Dimension reduction for finite trees in L_1
Metric Geometry
2011-09-07 v3 Data Structures and Algorithms
Abstract
We show that every n-point tree metric admits a (1+eps)-embedding into a C(eps) log n-dimensional L_1 space, for every eps > 0, where C(eps) = O((1/eps)^4 log(1/eps)). This matches the natural volume lower bound up to a factor depending only on eps. Previously, it was unknown whether even complete binary trees on n nodes could be embedded in O(log n) dimensions with O(1) distortion. For complete d-ary trees, our construction achieves C(eps) = O(1/eps^2).
Cite
@article{arxiv.1108.2290,
title = {Dimension reduction for finite trees in L_1},
author = {James R. Lee and Arnaud de Mesmay and Mohammad Moharrami},
journal= {arXiv preprint arXiv:1108.2290},
year = {2011}
}