Abstract
We introduce a new notion of ‘neighbors’ in geometric permutations. We conjecture that the maximum number of neighbors in a set S of n pairwise disjoint convex bodies in ℝd is O(n), and we prove this conjecture for d = 2. We show that if the set of pairs of neighbors in a set S is of size N, then S admits at most O(N d-1) geometric permutations. Hence we obtain an alternative proof of a linear upper bound on the number of geometric permutations for any finite family of pairwise disjoint convex bodies in the plane.
Work on this paper has been supported by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the U.S.-Israeli Binational Science Foundation, by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University.
The research by this author was done while the author was a Ph.D. student under the supervision of Micha Sharir.
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Sharir, M., Smorodinsky, S. (2002). On Neighbors in Geometric Permutations. In: Penttonen, M., Schmidt, E.M. (eds) Algorithm Theory — SWAT 2002. SWAT 2002. Lecture Notes in Computer Science, vol 2368. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45471-3_14
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DOI: https://doi.org/10.1007/3-540-45471-3_14
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