Abstract
The Distance Geometry Problem asks whether a given weighted graph has a realization in a target Euclidean space \(\mathbb {R}^K\) which ensures that the Euclidean distance between two realized vertices incident to a same edge is equal to the given edge weight. In this paper we look at the setting where the target space is the surface of the sphere \(\mathbb {S}^{K-1}\). We show that the Distance Geometry Problem is almost the same in this setting, as long as the distances are Euclidean. We then generalize a theorem of Gödel about the case where the distances are spherical geodesics, and discuss a method for realizing cliques geodesically on a K-dimensional sphere.
L. Liberti—Partly supported by the French national research agency ANR under the “Bip:Bip” project under contract ANR-10-BINF-0003.
C. Lavor—The support of FAPESP and CNPq is gratefully acknowledged.
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Liberti, L., Swirszcz, G., Lavor, C. (2016). Distance Geometry on the Sphere. In: Akiyama, J., Ito, H., Sakai, T., Uno, Y. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2015. Lecture Notes in Computer Science(), vol 9943. Springer, Cham. https://doi.org/10.1007/978-3-319-48532-4_18
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DOI: https://doi.org/10.1007/978-3-319-48532-4_18
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