Embedding the diamond graph in L _p and dimension reduction in L ₁

Abstract

We show that any embedding of the level k diamond graph of Newman and Rabinovich [NR] into L _p , 1 < p ≤ 2, requires distortion at least $\sqrt{k(p-1) + 1}$. An immediate corollary is that there exist arbitrarily large n-point sets $X \subseteq L_1$ such that any D-embedding of X into $\ell^d_1$ requires $d \geq n^{\Omega(1/D^2)}$. This gives a simple proof of a recent result of Brinkman and Charikar [BrC] which settles the long standing question of whether there is an L ₁ analogue of the Johnson-Lindenstrauss dimension reduction lemma [JL].