Quasisymmetric embeddings, the observable diameter, and expansion properties of graphs

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1 Quasisymmetric embeddings, the observable diameter, and expansion properties of graphs Assaf Naor Yuval Rabani Alistair Sinclair Abstract It is shown that the edges of any n-point vertex expander can be replaced by new edges so that the resulting graph is an edge expander, and such that any two vertices that are joined by a new edge are at distance O log n) in the original graph. This result is optimal, and is shown to have various geometric consequences. In particular, it is used to obtain an alternative perspective on the recent algorithm of Arora, Rao and Vazirani [] for approximating the edge expansion of a graph, and to give a nearly optimal lower bound on the ratio between the observable diameter and the diameter of doubling metric measure spaces which are quasisymmetrically embeddable in Hilbert space. Introduction Expansion properties of graphs are a fundamental tool in modern combinatorics. Questions related to expansion have found deep connections to numerous mathematical disciplines, such as number theory, Lie groups, measure theory, geometry and topology, mixing times of Markov chains, derandomization and coding theory. The various forms of graph expansion can be viewed as discrete analogs of isoperimetery, and are thus intimately related to classical analytic concepts. From a computational point of view, the Sparsest Cut Problem, which involves calculating the edge expansion of a graph, is a well known NP-hard problem, and hence not solvable in polynomial time unless P = NP). Whether it is possible to efficiently compute a good approximation to the edge expansion is arguably one of the most important outstanding questions in the field of approximation algorithms. A recent breakthrough in this direction, due to Arora, Rao and Vazirani [], yields a polynomial time algorithm which computes the edge expansion of an n-vertex graph within a factor of O log n). Previously the best known approximation guarantee had been Olog n) [].) The present paper builds on the remarkable ideas of [] to obtain new structural information on the relation between edge expansion and vertex expansion, which is shown to have applications to the theory of quasisymmetric embeddings. Additionally, we highlight a new perspective on the results of [] which we believe is at the core of the phenomenon discovered there. Specifically, we Theory Group, Microsoft Research. One Microsoft Way, Redmond WA anaor@microsoft.com Computer Science Department, Technion Israel Institute of Technology, Haifa 3000, Israel. rabani@cs.technion.ac.il. This work was done while the author was on sabbatical leave at Cornell University. Part of this work was done while the author was visiting Microsoft Research, Redmond. Computer Science Division, University of California, Berkeley CA , U.S.A. sinclair@cs.berkeley.edu. Supported in part by NSF ITR grant CCR Part of this work was done while the author was on sabbatical leave at Microsoft Research, Redmond.

2 formulate a geometric fact which implies the main results of [] without using negative type also known as squared L ) triangle inequality conditions see below for a definition). While the negative type condition is natural from the point of view of semidefinite programming, we find it to be an unnatural geometric assumption. Although the proofs in [] use this condition in an essential way, we show that the results of [] are actually based on a purely Euclidean geometric fact.. Vertex expansion, edge expansion, and the edge replacement theorem We begin by recalling some classical definitions. In what follows all graphs are unweighted, and allowed to have multiple edges and self loops. We shall use the following notation. Given a graph G =V, E) we denote by d G, ) the shortest path metric induced by G on V. For S V, its neighborhood in G is defined as N G S) =v V : d G v, S) = }. Given S, T V, es, T ) denotes the number of edges which intersect both S and T. Definition. Vertex expansion). Let G =V, E) be a graph. Its vertex expansion hg) is defined to be the largest constant h such that for every S V with S V / we have N G S) h S. Definition. Edge expansion). Let G =V, E) be a graph. The edge expansion of G, denoted αg), is the largest constant α such that for every S V with S V / we have es, V \ S) α S E. V These two notions of expansion play a central role in modern combinatorics. It is clear that for a graph G =V, E) of bounded average degree i.e., E = O V )), a lower bound on hg) implies a lower bound on αg). For graphs of unbounded average degree the same statement is clearly false in general. The main combinatorial result of this paper is the following: Theorem.3 Edge replacement theorem). There are absolute constants c, C > 0 with the following properties. For every n-vertex graph G =V, E) with hg), there is a set of edges E on V satisfying:. For every u, v} E, d G u, v) C log n.. αv, E ) c. On the other hand, there are arbitrarily large n-vertex graphs G =V, E) with hg) such that, for every c>0 and every set of edges E on V for which αv, E ) c, there is some u, v} E satisfying d G u, v) 0 c log n. Given a graph G =V, E) and r, denote by G r =V, E r ) the graph whose edges are E r = u, v} : d G u, v) r}. It is a standard fact that h G ) hg). Therefore, an immediate corollary of Theorem.3 is that the same result holds for arbitrary graphs, with the upper bound on the length of edges in E replaced by C hg) log n. The proof of Theorem.3 has two components: a geometric argument, presented in Section 3., which establishes the existence of a new edge set for which every large enough subset of the vertices has the appropriately large edge boundary, and a combinatorial argument, presented in Section 3., which takes care of the edge expansion of small subsets. The geometric component can be formally

