Measured descent: A new embedding method for finite metrics

Robert Krauthgamer^{∗}  James R. Lee^{†}  Manor Mendel^{‡}  Assaf Naor^{§}

Abstract

We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Fréchet embeddings for finite metrics, due to [Bourgain, 1985] and [Rao, 1999]. We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion O(\sqrt{α_{X} · log n}), where α_{X} is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O(\sqrt{(log λ_{X}) log n}) distortion embedding, where λ_{X} is the doubling constant of X. Since λ_{X} ≤ n, this result recovers Bourgain’s theorem, but when the metric X is, in a sense, “low-dimensional,” improved bounds are achieved.

Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volume-respecting embeddings for all 1 ≤ k ≤ n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in ℓ_{∞}^{O(log n)} with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n)^{2}).

1 Introduction

The theory of low-distortion embeddings of finite metric spaces into normed spaces has attracted a lot of attention in recent decades, due to its intrinsic geometric appeal, as well as its applications in Computer Science. A major driving force in this research area has been the quest for analogies between the theory of finite metric spaces and the local theory of Banach spaces. While being very successful, this point of view did not always result in satisfactory metric analogues of basic theorems from the theory of finite-dimensional normed spaces. An example of this is Bourgain’s embedding theorem [5], the forefather of modern embedding theory, which states that every n-point metric space embeds into a Euclidean space with distortion O(log n). This upper bound on the distortion is known to be optimal [27]. Taking the point of view that log n is a substitute for the dimension of an n-point metric space (see [5]; this approach is clearly natural when applied to a net in the unit ball of some normed space), an analogue of John’s theorem [17] would assert that n-point metrics embed into Hilbert space with distortion O(\sqrt{log n}). As this is not the case, the present work is devoted to a more refined analysis of the Euclidean distortion of finite metrics, and in particular to the role of a metric notion of dimension.

∗IBM Almaden Research Center, San Jose, CA 95120, USA. Email: robi@almaden.ibm.com †Computer Science Division, University of California, Berkeley, CA 94720. Supported by NSF grant CCR-0121555 and an NSF Graduate Research Fellowship. Email: jrl@cs.berkeley.edu

‡Seibel Center for Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. Email:

mendelma@uiuc.edu

§Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA. Email: anaor@microsoft.com

We introduce a new embedding method, called measured descent, which unifies and refines the known methods of Bourgain [5] and Rao [33] for constructing Fréchet-type embeddings (i.e. embeddings where each coordinate is proportional to the distance from some subset of the metric space). Our method yields an embedding of any n-point metric space X into ℓ_{2} with distortion O(\sqrt{α_{X} log n}), where α_{X} is a geometric estimate on the decomposability of X (see Definition 1.3 for details). As α_{X} ≤ O(log n), we obtain a refinement of Bourgain’s theorem, and when α_{X} is small (which includes several important families of metrics) improved distortion bounds are achieved. This technique easily generalizes to produce embeddings which preserve higher dimensional structures (i.e. not just distances between pairs of points). For instance, our embeddings can be made volume-respecting in the sense of Feige (see Section 1.2), and hence we obtain optimal volume-respecting embeddings for arbitrary n-point spaces.

Applications. In recent years, metric embedding has become a frequently used algorithmic tool. For example, embeddings into normed spaces have found applications to approximating the sparsest cut of a graph [27, 3, 1] and the bandwidth of a graph [12, 8], and to distance labeling schemes (see e.g. [16, Sec. 2.2]). The embeddings introduced in this paper refine our knowledge on these problems, and in some cases improve the known algorithmic results. For instance, they immediately imply an improved approximate max-flow/min-cut theorem (and algorithm) for graphs excluding a fixed minor, an improved algorithm for approximating the bandwidth of graphs whose metric has a small doubling constant, and so forth.

1.1 Notation

Let (X, d) be an n-point metric space. We denote by B(x, r) = {y ∈ X : d(x, y) < r} the open ball of radius r about x. For a subset S ⊆ X, we write d(x, S) = min_{y∈S} d(x, y), and define diam(S) = max_{x,y∈S} d(x, y). We recall that the doubling constant of X, denoted λ_{X}, is the least value λ such that every ball in X can be covered by λ balls of half the radius [23, 2, 29, 15]. We say that a measure μ on X is non-degenerate if μ(x) > 0 for all x ∈ X. For a non-degenerate measure μ on X define Φ(μ) = max_{x∈X} μ(X)/μ(x) to be the aspect ratio of μ.

Let (X, d_{X}) and (Y, d_{Y}) be metric spaces. A mapping f : X → Y is called C-Lipschitz if d_{Y} (f(x), f(y)) ≤ C · d_{X}(x, y) for all x, y ∈ X. The mapping f is called K-bi-Lipschitz if there exists a C > 0 such that

CK^{−1} · d_{X}(x, y) ≤ d_{Y} (f(x), f(y)) ≤ C · d_{X}(x, y),

for all x, y ∈ X. The least K for which f is K-bi-Lipschitz is called the distortion of f, and is denoted dist(f). The least distortion with which X may be embedded in Y is denoted c_{Y} (X). When Y = L_{p} we use the notation c_{Y} (·) = c_{p}(·). Finally, the parameter c_{2}(X) is called the Euclidean distortion of X.

Metric Decomposition. Let (X, d) be a finite metric space. Given a partition P = {C_{1}, …, C_{m}} of X, we refer to the sets C_{i} as clusters. We write P_{X} for the set of all partitions of X. For x ∈ X and a partition P ∈ P_{X} we denote by P (x) the unique cluster of P containing x. Finally, the set of all probability distributions on P_{X} is denoted D_{X}.

Definition 1.1 (Padded decomposition). A (stochastic) decomposition of a finite metric space (X, d) is a distribution Pr ∈ D_{X} over partitions of X. Given Δ > 0 and ε : X → (0, 1], a Δ-bounded ε-padded decomposition is one which satisfies the following two conditions.

1. For all P ∈ supp(Pr), for all C ∈ P, diam(C) ≤ Δ.

2. For all x ∈ X, Pr[B(x, ε(x)Δ) ⊈ P (x)] ≤ \frac{1}{2}.

We will actually need a collection of such decompositions, with the diameter bound Δ > 0 ranging over all integral powers of 2 (of course the value 2 is arbitrary).

Definition 1.2 (Decomposition bundle). Given a function ε : X × ℤ → (0, 1], an ε-padded decomposition bundle on X is a function β : ℤ → D_{X}, where for every u ∈ ℤ, β(u) is a 2^{u}-bounded ε(·, u)-padded stochastic decomposition of X.

Finally, we associate to every finite metric space an important “decomposability” parameter α_{X}. (See [26] for relationships to other notions of decomposability.)

Definition 1.3 (Modulus of padded decomposability). The modulus of padded decomposability of a finite metric space (X, d) is defined as

α_{X} = inf{α : there exists an ε-padded decomposition bundle on X with ε(x, u) ≡ 1/α}.

