Restricted accessResearch articleFirst published online 2019-1
A Simple Approach to the Reconstruction of a Set of Points from the Multiset of Pairwise Distances in n 2 Steps for the Sequencing Problem: III. Noise Inputs for the Beltway Case
The approach based on the removal of redundancy in inputs when reconstructing a set of points X from the set of their pairwise distances \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\Delta X$$
\end{document} is generalized for the beltway case by using integral transformations. It is shown that the generalized approach can be successfully used not only for complete and error-free sets of pairwise distances \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\Delta X$$
\end{document}, but also for sets \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\Delta \mathfrak{X} = \Delta X + \mathfrak{f}$$
\end{document} containing a large number of noise and missing data \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mathfrak{f}$$
\end{document}. The proposed approach allows to reconstruct X in n2 steps, where n is the cardinality of noise input sets.
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