nodes without additional devices and thus has been widely used. The multi-hop range-free localization methods perform rather well in an even and dense network, which frequently means that a large number of nodes are deployed in a limited area. They seriously depend on the network topology. Taking Figure 1 as an example, the dotted lines indicate the geometric distance between nodes A and B, while the red solid lines denote the corresponding hop distances between them. If the sensor nodes are distributed evenly, the geometric distances will roughly match the hop distances (refer to Figure 1a). However, this matching relationship will be totally undermined if there are some obstacles that bend the paths between nodes A and B, leading to a mismatch between the geometric distances and the hop distances (refer to Figure 1b). We usually call a network like that shown in Figure 1b an anisotropic network, which has different propagation models and irregular localization areas. A large deviation between the geometric distances and hop distances may probably cause a sharp drop in positioning accuracy, which is exactly the reason that the multi-hop range-free localization methods perform so poorly in anisotropic networks.