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 = {r1, ..., rn} and B = {b1, ..., bn} distributed uniformly and randomly on the m leaves of -Hierarchically Separated Trees with branching factor b such that each one of its leaves are of depth , we will prove that the expected weight of optimal matching between R and B is ( nb Ph k=1( b)k ), for h = min(, logb 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., t...

• Type: Conference
• Year: 2008
• Where: APPROX
• Authors: Jeff Abrahamson, Béla Csaba, Ali Shokoufandeh
• Keywords: Algorithms, APPROX 2008, Optimal Matching, Random Point Sets, Separated Trees