Dimension Reduction for Finite Trees in $$\varvec{\ell _1}$$ ℓ 1

Abstract

We show that every $$n$$ -point tree metric admits a $$(1+\varepsilon )$$ -embedding into $$\ell _1^{C(\varepsilon ) \log n}$$ , for every $$\varepsilon > 0$$ , where $$C(\varepsilon ) \le O\big ((\frac{1}{\varepsilon })^4 \log \frac{1}{\varepsilon })\big )$$ . This matches the natural volume lower bound up to a factor depending only on $$\varepsilon $$ . Previously, it was unknown whether even complete binary trees on $$n$$ nodes could be embedded in $$\ell _1^{O(\log n)}$$ with $$O(1)$$ distortion. For complete $$d$$ -ary trees, our construction achieves $$C(\varepsilon ) \le O\big (\frac{1}{\varepsilon ^2}\big )$$ .