Understanding Node Localizability in Barycentric Linear Localization

Abstract

The barycentric linear localization (BLL) methods provide a lightweight, distributed way to calculate locations for resource-limited IoT devices. A crucial requirement for BLL is that the nodes participating in the iterative location propagation are localizable. Otherwise, the unlocalizable nodes will continuously pose error information in the location propagation process, making even the theoretically localizable nodes converge to the wrong locations. However, the research on node localizability in BLL is much lacked, greatly limiting the application scope of BLL. In specific, BLL node localizability is detected on a generated graph <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{G^A}$</tex-math> </inline-formula> . For any node, its neighbors appear in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{G^A}$</tex-math> </inline-formula> only when the neighbors can form triangle(s), so that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{G^A}$</tex-math> </inline-formula> is much sparser than the original <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal G$</tex-math> </inline-formula> . Thus, the node localizability condition in BLL is harder to be satisfied than that in traditional localization methods. Moreover, the distributed algorithm to detect BLL localizable nodes is still open. This paper thoroughly investigates the node localizability conditions and distributed localizable node detection algorithms in BLL. At first, an efficient and fully distributed Negative Edge Inference (NEI) algorithm is proposed for each node to infer implicit edges in its neighborhood. NEI strengthens the distance graph by revealing more distance constraints so that enables more neighboring triangles. Then a new sufficient condition, i.e., the recursive three disjoint path condition (Recursive-3DP) on the strengthened distance graph is proposed to identify BLL localizable nodes much more accurately. Secondly, a distributed Path Extension and Pruning (PEP) algorithm is proposed for distributed localizable node detection. PEP is proved to detect all the theoretically Recursive-3DP nodes in the strengthened distance graph. A Fast-PEP algorithm is further proposed, which misses very limited Recursive-3DP nodes while bringing significant improvement in efficiency. PEP and Fast-PEP guarantee to identify BLL localizable nodes in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2H$</tex-math> </inline-formula> rounds, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H$</tex-math> </inline-formula> is the maximum hop number of the node disjoint paths. Finally, by using NEI and PEP (Fast-PEP), a localizability-aware BLL (LABEL) method is proposed, which correctly identifies localizable nodes and guarantees their correct location convergence. Extensive analysis and experiments show the advantages in localizability and location accuracy of the proposed schemes over the state-of-the-art methods.