3 proved via a duality argument presented in Section 3.) using the main result of [] as a black box ; however, for the purposes of Theorem.3 it turns out that it is possible to use a simpler proof than that of [], which is nevertheless strongly based on their ideas. On the other hand, as we shall show in Section, Theorem.3 is easily seen to have powerful geometric consequences. Firstly, it actually readily implies the geometric fact from [] see also [0]) that lies at the heart of the approximation algorithm for sparsest cut given in []; we present this fact, and explain its algorithmic role, in the next subsection. Secondly, as we discuss in Section.3 below, it gives a nearly optimal lower bound on the observable diameter of doubling metric measure spaces which are quasisymmetrically equivalent to subsets of Hilbert space.. The relation to algorithmic graph partitioning As stated above, the present paper is motivated by the recent algorithm of Arora, Rao and Vazirani [] which, given an n-vertex graph G =V, E), approximates in polynomial time its edge expansion up to a factor of O log n). In this subsection we explain how Theorem.3 leads to an alternative proof of the key geometric result of []; for the convenience of readers not familiar with [], we also indicate how this result gives an approximation algorithm for edge expansion. Let G be an n-vertex graph and define β = min S V S n/ es, V \ S). S Take S V with S n/ and es, V \ S) =β S. For every v V set x v = if v S and x v = otherwise. Then x u x v =4 S n S ) S n, and u,v} E x u x v = 4eS, V \ S) = 4β S. Moreover, since x v, }, we have for every u, v, w V, x v x u x v x w + x w x u. Hence, by normalization, if we let β be the minimum of n u,v} E z u z v over all choices of z,..., z n S n the unit Euclidean sphere in R n ) satisfying n z u z v = and, for all u, v, w V, z v z u z v z w + z w z u, then β β. The advantage of passing to β is that such a semidefinite minimization problem can be solved in polynomial time up to an arbitrarily small additive error) using the ellipsoid algorithm see, e.g., [8] for details on semidefinite programming). Hence, we can efficiently produce vectors z v S n satisfying the above constraints such that u,v} E z u z v + o))β n. Now, as we shall see below, there exists a universal constant c>0 such that β c log n β. ) Thus, it is possible to evaluate β within a factor of O log n) in polynomial time. This algorithm is one of the main results of []. Let z,,z n be a set of vectors as above such that u,v} E z u z v β n, and for u, v V denote du, v) = z u z v. Our constraints imply that V, d) is a metric space. Such metrics are commonly known as metrics of negative type, or squared L metrics.) Let diamv )= max du, v) be the diameter of V. The key geometric fact from [] that is used to deduce ) is the following: In addition, [] also gives an algorithm for finding a subset S V that achieves the desired approximation. 3

4 Theorem.4. Let V, d) be an n-point metric space of negative type with diameter. Assume that n du, v) for some > 0. Then there are A, B V with A, B κ 6n and da, B) log n, where κ> 0 depends only on. We will show in Section how to deduce Theorem.4 fairly painlessly from Theorem.3; in fact, we will deduce much more general versions Theorems.4 and.5) that apply to all metrics that are uniformly embeddable and all metrics that are quasisymmetrically embeddable in Hilbert space see Section.3 below for definitions). Thus the property in Theorem.4 is quite general and not special to metrics of negative type. For completeness, we now indicate how to derive ) from Theorem.4; here we are essentially repeating the argument of []. Let Bv, r) denote the open ball of radius r centered at v, i.e., Bv, r) =u V : du, v) <r}. Assume first that for every v V, Bv, /8) n/8. Since n du, v) = there is some vertex w V for which Bw, ) n/4. Moreover, by our assumption we have that Bw, ) u,v Bw,) du, v) Bw, ) Bw, ) u Bw,) v Bw,)\Bu,/8) Bw, ) n 8 ) 6. du, v) Hence, by Theorem.4 there are universal constants a, b 0, /) and A, B V with A, B an and da, B) > b/ log n. For t [0, b/ log n] define S t = v V : dv, A) t}. Then for all t, an S t a)n. Moreover, by a simple computation, for every u, v V, implying that log n b b/ log n 0 b/ log n 0 u,v} E St u) St v) dt du, A) dv, A) du, v), St u) St v) log n dt b We deduce that there is some t [0, b/ log n] for which u,v} E u,v} E St u) St v) = es t,v \ S t ) log n b z u z v log n b β n. β n. Since V \ S t n/, es t,v \ S t ) β V \ S t βan. We conclude that β ab log n β. It remains to deal with the case in which there exists w V such that Bw, /8) > n/8. Since n du, v) =, and diamv ) =, there are at least n / pairs u, v) V V for which du, v). By the triangle inequality, for such pairs u, v) we have maxdu, w),dv, w)} /4, so that V \ Bw, /4) n/. Setting A = Bw, /8) and B = V \ Bw, /4), we have da, B) /8 and A, B n/8, so we are again in the situation of the above argument in this case we actually get that β = Oβ )). 4

