  Net-trees are a general purpose data structure for metric data that have been
used to solve a wide range of algorithmic problems. We give a simple randomized
algorithm to construct net-trees on doubling metrics using $O(n\log n)$ time in
expectation. Along the way, we define a new, linear-size net-tree variant that
simplifies the analyses and algorithms. We show a connection between these
trees and approximate Voronoi diagrams and use this to simplify the point
location necessary in net-tree construction. Our analysis uses a novel
backwards analysis that may be of independent interest.
