Embedding the diamond graph in Lp and dimension reduction in L1

We show that any embedding of the level k diamond graph of Newman and Rabinovich [6] into Lp, 1 < p ≤ 2, requires distortion at least √ k(p− 1) + 1. An immediate corollary is that there exist arbitrarily large n-point sets X ⊆ L1 such that any D-embedding of X into l1 requires d ≥ n Ω(1/D2). This gives a simple proof of a recent result of Brinkman and Charikar [2] which settles the long standing question of whether there is an L1 analogue of the Johnson-Lindenstrauss dimension reduction lemma [4]. 1 The diamond graphs, distortion, and dimension We recall the definition of the diamond graphs {Gk} ∞ k=0 whose shortest path metrics are known to be uniformly bi-lipschitz equivalent to a subset of L1 (see [3] for the L1 embeddability of general series-parallel graphs). The diamond graphs were used in [6] to obtain lower bounds for the Euclidean distortion of planar graphs and similar arguments were previously used in a different context by Laakso [5]. G0 consists of a single edge of length 1. Gi is obtained from Gi−1 as follows. Given an edge (u, v) ∈ E(Gi−1), it is replaced by a quadrilateral u, a, v, b with edge lengths 2 −i. In what follows, (u, v) is called an edge of level i − 1, and (a, b) is called the level i anti-edge corresponding to (u, v). Our main result is a lower bound on the distortion necessary to embed Gk into Lp, for 1 < p ≤ 2. Theorem 1.1. For every 1 < p ≤ 2, any embedding of Gk into Lp incurs distortion at least