localized neighbors of every node and inserting a node when the count hits the score of three. An effective three is the smallest count $\geq 4$ with three non-coplanar neighbors. When we reach the count $4$ for the first time, we check coplanarity. From then on, every new increment checks only the new one with any two previous neighbors. Therefore, it terminates after at most $\OO(n^4m)$ steps.

As seen above, the worst-case analysis gives us quartic running time. But notice that when the sensor nodes are deployed to form coplanar groups in 3D, and the communication between groups are held by a limited number of nodes (i.e. the number of interplanar edges are low), most of the neighbors of an unlocalized node will be coplanar. On the other hand, in the algorithm that we describe to utilize grouping function, this type of deployment is an advantage. Thus, it is safe to say that these two algorithms complete each other, in terms of their functionalities with respect to sparsity.

While describing our algorithms, we use the following data structure for a vertex of the network graph. A sensor node $s_u$, represented by a vertex $u$ in the network graph has three 