given network is initially *unlocalized*, i.e. it is not aware of its position. When we fix the position of a node, we mark it as *localized*. To fix the position, we need to reduce its *degrees of freedom* to zero by obtaining $d+1$ Euclidean distances from its localized neighbors in general position. Intuitively, one expects a localizable network to be dense enough. Our algorithm, however, is able to perform localization in very sparse graphs if the grouping function is provided.

Observe that each hyperplanar group is an induced subgraph of the network graph $G = (V, E)$, and each pair $(u,v)$ of vertices in that induced subgraph satisfies $h(u) = h(v)$. The set of all vertices in a particular hyperplanar group can be retrieved by taking the inverse of $h$. In other words, $h^{-1}(i)$ gives us the set of all vertices in the hyperplanar group $i$.

We refer to the hyperplanar groups in 2D and 3D as *collinear groups*, and *coplanar groups* respectively. An edge is called *interlinear* (respectively, *interplanar*) edge, if it corresponds to a connection between two nodes from different collinear (respectively, *coplanar*) groups. Otherwise, it is a collinear (respectively, *coplanar*) edge.

Our contributions
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Our main contributions 