Dimension reduction for finite trees in L_1

Details Dimension reduction for finite trees in L_1 Dimension reduction for finite trees in L_1

2011-08-10

arxiv.org/abs/1108.2290v3

Mathematics

Metric Geometry

Scientific paper

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).

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