Abstract
In this paper we will consider tight upper and lower bounds on the weight of the optimal matching for random point sets distributed among the leaves of a tree, as a function of its cardinality. Specifically, given two n sets of points R = {r 1,...,r n } and B = {b 1,...,b n } distributed uniformly and randomly on the m leaves of λ-Hierarchically Separated Trees with branching factor b such that each of its leaves is at depth δ, we will prove that the expected weight of optimal matching between R and B is \(\Theta(\sqrt{nb}\sum_{k=1}^h(\sqrt{b}\l)^k)\), for h = min (δ,log b n). Using a simple embedding algorithm from ℝd to HSTs, we are able to reproduce the results concerning the expected optimal transportation cost in [0,1]d, except for d = 2. We also show that giving random weights to the points does not affect the expected matching weight by more than a constant factor. Finally, we prove upper bounds on several sets for which showing reasonable matching results would previously have been intractable, e.g., the Cantor set, and various fractals.
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Abrahamson, J., Csaba, B., Shokoufandeh, A. (2008). Optimal Random Matchings on Trees and Applications. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2008 2008. Lecture Notes in Computer Science, vol 5171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85363-3_21
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DOI: https://doi.org/10.1007/978-3-540-85363-3_21
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