Geometric permutations of disjoint translates of convex sets
Discrete Mathematics
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Discrete & Computational Geometry
Discrete & Computational Geometry
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Discrete & Computational Geometry
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SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
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SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
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Improved Bounds for Geometric Permutations
SIAM Journal on Computing
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We study combinatorial bounds for geometric permutations of balls with bounded size disparity in d-space. Our main contribution is the following theorem: given a set S of n disjoint balls in Rd, if n is sufficiently large and the radius ratio between the largest and smallest balls of S is γ, then the maximum number of geometric permutations of S is O(γlogγ). When d = 2, we are able to prove the tight bound of 2 on the number of geometric permutations for S, which is the best possible bound because it holds even when γ = 1. Our theorem shows how the number of permutations varies as a function of the size disparity among balls, thus gracefully bridging the gap between two extreme bounds that were known before: the O(1) bound for congruent balls, and the Θ(nd-1) bound for arbitrary balls.