It is known that α_{X} = O(log n) [28, 4], and furthermore α_{X} = O(log λ_{X}) [14]. Additionally, if X is the shortest-path metric on an n-point constant-degree expander, then α_{X} = Ω(log n) [4]. For every metric space X induced by an edge-weighted graph which excludes K_{r,r} as a minor, it is shown in the sequence of papers [19, 33, 11] that α_{X} = O(r^{2}).

Volume-respecting embeddings. We recall the notion of volume-respecting embeddings, which was introduced by Feige [12] as a tool in his study of the graph bandwidth problem. Let S ⊆ X be a k-point subset of X. We define its volume by

vol(S) = sup{vol_{k−1}(conv(f(S))) : f : S → L_{2} is 1-Lipschitz},

where for A ⊆ L_{2}, conv(A) denotes its convex hull and the (k − 1)-dimensional volume above is computed with respect to the Euclidean structure induced by L_{2}. A mapping f : X → L_{2} is called (k, η)-volume-respecting if it is 1-Lipschitz and for every k-point subset S ⊂ X,

[\frac{vol(S)}{vol_{k−1}(conv(f(S)))}]^{\frac{1}{k−1}} ≤ η.

It is easy to see that a 1-Lipschitz map f : X → L_{2} has distortion D if and only if it is (2, D)- volume-respecting. Thus the volume-respecting property is a generalization of distortion to larger subsets.

1.2 Results

The following theorem refines Bourgain’s result in terms of the decomposability parameter.

Theorem 1.4 (Padded embedding theorem). For every n-point metric space (X, d), and every 1 ≤ p ≤ ∞,

c_{p}(X) ≤ O(α_{X}^{1−1/p}(log n)^{1/p}).  (1)

The proof appears in Sections 1.3 and 2. Since α_{X} = O(log λ_{X}), it implies in particular that c_{2}(X) ≤ O(\sqrt{(log λ_{X}) · log |X|}), for any metric space X. This refines Bourgain’s embedding theorem [5], and improves upon previous embeddings of doubling metrics [14]. It is tight for λ_{X} = O(1) [21, 22, 14], and λ_{X} = n^{Ω(1)} [27]. The question of whether this bound is tight up to a constant factor for the range λ_{X} ∈ {c_{1}, …, |X|^{c_{2}}}, where c_{1} ∈ ℕ, 0 < c_{2} < 1 are some constants, is an interesting open problem.

For 1 ≤ p < 2, the bound O(\sqrt{α_{X} log n}) is better than (1), and thus, in these cases, it makes sense to construct the embedding first into L_{2}.

A more careful analysis of the proof of Theorem 1.4 yields the following result, proved in Section 2.2, which answers a question posed by Feige in [12].

Theorem 1.5 (Optimal volume-respecting embeddings). Every n-point metric space X admits an embedding into L_{2} which is (k, O(\sqrt{α_{X} log n})) volume-respecting for every 2 ≤ k ≤ n.

Since α_{X} = O(log n), this provides (k, O(log n))-volume-respecting embeddings for every 2 ≤ k ≤ n. This is optimal; a matching lower bound is given in [20] for all k < n^{1/3}. We note that the previous best bounds were due to Feige [12], who showed that a variant of Bourgain’s embedding achieves distortion O(\sqrt{log n} · \sqrt{log n + k log k}) (note that this is Ω(\sqrt{n}) for large values of k), and to Rao who showed that O((log n)^{3/2}) volume distortion is achievable for all 1 ≤ k ≤ n (this follow indirectly from [33], and was first observed in [13]).

This also improves the dependence on r in Rao’s volume-respecting embeddings of K_{r,r}-excluded metrics, from (k, O(r^{2}\sqrt{log n})), due to [33, 11], to (k, O(r\sqrt{log n})).^{1} As a corollary, we obtain an improved O(r\sqrt{log n})- approximate max-flow/min-cut algorithm for graphs which exclude K_{r,r} as a minor.

ℓ_{∞} embeddings. It is not difficult to see that every n-point metric space (X, d) embeds isometrically into ℓ_{∞}^{n} via the map y ↦ {d(x, y)}_{x∈X}. And for some spaces, like the shortest-path metrics on expanders, or on the log n-dimensional hypercube (see, e.g. [24]), it is known that n^{Ω(1)} dimensions are required to obtain any map with O(1) distortion. On the other hand, a simple variant of Rao’s embedding shows that every planar metric O(1)-embeds into ℓ_{∞}^{O((log n)^{2})}. Thus the dimension required to embed a family of metrics into ℓ_{∞} with low distortion is a certain measure of the family’s complexity (see [31]).

In Section 3 we use a refinement of measured descent to prove the following theorem, which answers positively a question posed by Y. Rabinovich [32], and improves Rao’s result to obtain the optimal bound.

Theorem 1.6. Let X be an n-point edge-weighted planar graph, equipped with the shortest path metric. Then X embeds into ℓ_{∞}^{O(log n)} with O(1) distortion.

The O(log n) bound on the dimension is clearly optimal (by simple volume arguments). Furthermore, this result is stronger than the O(\sqrt{log n}) distortion bound on Euclidean embeddings of planar metrics, due to Rao [33]. The embedding is produced by “derandomizing” both the decomposition bundle of [33, 19] and the proof of measured descent (applied to this special decomposition bundle).

1This bound is tight for the path even for k = 3, see [8, 20].

1.3 The main technical lemmas

The following lemma is based on a decomposition of [6], with the improved analysis of [9, 10]. The extension to general measures was observed in [25]. Since this lemma is central to our techniques, its proof is presented in Section 2 for completeness. Throughout, log x denotes the natural logarithm of x.

Lemma 1.7. Let (X, d) be a finite metric space and let μ be any non-degenerate measure on X. Then there exists an ε(x, u)-padded decomposition bundle on X where

ε(x, u) = [16 + 16 log \frac{μ(B(x, 2^{u}))}{μ(B(x, 2^{u−3}))}]^{−1} .  (2)

Remark 1.1. In [34] it was shown that X admits a doubling measure, i.e. a non-degenerate measure μ such that for every x ∈ X and every r > 0 we have \frac{μ(B(x,2r))}{μ(B(x,r))} = λ_{X}^{O(1)}. We thus recover the fact, first proved in [14], that for every metric space X, α_{X} = O(log λ_{X}). In particular, for every d-dimensional normed space Y, α_{Y} = O(d). In [7], it is argued that α_{Y} = Ω(d) when Y = ℓ_{1}^{d}. The same lower bound was shown to hold for every d dimensional normed space Y in [26].

The main embedding lemma. Let (X, d) be a finite metric space, and for ε : X × ℤ → ℝ define for all x, y ∈ X,

δ_{ε}(x, y) = min \{ε(x, u) : u ∈ ℤ and \frac{d(x, y)}{32} ≤ 2^{u} ≤ \frac{d(x, y)}{2}\} .

