The MNCE algorithm divides the multi-hop distance into several small hops; then calculates the small hops one by one to get the estimated distance. The algorithm is described from these three stages.

#### 3.1. Distance Estimation

We suppose that the quantity of nodes in a WSN is

N, the quantity of anchor nodes is

n. The communication radius of all nodes is

$R$, nodes get their own information with neighbor nodes by forwarding messages. The neighbor relationship model is as follows:

The i and j in Formula (2) symbolize two sensor nodes; d_{ij} represents the Euclidean distance between the two nodes. The general idea of this type of localization algorithms is to propose a measure which is positively correlated with the distance between nodes according to the node distribution and node connectivity information, and then to represent the measure as accurately as possible according to the locality of the node distribution. As an example of DV-hop algorithm: The estimated distance is calculated by the minimum hop count and the single hop correction.

As shown in

Figure 1, this localization method will have a large error. The hop count between nodes will be defined as one hop, if the actual distance between the two sensor nodes is between 0 and

R. We propose the MNCE algorithm, which uses the relationship of the distance between adjacent nodes and the area of communication overlapped region between adjacent nodes to obtain the estimated distance between nodes more accurately.

We divide the estimation distance process into several steps: first, we divide the multi-hop distance into the accumulation of multiple single-hop distances; second, we estimate the distance for each single-hop distance. The node

i and node

j in

Figure 2 are neighbor nodes with one hop in the sensor network. The black solid points in

Figure 2 represent the different neighbor nodes of the two nodes, and the red solid points represent the common neighbor node of two nodes.

The area of the communication overlap region

A_{ij} is obtained by geometric calculation:

The

d_{ij} in Formula (2) is the distance between the two nodes, so we can get the area ratio between node communication overlap region and node communication region as shown in Formula (3):

There are two unknown terms in Formula (3): Area ratio and

d_{ij}, so we cannot work out the

d_{ij} according to one formula. Formula (4) is the inverse function of Formula (3). We assume that there are a large number of sensor nodes in the WSN, and the nodes deployment characteristics in the local region of the node communication range are approximately the same. In addition, the ratio of the two regions is approximately equal to the ratio of the quantity of nodes in two regions. The MNCE algorithm does not require additional measurement techniques, as long as the neighbor nodes can communicate. Therefore, this type of localization algorithms has low power consumption and nodes can be deployed with the high density in such algorithms. The node distribution of local neighboring regions is approximately the same in WSNs with the node high density deployment. Therefore, we can approximate the area ratio of the two regions by the ratio of the quantity of nodes in the two local neighboring regions. Therefore, the distance

d_{ij} between neighbor nodes can be calculated using Formula (3):

The more nodes are deployed in the network, the closer the proportion of the quantity of nodes is to the area ratio of the regions where the node is located. Therefore, in Formula (6), we choose the maximum value of the quantity of nodes in the communication area of two nodes to replace the denominator of the area ratio.

Suppose the communication path between two nodes is shown in

Figure 3. The estimated distance is the superposition of each hop distance. We assume the minimum hop count is

n and the distance between nodes of the i-th hop is

d_{i}, obviously the initial estimated distance is calculated using the above method:

However, the shortest path of communication is not a straight line under most circumstances. When the shortest path of two nodes is a tortuous route, the error will gradually become larger when the estimated distance is calculated by using a single hop distance accumulation method. As shown in

Figure 4 below, the actual distance

d_{ij} between the two hops is much less than

d_{ik} + d_{kj.}In order to find out the estimated distance between nodes more accurately and reduce the error, we should split the multi-hop distance between nodes up small hops with as little as possible. The relationship between distance

d_{ij} and area

A_{ij} must satisfy the Formula (2) if the distance between nodes is calculated by the method in this study. When the hop count is greater than 2, there is no overlapped area in the communication area between two nodes. Therefore, it is reasonable to divide the multi-hop distance into multiple accumulations in units of two hops. The node distance and area of two hops also apply to the function

$\phi ({d}_{ij})$:

We assume the minimum hop count is

n. When

n is even, the estimated distance is shown in Formula (9). when

n is odd, the estimated distance is shown in Formula (10):