Network location is one of the critical issues and a challenge in wireless sensor and ad hoc networks, in particular when networks are sparse. However, even in highly sparse networks, there exist well-connected subgraphs while the distribution of the networks is random. This paper introduces weighted component stitching (WCS) to find redundantly rigid components with high redundant ratios, which can be used to generate reliable local realization. Finding and ranking the redundantly rigid components is an NP-hard problem (a reduction from maximum quasi-clique). Here, we introduce a series of theorems and algorithms to carry out WCS efficiently. More precisely, we prove that each graph has a determinant number of redundantly rigid components, each redundantly rigid component is covered by a set of basic redundant components (BRCs), and each BRC contains one redundant edge. We apply constraints to merge the BRCs to form components with higher redundancy ratio and develop a greedy algorithm to merge BRCs to form locally mostly redundant components (LMRCs). Finally, we give the approximation ratio. The local coordinates of nodes are calculated by optimization in each LMRC and are synchronized with weights to produce the global coordinates of nodes in the network to overcome the sparseness of subgraphs. Extensive experiments demonstrate significant improvements in accuracy (45%-64%) using our WCS method over the state-of-the-art algorithms under various settings of network sparseness and ranging noises.
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