Given a non-degenerate measure μ on X denote for x, y ∈ X:

V_{μ}(x, y) = max \{log \frac{μ(B(x, 2d(x, y)))}{μ(B(x, d(x, y)/512))}, log \frac{μ(B(y, 2d(x, y)))}{μ(B(y, d(x, y)/512))}\} .  (3)

In what follows we use the standard notation c_{00} for the space of all finite sequences of real numbers. The following result is the main embedding lemma of this paper.

Lemma 1.8 (Main embedding lemma). Let X be an n-point metric space, μ a non-degenerate measure on X, and β : ℤ → D_{X} an ε(x, u)-padded decomposition bundle on X. Then there exists a map φ : X → c_{00} such that for every 1 ≤ p ≤ ∞ and for all distinct x, y ∈ X,

[V_{μ}(x, y)]^{1/p} · min\{δ_{ε}(x, y), δ_{ε}(y, x)\} ≤ \frac{||φ(x) − φ(y)||_{p}}{d(x, y)} ≤ C [log Φ(μ)]^{1/p} .

Here C is a universal constant.

Using Lemma 1.8 we are in a position to prove Theorem 1.4. We start with the following simple observation, which bounds V_{μ}(x, y) from below, and which will be used several times in what follows.

Lemma 1.9. Let μ be any non-degenerate measure on X and x, y ∈ X, x ≠ y. Then

max \{\frac{μ(B(x, 2d(x, y)))}{μ(B(x, d(x, y)/2))}, \frac{μ(B(y, 2d(x, y)))}{μ(B(y, d(x, y)/2))}\} ≥ 2.

Proof. Assume without loss that μ(B(x, 2d(x, y))) ≤ μ(B(y, 2d(x, y))). Noticing that the two balls B(x, d(x, y)/2) and B(y, d(x, y)/2) are disjoint, and that both are contained in B(x, 2d(x, y)); the proof follows.

Proof of Theorem 1.4. Fix p ∈ [1, ∞] and let μ = | · | be the counting measure on X. Let ε(x, u) be as in (2), and observe that in this case for all x, y ∈ X we have δ_{ε}(x, y) ≥ [16 + 16V_{μ}(x, y)]^{−1}. Applying Lemma 1.8 to the decomposition bundle of Lemma 1.7 we get a mapping φ_{1} : X → L_{p} such that for all x, y ∈ X,

\frac{[V_{μ}(x, y)]^{1/p}}{16 + 16V_{μ}(x, y)} ≤ \frac{||φ_{1}(x) − φ_{1}(y)||_{p}}{d(x, y)} ≤ C(log n)^{1/p}.

On the other hand, Lemma 1.8 applied to the decomposition bundle ensured by the definition of α_{X} yields a mapping φ_{2} : X → L_{p} for which

\frac{[V_{μ}(x, y)]^{1/p}}{α_{X}} ≤ \frac{||φ_{2}(x) − φ_{2}(y)||_{p}}{d(x, y)} ≤ C(log n)^{1/p}.

Finally, for φ = φ_{1} ⊕ φ_{2} we have

\frac{||φ(x) − φ(y)||_{p}^{p}}{d(x, y)^{p}} ≥ \frac{V_{μ}(x, y)}{[16 + 16V_{μ}(x, y)]^{p}} + \frac{V_{μ}(x, y)}{α_{X}^{p}} ≥ Ω (\frac{1}{α_{X}^{p−1}} ) ,

where we have used the fact that Lemma 1.9 implies that V_{μ}(x, y) ≥ Ω(1).

2 Measured descent

In this section, we prove the main embedding lemma and exhibit the existence of optimal volumerespecting embeddings. We use decomposition bundles to construct random subsets of X, the distances from which are used as coordinates of an embedding into c_{00}. As the diameters of the decompositions become smaller, our embedding “zooms in” on the increasingly finer structure of the space. Our approach is heavily based on the existence of good decomposition bundles; we thus start by proving Lemma 1.7, which is essentially contained in [10].

Proof of Lemma 1.7. By approximating the values {μ(x)}_{x∈X} by rational numbers and duplicating points, it is straightforward to verify that it is enough to prove the required result for the counting measure on X, i.e. when μ(S) = |S|.

Let Δ = 2^{u} for some u ∈ ℤ. We now describe the distribution β(u). Choose, uniformly at random, a permutation π of X and a value α ∈ [\frac{1}{4}, \frac{1}{2}]. For every point y ∈ X, define a cluster

C_{y} = B(y, αΔ) \ ⋃_{z:π(z)<π(y)} B(z, αΔ).

In words, a point x ∈ X is assigned to C_{y} where y is the minimal point according to π that is within distance αΔ from x.

Clearly the set P = {C_{y}}_{y∈X} constitutes a partition of X. Furthermore, C_{y} ⊆ B(y, αΔ), thus diam(C_{y}) ≤ Δ, so requirement (1) in Definition 1.1 is satisfied for every partition P arising from this process. It remains to prove requirement (2).

Fix a point x ∈ X and some value t ≤ Δ/8. Let a = |B(x, Δ/8)|, b = |B(x, Δ)|, and arrange the points w_{1}, …, w_{b} ∈ B(x, Δ) in increasing distance from x. Let I_{k} = [d(x, w_{k}) − t, d(x, w_{k}) + t] and write E_{k} for the event that αΔ ≤ d(x, w_{k}) + t and w_{k} is the minimal element according to π such that αΔ ≥ d(x, w_{k}) − t. Note that if w_{k} ∈ B(x, Δ/8), then Pr[E_{k}] = 0 since in this case d(x, w_{k}) + t < Δ/8 + t ≤ Δ/4 ≤ αΔ. We claim that the event {B(x, t) ⊈ P (x)} is contained in the event ⋃_{k=1}^{b} E_{k}. Indeed, let w_{j} be the minimal element according to π such that C_{w_{j}} ∩ B(x, t) ≠ ∅. It follows that d(w_{j}, x) < αΔ + t. Furthermore, d(x, w_{j}) ≥ αΔ − t, since otherwise B(w_{j}, αΔ) ⊇ B(x, t), implying that P (x) = C_{w_{j}} ⊇ B(x, t). Hence there exists k for which the event E_{k} occurs.

Now

Pr[B(x, t) ⊈ P (x)] ≤ ∑_{k=a+1}^{b} Pr[E_{k}] = ∑_{k=a+1}^{b} Pr[αΔ ∈ I_{k}] · Pr[E_{k} | αΔ ∈ I_{k}]

≤ ∑_{k=a+1}^{b} \frac{2t}{Δ/4} · \frac{1}{k} ≤ \frac{8t}{Δ} (1 + log \frac{b}{a}) ,

where we used the fact that Pr[E_{k} | αΔ ∈ I_{k}] ≤ Pr[j < k ⟹ π(j) > π(k)] = 1/k. Setting t = ε(x, u)Δ ≤ Δ/8, where ε(x, u) is as in (2), the righthand side is at most \frac{1}{2}, proving requirement (2) in Definition 1.1.

