# Efficient Distributed Method for NLOS Cooperative Localization in WSNs
## 5 Numerical Simulations
### Performance Comparison
#### 5.1.2 Complexity Analysis  
We compared computational complexity in Table 1, in which we used \(N_{c}\) to represent the connective numbers of pair nodes \((i,j)\in\mathcal{Z}_{\mathcal{S}}\cup\mathcal{Z}_{\mathcal{A}}\). We could find that ADMM_P is more suitable for practical situations because it has much lower computational complexity and smaller problem size. The parallel distributed framework plays a key role in decreasing complexity.  
The SDPH method in Reference [35] and the SDP method in Reference [36] were implemented in the centralized framework. The problem sizes of SDPH and SDP were, respectively, \(dN+1/2*N(N+1)\)  
Figure 4: Performance comparison in varying environments with a \(M=10,N=40\) sensor network. Probability of NLOS connections was set to \(P_{NLOS}=95\%\). **(a)** Root Mean Square Error (RMSE) comparison in varying noise environments; **(b)** RMSE comparison with different NLOS parameters.  
Figure 3: Cumulative Distribution Function (CDF) comparison of the four aforementioned methods for a \(M=10,N=40\) sensor network. Anchors were randomly distributed. Probability of NLOS connections was set to \(P_{NLOS}=95\%\).  
and \(dN+1/2N(N+1)+N^{2}\), both depending on the whole scale of the WSN \(N\). The EDM method in Reference [37] applied a three-block ADMM, but problem size also depends on \(N\). According to Table 1, we can see that the problem size of the ADMM_P algorithm was determined by the decomposed local minimization and could be reduced to \(d(\hat{N}_{i}+1)+\hat{N}_{i}\), where \(\hat{N}_{i}\) represents the number of neighbors. Because of the limitation of \(r_{\mu}\), the size of the neighbor set was much smaller than the whole scale of the WSN, i.e., \(\hat{N}_{i}\ll N\). This conclusion can theoretically prove that our method can drastically reduce the precessing burden, no matter the problem size or computational complexity.  
Our method decomposes the original large-scale problem into \(N+M\) small-scale subproblems via consensus form, and handles these subproblems in a parallel way. Hence, when compared with the centralized algorithm, our method can dramatically decrease computational complexity. In Sections 3 and 4, we could find that the total complexity of the proposed algorithm almost depends on solving the subproblems. Section 4 provides an iterative method to solve the subproblems, which are nonquadratic differentials. In this iterative method, with the guarantee of objective-function convexity, the optimal value is finite and attained at \(\mathbf{z}_{i}^{*}\). The derived method is known to converge at a rate of \(O(1/k^{2})\)[44]:  
\[F_{i}(\mathbf{z}_{i}^{(k)})-F_{i}^{*}\leq\frac{2}{(k+1)^{2}\mu}\|\mathbf{z}_{i}^{(0)}- \mathbf{z}_{i}^{*}\|^{2} \tag{43}\]  
Subupdates require very simple operations, no matter if they are at anchor nodes or at source nodes. The inversion of matrix \(\mathbf{H}_{i}\) and \(\mathbf{H}_{i}\mathbf{Q}_{i}\) is easy to precompute because \(\mathbf{H}_{i}\) and \(\mathbf{Q}_{i}\) only consist of identity matrices and unit vectors. Projection on set \(\mathbf{C}_{i}\) can be decomposed into projections on sets \(\mathbf{C}_{i,j},j\in\mathcal{N}_{i}\), which involves very limited dimension \(d\leq 3\) and is easy to compute. Therefore, the local update given by Equations (34) and (41) has much lower complexity than the three other algorithms. However, because every node shares the processing burden and plays the same role in the parallel framework, the communication cost of each node is higher than that of centralized frameworks.