Optimal Random Matchings on Trees and Applications

• Authors: Jeff Abrahamson;Béla Csaba;Ali Shokoufandeh
• Venue: APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
• Year: 2008

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 nsets of points R= {r1,...,rn} and B= {b1,...,bn} distributed uniformly and randomly on the mleaves of 茂戮驴-Hierarchically Separated Trees with branching factor bsuch that each of its leaves is at depth 茂戮驴, we will prove that the expected weight of optimal matching between Rand Bis $\Theta(\sqrt{nb}\sum_{k=1}^h(\sqrt{b}\l)^k)$, for h= min (茂戮驴,logbn). Using a simple embedding algorithm from 茂戮驴dto 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.