2.1 Proof of main embedding lemma

We first introduce some notation. Define two intervals of integers I, T ⊆ ℤ by

I = {−6, −5, −4, −3, −2, −1, 0, 1, 2, 3} and T = {0, 1, …, ⌈log_{2} Φ(μ)⌉}.

For t > 0 write κ(x, t) = max{κ ∈ ℤ : μ(B(x, 2^{κ})) < 2^{t}}. For each u ∈ ℤ let P_{u} be chosen according to the distribution β(u). Additionally, for u ∈ ℤ let {σ_{u}(C) : C ⊆ X} be i.i.d. symmetric {0, 1}-valued Bernoulli random variables. We assume throughout the ensuing argument that the random variables {σ_{u}(C) : C ⊆ X, u ∈ ℤ}, {P_{u} : u ∈ ℤ} are mutually independent. For every t ∈ T and i ∈ I define a random subset W_{t}^{i} ⊆ X by

W_{t}^{i} = {x ∈ X : σ_{κ(x,t)−i}(P_{κ(x,t)−i}(x)) = 0}.

Our random embedding f : X → c_{00} is defined by f(x) = (d(x, W_{t}^{i}) : i ∈ I, t ∈ T). In the sequel, we assume that p < ∞; the case p = ∞ follows similarly. Since each of the coordinates of f is Lipschitz with constant 1, we have for all x, y ∈ X,

∥f(x) − f(y)∥_{p}^{p} ≤ |I| · |T | · d(x, y)^{p} ≤ 50[log Φ(μ)] d(x, y)^{p}.  (4)

The proof will be complete once we show that for all x, y ∈ X

𝔼∥f(x) − f(y)∥_{p}^{p} ≥ [Ω(min{δ_{ε}(x, y), δ_{ε}(y, x)} · d(x, y))]^{p} · V_{μ}(x, y).  (5)

Indeed, denote by (Ω, Pr) the probability space on which the above random variables are defined, and consider the space L_{p}(Ω, c_{00}), i.e. the space of all c_{00} valued random variables ζ on Ω equipped with the L_{p} norm ∥ζ∥_{p} = (𝔼∥ζ∥_{p}^{p})^{1/p}. Equations (4) and (5) show that the mapping x ↦ f(x) is the required embedding of X into L_{p}(Ω, c_{00}). Observe that all the distributions are actually finitely supported, since X is finite, so that this can still be viewed as an embedding into c_{00}. See Remark 2.2 below for more details.

To prove (5) fix x, y ∈ X, x ≠ y. Without loss of generality we may assume that the maximum in (3) is attained by the first term, namely, \frac{μ(B(x,2d(x,y)))}{μ(B(x,d(x,y)/512))} ≥ \frac{μ(B(y,2d(x,y)))}{μ(B(y,d(x,y)/512))}. Using Lemma 1.9, it immediately follows that

\frac{μ(B(x, 2d(x, y)))}{μ(B(x, d(x, y)/512))} ≥ 2.  (6)

Setting R = \frac{1}{4}d(x, y), denote for i ∈ ℤ, s_{i} = log_{2} μ(B(x, 2^{i}R)). We next extend some immediate bounds on κ(x, t) (in terms of R) to any nearby point z ∈ B(x, R/256).

Claim 2.1. For i ∈ I and all t ∈ ℤ ∩ [s_{i−1}, s_{i}], every z ∈ B(x, R/256) satisfies \frac{R}{8} ≤ 2^{κ(z,t)−i} < \frac{5R}{4}.

Proof. By definition, μ (B(z, 2^{κ(z,t)})) < 2^{t} ≤ μ (B(z, 2^{κ(z,t)+1})) . For the upper bound, we have

μ (B(x, 2^{κ(z,t)} − R/256)) ≤ μ (B(z, 2^{κ(z,t)})) < 2^{t} ≤ 2^{s_{i}} = μ (B(x, 2^{i}R)) ,

implying that 2^{κ(z,t)} − \frac{R}{256} < 2^{i}R, which yields 2^{κ(z,t)−i} < \frac{5R}{4}. For the lower bound, we have

μ (B(x, 2^{κ(z,t)+1} + R/256)) ≥ μ (B(z, 2^{κ(z,t)+1})) ≥ 2^{t} ≥ 2^{s_{i−1}} = μ (B(x, 2^{i−1}R)) .

We conclude that 2^{κ(z,t)+1} + \frac{R}{256} ≥ 2^{i−1}R, which implies that \frac{R}{8} ≤ 2^{κ(z,t)−i}.

Consider the following events

1. E_{1} = \{d(x, X \ P_{u}(x)) ≥ δ_{ε}(x, y)\frac{R}{8} for all u ∈ ℤ with 2^{u} ∈ [R/8, 5R/4]\},
2. E_{2} = {σ_{u}(P_{u}(x)) = 1 for all u ∈ ℤ with 2^{u} ∈ [R/8, 5R/4]},
3. E_{3} = {σ_{u}(P_{u}(x)) = 0 for all u ∈ ℤ with 2^{u} ∈ [R/8, 5R/4]},
4. E_{i,t}^{big} = \{d(y, W_{t}^{i}) ≥ \frac{1}{512}δ_{ε}(x, y)R\},
5. E_{i,t}^{small} = \{d(y, W_{t}^{i}) < \frac{1}{512}δ_{ε}(x, y)R\}.

The basic properties of these events are described in the following claim.

Claim 2.2. The following assertions hold true:

(a). Pr[E_{1}], Pr[E_{2}], Pr[E_{3}] ≥ 2^{−4}.

(b). For all i ∈ I and t ∈ ℤ ∩ [s_{i−1}, s_{i}], the event E_{3} is independent of E_{i,t}^{big}.

(c). For all i ∈ I and t ∈ ℤ ∩ [s_{i−1}, s_{i}], the event E_{2} is independent of E_{1} ∩ E_{i,t}^{small}.

(d). For all i ∈ I and t ∈ ℤ ∩ [s_{i−1}, s_{i}], if the event E_{3} occurs then x ∈ W_{i}^{t}.

(e). For all i ∈ I and t ∈ ℤ ∩ [s_{i−1}, s_{i}], if the event E_{1} ∩ E_{2} occurs then d(x, W_{i}^{t}) ≥ \frac{1}{256}δ_{ε}(x, y)R.

