On Neighbors in Geometric Permutations

Authors:
Micha Sharir;Shakhar Smorodinsky
Affiliations:
-;-
Venue:
SWAT '02 Proceedings of the 8th Scandinavian Workshop on Algorithm Theory
Year:
2002

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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(Nd-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.