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On the Size of Some Trees Embedded in ${\mathbb R}^d$

Pedro Machado Manhães de Castro 1 Olivier Devillers 1
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : We show that, for an Euclidean minimal k-insertion tree (EMITk), if the weight w of an edge e is its Euclidean length to the power of α, the sum on all edges of EMITk of their weights w(e) is O(n * k−α/d) in the worst case, where d is the dimension, for d ≥ 2 and 0 < α < d. We also analyze the expected size of EMITk and some stars, when points are evenly distributed inside the unit ball, for any α > 0.
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Pedro Machado Manhães de Castro, Olivier Devillers. On the Size of Some Trees Embedded in ${\mathbb R}^d$. Operations Research Letters, Elsevier, 2011, 39, pp.44-48. ⟨10.1016/j.orl.2010.10.005⟩. ⟨hal-00991081⟩



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