Proof. For the first assertion, fix u such that 2^{u} ∈ [R/8, 5R/4]. Since δ_{ε}(x, y) ≤ ε(x, u), and P_{u} is chosen from β(u) which is ε(x, u)-padded, Pr[d(x, X \ P_{u}(x)) ≥ δ_{ε}(x, y)2^{u}] ≥ \frac{1}{2}. In addition, Pr[σ_{u}(P_{u}(x)) = 1] = Pr[σ_{u}(P_{u}(x)) = 0] = 1/2. Furthermore, the number of relevant values of u in each of the events E_{1}, E_{2}, E_{3} is at most four for each event, and the outcomes for different values of u are mutually independent. This implies assertion (a).

To prove the second and third assertions note that for 2^{u} ∈ [R/8, 5R/4], we always have diam(P_{u}(x)) ≤ \frac{5R}{4} < \frac{1}{2} d(x, y), and thus P_{u}(x) ≠ P_{u}(y). Furthermore, for every z ∈ B(y, \frac{1}{512}δ_{ε}(x, y)R), we always have d(x, z) ≥ 3R > diam(P_{u}(x)) + diam(p_{u}(z)), thus P_{u}(x) ≠ P_{u}(z), and the choices of σ_{u}(P_{u}(x)) and σ_{u}(P_{u}(z)) are independent. It follows that the value of σ_{u}(P_{u}(x)) is independent of the data determining whether d(y, W_{t}^{i}) < \frac{1}{512}δ_{ε}(x, y)R, which proves (b). Assertion (c) follows similarly observing that σ_{u}(P_{u}(x)) is independent also of the value of d(x, X \ P_{u}(x)).

To prove the last two assertions fix i ∈ I and t ∈ ℤ ∩ [s_{i−1}, s_{i}]. An application of Claim 2.1 to z = x shows that 2^{κ(x,t)−i} ∈ [R/8, 5R/4]. Now, by the construction of W_{t}^{i}, if E_{3} occurs then x ∈ W_{t}^{i}; this proves assertion (d). Finally, fix any z ∈ B (x, \frac{1}{256}δ_{ε}(x, y)R). Since z ∈ B(x, R/256), Claim 2.1 implies that 2^{κ(z,t)−i} ∈ [R/8, 5R/4]. The event E_{1} implies that for u = κ(z, t) − i, P_{u}(x) = P_{u}(z), and thus σ_{κ(z,t)−i}(P_{κ(z,t)−i}(z)) = σ_{κ(z,t)−i}(P_{κ(z,t)−i}(x)). Now E_{2} implies that the latter quantity is 1, and hence z ∉ W_{t}^{i}. Assertion (e) follows.

We can now conclude the proof of Lemma 1.8. Fix i ∈ I and t ∈ [s_{i−1}, s_{i}]. By assertions (d) and (e), if either of the (disjoint) events E_{3} ∩E_{i,t}^{big} and E_{1} ∩E_{2} ∩E_{i,t}^{small} occurs then |d(x, W_{t}^{i})−d(y, W_{t}^{i})| ≥ \frac{1}{512}δ_{ε}(x, y)R. The probability of this is Pr[E_{3}] · Pr[E_{i,t}^{big}] + Pr[E_{2}] · Pr[E_{1} ∩ E_{i,t}^{small}] ≥ 2^{−4} Pr[E_{1}] = Ω(1), where we have used assertions (a), (b) and (c), and the fact that E_{i,t}^{big} ∪ (E_{1} ∩ E_{i,t}^{small}) ⊇ E_{1}. It follows that 𝔼|d(x, W_{t}^{i}) − d(y, W_{t}^{i})|^{p} = [Ω(δ_{ε}(x, y) · d(x, y))]^{p}, and hence

𝔼∥f(x) − f(y)∥_{p}^{p} ≥ ∑_{i∈I} ∑_{t∈ℤ∩[s_{i−1},s_{i}]}𝔼|d(x, W_{t}^{i}) − d(y, W_{t}^{i})|^{p}

≥ [Ω(δ_{ε}(x, y) · d(x, y))]^{p} ∑_{i=−6}^{3} |Z ∩ [s_{i−1}, s_{i}]|
≥ [Ω(δ_{ε}(x, y) · d(x, y))]^{p} · \frac{s_{3} − s_{−7}}{2}  (7)
≥ [Ω(δ_{ε}(x, y) · d(x, y))]^{p} · V_{μ}(x, y),

where in (7) we used the fact that (6) implies that s_{3} − s_{−7} ≥ 1.

This completes the proof of Lemma 1.8.

Remark 2.1. The above proof actually yields an embedding φ, such that for all x, y ∈ X satisfying (6),

[log \frac{μ(B(x, 2d(x, y)))}{μ(B(x, d(x, y)/512))}]^{1/p} · δ_{ε}(x, y) ≤ \frac{||φ(x) − φ(y)||_{p}}{d(x, y)} ≤ C [log Φ(μ)]^{1/p} .

Remark 2.2. If in the above proof we use sampling and a standard Chernoff bound instead of taking expectations, we can ensure that the embedding takes values in ℝ^{k}, where k = O[(log n) log Φ(μ)]. (This is because the lower bound on 𝔼|d(x, W_{t}^{i}) − d(y, W_{t}^{i})|^{p} relies on an event that happens with constant probability, similar to [27].) In particular, when μ is the counting measure on X we get that k = O[(log n)^{2}]. It would be interesting to improve this bound to k = O(log n) (if p = 2 then this follows from the Johnson-Lindenstrauss dimension reduction lemma [18]).

2.2 Optimal volume-respecting embeddings

Here we prove Theorem 1.5. Let f : X → ℓ_{2} be the embedding constructed in the previous section and g = f/\sqrt{50 log Φ(μ)}, so that g is 1-Lipschitz. For concreteness, we denote by (Ω, Pr) the probability space over which the random embedding g is defined.

Lemma 2.3. Fix a subset Y ⊆ X, x ∈ X \ Y and let y_{0} ∈ Y satisfy d(x, Y) = d(x, y_{0}). Let Z be any ℓ_{2} valued random variable on (Ω, Pr) which is measurable with respect to the sigma algebra generated by the random variables {g(y)}_{y∈Y}. Then

\frac{\sqrt{𝔼||Z − g(x)||_{2}^{2}}}{d(x, Y)} ≥ C · δ_{ε}(x, y_{0}) · \sqrt{\frac{1}{log Φ(μ)} ⌊log (\frac{μ(B(x, 2d(x, y_{0})))}{μ(B(x, d(x, y_{0})/512))})⌋},

where C is a universal constant.

