are listed as follows.

-   We exploit the fact that the sensor nodes tend to form hyperplanar groups in real-world scenarios.

-   We prove that extracting the grouping information is a hard task.

-   We give an efficient and sustainable localization algorithm to utilize provided grouping function. Our algorithm does not require a dense network graph.

Hyperplanar substructures recognition {#sec:reduction}
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In this section, we formally define the problem of recognizing hyperplanar substructures, and then prove that the problem is -hard.

Problem definition
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Consider a set $\mathcal{H} = \{\ell_1, \ell_2, \dots, \ell_k\}$ of $k$ hyperlanes in $\mathbb{R}^d$, given by their equations $\ell_i : a_{i1}x_1 + a_{i2}x_2 + \dots + a_{in}x_d = b$, and a unit disk graph $G = (V,E)$, where $V = \{1,2,\dots,n\}$. The task is to find a mapping $h : V \rightarrow \{1,\dots,k\}$ such that there is a realization of $G$ in which $v \in \ell_{f(v)}$ for each $v \in V$.

NP-hardness reduction for 2D environment {#sec:hard2d}
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We first show that Hyperplanar substructures recognition is -hard when the sensor nodes are in 2D, and then move onto higher dimensions.

\[lem:hard2d\] Hyperplanar substructures recognition is -hard in 2D.

We give a reduction of this problem from 