Optimal Random Matchings on Trees and Applications

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 ℝ d 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.