Proof. We use the notation of Section 2.1. Denote R = \frac{1}{4}d(x, y_{0}) and write Z = (Z_{t}^{i} : i ∈ I, t ∈ J). Consider the events Ẽ_{i,t}^{big} = \{Z_{t}^{i} ≥ \frac{1}{512}δ_{ε}(x, y_{0})R\} and Ẽ_{i,t}^{small} = \{Z_{t}^{i} < \frac{1}{512}δ_{ε}(x, y_{0})R\}. Arguing as in Section 2.1, it is enough to check that E_{3} is independent of Ẽ_{i,t}^{big} and that E_{2} is independent of E_{1} ∩ Ẽ_{i,t}^{small}. Observe that the proof of assertions (b) and (c) in Claim 2.2 uses only the fact that d(x, y) ≥ 4R (when considering z ∈ B(y, \frac{1}{512}δ_{ε}(x, y)R)), and this now holds for all y ∈ Y. Since we assume that Z_{t}^{i} is measurable with respect to {d(y, W_{t}^{i})}_{y∈Y}, the required independence follows.

We remark that we will apply Lemma 2.3 to random variables of the form Z = ∑_{y∈Y} c_{y}g(y), where the c_{y}’s are scalars. However, the same statement holds for Z’s which are arbitrary functions of the variables {g(y)}_{y∈Y}.

The following lemma is based on a Bourgain-style embedding.

Lemma 2.4. Define a subset S ⊆ X by S = {x ∈ X : |B(x, R)| ≤ e|B(x, R/2)|}. Then there exists a 1-Lipschitz map F : X → L_{2} such that if x ∈ S and Y ⊆ X with d(x, Y) ≥ R, then

d_{L_{2}} (F (x), span\{F (y)\}_{y∈Y}) ≥ \frac{C′R}{\sqrt{log n}},

where C′ > 0 is a universal constant.

Proof. For each t ∈ {1, 2, …, ⌈log n⌉}, let W_{t} ⊆ X be a random subset which contains each point of X independently with probability e^{−t}. Let g_{t}(x) = min{d(x, W_{t}), R/4} and define the random map f = \frac{1}{\sqrt{⌈log n⌉}} (g_{1} ⊕ ⋯ ⊕ g_{⌈log n⌉}) so that ||f||_{Lip} ≤ 1. Finally, we define F : X → L_{2}(ν) by

F (x) = f(x), where ν is the distribution over which the random subsets {W_{t}} are defined.

Now fix x ∈ S and let t ∈ ℕ be such that e^{t} ≤ |B(x, R/2)| ≤ e^{t+1}. Let E_{far} be the event {d(x, W_{t}) ≥ R/4} and let E_{close} be the event {d(x, W_{t}) ≤ R/8}. Clearly both such events are independent of the values {g_{t}(y) : d(x, y) ≥ R} ⊇ {g_{t}(y)}_{y∈Y} (this relies crucially on the use of min{·, R/4} in the definition of g_{t}). Fixing c_{y} ∈ ℝ for each y ∈ Y, we see that

∥F (x) − ∑_{y∈Y} c_{y}F (y)∥_{L_{2}(ν)}^{2} = 𝔼_{ν} (f(x) − ∑_{y∈Y} c_{y}f(y))^{2}

≥ \frac{1}{2 log n}𝔼_{ν} (g_{t}(x) − ∑_{y∈Y} c_{y}g_{t}(y))^{2}

= Ω (\frac{R^{2}}{log n} · min \{ν(E_{far}), ν(E_{close})\}) .

Finally, we observe that by the definition of S, ν(E_{far}) and ν(E_{close}) can clearly be lower bounded by some universal constant.

Proof of Theorem 1.5. Using the notation of Lemma 2.3, consider the Hilbert space H = L_{2}(Ω, ℓ_{2}), i.e. the space of all square integrable ℓ_{2} valued random variables ζ on Ω equipped with the Hilbertian norm ∥ζ∥_{2} = \sqrt{𝔼∥ζ∥_{2}^{2}}. Defining G : X → H via G(x) = g(x), Lemma 2.3 implies that for every Y ⊆ X and x ∈ X \ Y,

\frac{d_{H}(G(x), span(\{G(y)\}_{y∈Y}))}{d(x, Y)} ≥ C · δ_{ε}(x, y_{0}) · \sqrt{\frac{1}{log Φ(μ)} ⌊log (\frac{μ(B(x, 2d(x, y_{0})))}{μ(B(x, d(x, y_{0})/512))})⌋}. (8)

We now argue as in the proof of Theorem 1.4. Let H_{1}, H_{2} be Hilbert spaces and G_{1} : X → H_{1}, G_{2} : X → H_{2} be two 1-Lipschitz mappings satisfying for every Y ⊆ X and x ∈ X \ Y,

\frac{d_{H_{1}}(G_{1}(x), span(\{G_{1}(y)\}_{y∈Y}))}{d(x, Y)}

≥ \frac{C}{16 + 16 log (\frac{|B(x,2d(x,y_{0}))|}{|B(x,d(x,y_{0})/512)|})}\sqrt{\frac{1}{log n} ⌊log (\frac{|B(x, 2d(x, y_{0}))|}{|B(x, d(x, y_{0})/512)|})⌋},

and

\frac{d_{H_{2}}(G_{2}(x), span(\{G_{2}(y)\}_{y∈Y}))}{d(x, Y)} ≥ \frac{C}{α_{X}} \sqrt{\frac{1}{log n} ⌊log (\frac{|B(x, 2d(x, y_{0}))|}{|B(x, d(x, y_{0})/512)|})⌋},

where d(x, y_{0}) = d(x, Y), and we used (8) with μ being the counting measure on X. Also, by Lemma 2.4 there is a Hilbert space H_{3} and a mapping G_{3} : X → H_{3} such that for every such Y, x, y_{0}

\frac{d_{H_{3}}(G_{3}(x), span(\{G_{3}(y)\}_{y∈Y}))}{d(x, Y)} ≥ \frac{C′}{\sqrt{log n}} · 1_{\{|B(x,d(x,y_{0})/2)|≤e|B(x,d(x,y_{0})/8)|\}}.

Denoting H = H_{1} ⊕ H_{2} ⊕ H_{3} and G = \frac{1}{\sqrt{3}}(G_{1} ⊕ G_{2} ⊕ G_{3}), similar argument as in the proof of Theorem 1.4 implies that

\frac{d_{H}(G(x), span(\{G(y)\}_{y∈Y}))}{d(x, Y)} ≥ Ω (\frac{1}{\sqrt{α_{X} log n}}) .

Now, Feige’s argument (see the proof of Lemma 15 in [12]) yields the required result.

Remark 2.3. Note that, by general dimension reduction techniques which preserve distance to affine hulls [30], the dimension of the above embedding can be reduced to O(k log n) while maintaining the volume-respecting property for k-point subsets.

3 Low-dimensional embeddings of planar metrics

In this section we refine the ideas of the previous section and prove Theorem 1.6. We say that a metric (X, d) is planar (resp. excludes K_{s,s} as a minor) if there exists a graph G = (X, E) with positive edge weights, such that G is planar (resp. does not admit the complete bipartite graph K_{s,s} as a minor) and d(·, ·) is the shortest path metric on a subset of G. We shall obtain optimal low-dimensional embeddings of planar metrics into ℓ_{∞} by proving the following more general result.

