Flipping Free Conditions and Their Application in Sparse Network Localization


An essential challenge involves determining the topology of a network based on the distances between its nodes. When there are few distance measurements available, it is difficult because the absence of edge constraints may result in ambiguous realizations that are very different from the ground truth. This presents a challenge. In 2D, the flipping ambiguities are brought on by binary vertex cut sets, and in 3D, the triple vertex cut sets that are called separators are responsible for them. In this paper, we investigate the conditions that determine whether or not the flipping ambiguities that are caused by these separators can be disambiguated by using neighborhood, full graph, and component-level conditions. In light of this, we propose the local flipping-free condition (LFFC), the global flipping-free condition (GFFC), and the component-based flipping free condition (CFFC). After that, a disambiguating framework that is proposed, and it is based on a combinatorial application of these conditions. It does this by first performing a local LFFC to disambiguate the separators, which then converts the graph into a binary tree with leaf nodes that are flipping-free components and edges that are LFFC unsolvable separators. After that, the CFFC condition is applied once more in order to disambiguate the LFFC unsolvable separators that exist between the components. The number of ambiguous solutions for network localization will be cut in half if LFFC and CFFC are used to disambiguate the k and g separators, respectively. This will result in a reduction of 2k+g times. Finally, the flipping-free components realize node coordinates in their own local coordinate systems, and a residue-based weighted component stitching algorithm (RWCS) is proposed in order to iteratively synchronize the local coordinates of the components in order to generate the global coordinates of the network. Extensive simulations demonstrate that the LFFC, CFFC, and RWCS frameworks are effective. These frameworks resolve a major portion of flipping ambiguities and significantly improve the localization accuracy in comparison to the state-of-the-art algorithms in a variety of settings that involve sparse networks.

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