Dimension reduction for finite trees in l1

Authors:
James R. Lee;Arnaud de Mesmay;Mohammad Moharrami
Affiliations:
University of Washington;Ecole Normale Supérieure;University of Washington
Venue:
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Year:
2012

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Abstract

We show that every n-point tree metric admits a (1 + ε)-embedding into l1C(ε) log n, for every ε 0, where C(ε) ≤ O ((1/ε)4 log 1/ε)). This matches the natural volume lower bound up to a factor depending only on ε. Previously, it was unknown whether even complete binary trees on n nodes could be embedded in l1O(log n) with O(1) distortion. For complete d-ary trees, our construction achieves C(ε) ≤ O (½ε2).