Theorem 3.1. Let (X, d) be an n-point metric space that excludes K_{s,s} as a minor. Then X embeds into ℓ_{∞}^{O(3^{s}(log s) log n)} with distortion O(s^{2}).

We will need three lemmas. The first one exhibits a family of decompositions with respect to a diameter bound Δ > 0; it follows easily from [19], with improved constants due to [11]. Note that in contrast to Definition 1.1 (and also to Rao’s embedding [33]), we require that x and y are padded simultaneously.

Lemma 3.2. There exists a constant c such that for every metric space (X, d) that excludes K_{s,s} as a minor, and for every Δ > 0, there exists a set of k = 3^{s} partitions P_{1}, …, P_{k} of X, such that

1. For every C ∈ P_{i}, diam(C) < Δ.

2. For every pair x, y ∈ X, there exists an i such that for T = cs^{2},

B(x, Δ/T) ⊆ P_{i}(x) and B(y, Δ/T) ⊆ P_{i}(y).

Proof. Fix a edge-weighted graph G that does not admit K_{s,s} as a minor and whose shortest path distance is d(·, ·). Fix also some x_{0} ∈ X and δ > 0. For i ∈ {0, 1, 2} and j ∈ {0} ∪ ℕ define:

A_{j}^{i} = \{x ∈ X : 9(j − 1) + 3i ≤ \frac{d(x, x_{0})}{δ} < 9j + 3i\} .

For every i, P^{i} = {A_{j}^{i}}_{j≥0} clearly forms a partition of X. Let us say that a subset S ⊆ X cuts a subset S′ ⊆ X if S ∩ S′ ≠ ∅ and S′ ⊈ S. Observe that for every x ∈ X at most one of the sets {A_{j}^{i} : i = 0, 1, 2 ; j = 0, 1, 2, …} cuts B(x, δ), as otherwise there exist z_{1}, z_{2} ∈ B(x, δ) for which d(z_{1}, z_{2}) ≥ d(x_{0}, z_{1}) − d(x_{0}, z_{2}) ≥ 3δ. Thus, for every x, y ∈ X, for one of the partitions P^{0}, P^{1}, P^{2} both B(x, δ) and B(y, δ) are contained in one of its clusters. For each cluster C of the partitions P^{0}, P^{1}, P^{2}, consider the subgraph of G induced on the points of C, partition C into its connected components, and apply the above process again to each such connected component. Continuing this way a total of s times, we end up with 3^{s} partitions, and in at least one of them, neither B(x, δ) nor B(y, δ) is cut. The results of [19, 11] show there exists a constant c > 0 such that the diameter of each cluster in the resulting partitions is at most cs^{2}δ, and the lemma follows by setting δ = Δ/(cs^{2}).

We next consider a collection of such decompositions, with diameter bounds Δ > 0 that are proportional to the integral powers of 4T. Furthermore, we need these decompositions to be nested.

Lemma 3.3. Let (X, d) be a metric space that excludes K_{s,s} as a minor, and let T = O(s^{2}) be as in Lemma 3.2. Then for every a > 0 there exists k = 3^{s} families of partitions of X, {P_{u}^{i}}_{u∈ℤ}, i = 1, …, k with the following properties:

1. For each i the partitions {P_{u}^{i}}_{u∈ℤ} are nested, i.e. P_{u−1}^{i} is a refinement of P_{u}^{i} for all u.

2. For each i, every C ∈ P_{u}^{i} satisfies diam(C) < a(4T)^{u}.

3. For each u ∈ ℤ and every pair x, y ∈ X, there exists an i such that,

B(x, a(4T)^{u}/(2T)) ⊆ P_{u}^{i}(x) and B(y, a(4T)^{u}/(2T)) ⊆ P_{u}^{i}(y).

Proof. Let P_{1}, …, P_{k} be partitions as in Lemma 3.2 with Δ = a(4T)^{u} and let Q_{1}, …, Q_{k} be partitions as in Lemma 3.2 with Δ = a(4T)^{u−1}. Fix j and C ∈ P_{j}, let S_{C} = {A ∈ Q_{j} : A ∩ C ≠ ∅, but A ⊈ C}, and replace every C ∈ P_{j} by the sets A ∈ S_{C} and the set C′ = C \ ⋃_{A∈S_{C}} A. Continuing this process we replace the partition P_{j} by a new partition P′_{j} such that Q_{j} is a refinement of P′_{j}. Note that we do not alter Q_{j}. Since diam(A) ≤ a(4T)^{u−1}, we have that if C ∈ P_{j} and B(x, a(4T)^{u}/T) ⊆ C, then B(x, 2a(4T)^{u−1}) ⊆ C′. Continuing this process inductively we obtain the required families of nested partitions.

We next use a nested sequence of partitions {P_{u}}_{u∈ℤ} to form a mapping ψ : X → ℝ^{O(log |X|)}.

Lemma 3.4. Let {P_{u}}_{u∈ℤ} be a sequence of partitions of X that is nested (i.e. P_{u−1} is a refinement of P_{u}), and let m ≥ 0 and D ≥ 2 be such that for all C ∈ P_{u}, diam(C) < 2^{m}D^{u}. Assume further that P_{u_{1}} = {X}, P_{u_{2}} = {{x} : x ∈ X}. Then for all u_{2} ≤ u ≤ u_{1} and all A ∈ P_{u} there exists a mapping ψ : A → ℝ^{2⌈log_{2} |A|⌉} that satisfies:

(a). For every x ∈ A and every 1 ≤ j ≤ 2⌈log_{2} |A|⌉ there exists u′ < u for which |ψ(x)_{j}| = min{d(x, X \ P_{u′}(x)), 2^{m}D^{u′}},

(b). For all x, y ∈ A, ||ψ(x) − ψ(y)||_{∞} ≤ 2 d(x, y),

(c). If x, y ∈ A are such that for some u′ ≤ u, d(x, y) ∈ [2^{m}D^{u′−1}, 2^{m+1}D^{u′−1}) and there exists a cluster C ∈ P_{u′} for which x, y ∈ C, B(x, 2^{m+1}D^{u′−2}) ⊆ P_{u′−1}(x) and B(y, 2^{m+1}D^{u′−2}) ⊆

P_{u′−1}(y), then ||ψ(x) − ψ(y)||_{∞} ≥ \frac{d(x,y)}{2D}.

Proof. Proceed by induction on u. The statement is vacuous for u = u_{2}, so we assume it holds for u and construct the required mapping for u + 1. Fix A ∈ P_{u+1} and assume that H = {A_{1}, …, A_{r}} ⊆ P_{u} is a partition of A. By induction there are mappings ψ_{i} : A_{i} → ℝ^{2⌈log_{2} |A_{i}|⌉} satisfying (a)-(c) above (with respect to A_{i} and u).

For h ∈ ℕ denote C_{h} = {A_{i} ∈ H : 2^{h−1} < |A_{i}| ≤ 2^{h}}. We claim that for every i = 1, …, r there is a choice of a string of signs σ^{i} ∈ {−1, 1}^{2⌈log_{2} |A|⌉−2⌈log_{2} |A_{i}|⌉} such that for all h and for all distinct A_{i}, A_{j} ∈ C_{h}, σ^{i} ≠ σ^{j}. Indeed, fix h; if h ≥ log_{2} |A| then for A_{i} ∈ C_{h}, |A_{i}| > 2^{h−1} ≥ |A|/2; thus |C_{h}| = 1 and there is nothing to prove. So, assume that h < log_{2} |A| and note that |C_{h}| ≤ |A|/2^{h−1}. Hence, the required strings of signs exist provided 2^{2⌈log_{2} |A|⌉−2h} ≥ |A|/2^{h−1}, which is true since h + 1 ≤ ⌈log_{2} |A|⌉.

Now, for every i = 1, …, r define a mapping ζ_{i} : A_{i} → ℝ^{2⌈log_{2} |A|⌉−2⌈log_{2} |A_{i}|⌉} by

ζ_{i}(x) = min{d(x, X \ A_{i}), 2^{m}D^{u}} · σ^{i}.

Finally, define the mapping ψ : X → ℝ^{2⌈log_{2} |A|⌉} by ψ|_{A_{i}} = ψ_{i} ⊕ ζ_{i}. Requirement (a) holds for ψ by construction. To prove requirement (b), i.e. that ψ is 2-Lipschitz, fix x, y ∈ A. If for some i, both x, y ∈ A_{i} then by the inductive hypothesis ψ_{i} is 2-Lipschitz, and clearly ζ_{i} is 1-Lipschitz, so ∥ψ(x) − ψ(y)∥_{∞} ≤ 2d(x, y). Otherwise, fix a coordinate 1 ≤ j ≤ 2⌈log_{2} |A|⌉ and use (a) to take u′ ≤ u such that |ψ(x)_{j}| = d(x, X \ P_{u′}(x)); since y ∉ P_{u′}(x), this is at most d(x, y). It similarly follows that |ψ(y)_{j}| ≤ d(x, y), and hence |ψ(x)_{j} − ψ(y)_{j}| ≤ 2d(x, y).

To prove that requirement (c) holds for ψ, take x, y ∈ A and u′ ≤ u + 1 such that d(x, y) ∈ [2^{m}D^{u′−1}, 2^{m+1}D^{u′−1}) and there exists a cluster C ∈ P_{u′} for which x, y ∈ C, B(x, 2^{m+1}D^{u′−2}) ⊆ P_{u′−1}(x) and B(y, 2^{m+1}D^{u′−2}) ⊆ P_{u′−1}(y). The case u′ ≤ u follows by induction, so assume that u′ = u + 1. Let i, j ∈ {1, …, r} be such that x ∈ A_{i}, y ∈ A_{j}; then i ≠ j, since diam(A_{i}) < 2^{m}D^{u} ≤ d(x, y). Assume first ⌈log_{2} |A_{i}|⌉ ≠ ⌈log_{2} |A_{j}|⌉, and without loss of generality suppose ⌈log_{2} |A_{i}|⌉ < ⌈log_{2} |A_{j}|⌉; then there is a coordinate ℓ = 2⌈log_{2} |A_{i}|⌉ + 1 for which

|ψ(x)_{ℓ}| = |ζ_{i}(x)_{1}| = min{d(x, X \ A_{i}), 2^{m}D^{u}},

and, for some u′′ < u,

|ψ(y)_{ℓ}| = |ψ_{j}(y)_{ℓ}| = min{d(y, X \ P_{u′′}(y)), 2^{m}D^{u′′}}.

It follows that |ψ(x)_{ℓ}| ≥ 2^{m+1}D^{u−1} (since we assumed B(x, 2^{m+1}D^{u−1}) ⊆ A_{i}), and that |ψ(y)_{ℓ}| ≤ 2^{m}D^{u−1}, and therefore

|ψ(x)_{ℓ} − ψ(y)_{ℓ}| ≥ 2^{m}D^{u−1} ≥ \frac{d(x, y)}{2D}.

It remains to deal with the case ⌈log_{2} |A_{i}|⌉ = ⌈log_{2} |A_{j}|⌉. By our choice of sign sequences, in this case there is an index ℓ for which σ_{ℓ}^{i} ≠ σ_{ℓ}^{j}, and thus, for ℓ′ = ℓ + 2⌈log_{2} |A_{i}|⌉, |ψ(x)_{ℓ′} − ψ(y)_{ℓ′}| = |ψ(x)_{ℓ′}| + |ψ(y)_{ℓ′}|. Since we assumed B(x, 2^{m+1}D^{u−1}) ⊆ A_{i} and B(y, 2^{m+1}D^{u−1}) ⊆ A_{j}, we get

|ψ(x)_{ℓ′} − ψ(y)_{ℓ′}| ≥ 2^{m+2}D^{u−1} ≥ \frac{2d(x, y)}{D}.

Finally, we prove the main result of this section by a concatenating several of the above maps ψ.

Proof of Theorem 3.1. For each m ∈ {0, 1, …, ⌈log_{2}(4cs^{2})⌉} set a = 2^{m}, apply Lemma 3.3 to obtain 3^{s} families of nested partitions {P_{1,u}^{m}}_{u∈ℤ}, …, {P_{3^{s},u}^{m}}_{u∈ℤ} that satisfy the conclusion of Lemma 3.3 with T = cs^{2}. For every i = 1, …, 3^{s}, let ψ_{i}^{m} be the mapping that Lemma 3.4 yields for {P_{i,u}^{m}}_{u∈ℤ} when setting A = X and D = 4cs^{2}. Consider the map Ψ = ⊕_{m,i}ψ_{i}^{m}, which takes values in ℓ_{∞}^{O(3^{s}(log s) log n)}. Clearly Ψ is 2-Lipschitz. Moreover, for every x, y ∈ X there is m ∈ {0, 1, …, ⌈log_{2}(4cs^{2})⌉} and u ∈ ℤ such that d(x, y) ∈ [2^{m}D^{u}, 2^{m+1}D^{u}). By Lemma 3.3, there is i ∈ {1, …, 3^{s}} for which B(x, 2^{m+1}D^{u−1}) ⊆ P_{u}(x) and B(y, 2^{m+1}D^{u−1}) ⊆ P_{u}(y); it then follows using Lemma 3.4 that

∥Ψ(x) − Ψ(y)∥_{∞} ≥ ∥ψ_{i}^{m}(x) − ψ_{i}^{m}(y)∥_{∞} = Ω(d(x, y)/s^{2}),

as required.

Acknowledgments

The third author is grateful to Y. Bartal for discussions on related issues during a preliminary stage of this work.

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