5 .3 Uniform and quasisymmetric embeddings and the observable diameter A metric measure space is a triple X, d, µ) consisting of a metric space X, d) and a Borel probability measure µ on X. Let Bx, r) denote the open ball of radius r centered at x and, for A X and ε> 0, define A ε = x X : dx, A) <ε}. In what follows all subsets of metric spaces are assumed to be Borel measurable. The measure µ is said to be doubling with constant λ if for every x X and r>0, µbx, r)) λµbx, r)). The isoperimetric function of µ is defined as: I X,d) µ ε) = sup µx \ A ε ): µa) Following M. Gromov see [7] and the references therein) we recall the notion of observable diameter of a metric measure space: the observable diameter of X, d, µ) with parameter κ> 0, denoted Obs µ X, d; κ), is defined by }. Obs µ X, d; κ) = supε > 0: I X,d) µ ε) κ}. The motivation for this nomenclature is as follows. Assume that we are trying to measure the size of X, d, µ). We make observations which consist of real valued -Lipschitz functions on X, i.e. we assign to each point of X a real number in a Lipschitz smooth way. We plot the distribution of these observations, and account for possible observational errors by discarding the part of the distribution which does not belong to a symmetric interval around its median of mass at least κ. The length of this central interval will never exceed Obs µ X, d; κ). Let S d R d be the unit Euclidean sphere, equipped with the standard Euclidean metric, and let σ be the normalized surface area measure on S d. Levy s isoperimetric inequality see, e.g., [3]) states that for every 0 < ε < π/, I Sd, ) σ ε) π 8 e dε /. It follows that for every κ, ) log/κ) Obs σ S d, ; κ) =O, d while the diameter of S d equals. Spaces for which the observable diameter is much smaller than the diameter are sometimes following V. Milman) said to have small isoperimetric constant. In order to state our main geometric result we require the following classical definitions: Definition.5 Uniform embedding). Let X, d X ), Y, d Y ) be metric spaces and α, β : [0, ) [0, ) be strictly increasing functions. A one to one mapping f : X Y is called a uniform embedding with moduli α and β if for every x, y X, αd X x, y)) d Y fx),fy)) βd X x, y)). When the moduli α and β are of the form αt) =Ct and βt) =C L t we say that the embedding f is L-bi-Lipschitz. We now recall the important notion of quasisymmetric embeddings, which was first introduced by Beurling and Ahlfors in [4]. Definition.6 Quasisymmetric embedding). Let X, d X ), Y, d Y ) be metric spaces and η : [0, ) [0, ) a strictly increasing function. A one to one mapping f : X Y is called a quasisymmetric embedding with modulus η if for every x, a, b X such that x b, d Y fx),fa)) d Y fx),fb)) η dx x, a) d X x, b) ). 5

6 Uniform and quasisymmetric embeddings are central notions in modern geometric analysis see [7, 9, 4]). Roughly speaking, bi-lipschitz embeddings preserve distances, while quasisymmetric embeddings preserve thickness of triangles, and hence, in a sense, preserve shape quasisymmetric embeddings are a natural metric analog of quasiconformal mappings). As an example, consider the classical isometric embedding of L, equipped with the metric x y, into L. The image of such an embedding consists of a set X L on which the function x y is a metric. These are just the metrics of negative type as defined in the previous subsection.) This embedding is both uniform with αt) =βt) = t) and quasisymmetric in fact, any uniform embedding with moduli αt),βt) = Θt a ) is clearly a quasisymmetric embedding). Additional examples, showing that the notions of uniform and quasisymmetric embeddings are incomparable, can be found in [9]. Although we formulate our results both for uniform and quasisymmetric embeddings, quasisymmetric embeddings are more natural to work with in the context of studying isoperimetric functions. The significance of the metrics of negative type in [] stems from the fact that they quasisymmetrically embed in Hilbert space; see Section for more details on this point. The following result is deduced from Theorem.3 in Section. It states that up to double logarithmic factors) any non-degenerate metric measure space which is doubling with constant λ and is quasisymmetrically equivalent to a subset of Hilbert space cannot have an observable diameter which is smaller than the observable diameter of the Euclidean sphere of dimension Olog λ). Theorem.7. Let X, d) be a bounded metric space and µ a Borel probability measure on X which is doubling with constant λ> 3, i.e., for every x X and r>0, µbx, r)) λµbx, r). Assume that dx, y)dµx)dµy) diamx) ) X X for some > 0. Let f : X l be a quasisymmetric embedding with modulus η. Then Obs µ X, d; κ) diamx) τ log λ)log log λ) where κ = κ, η) and τ = τ, η) depend only on and η. There is a natural analog of Theorem.7 in the case of uniform embeddings see the remarks at the end of Section ). However, in this case we need some restriction on the diameter of X, since it is typically impossible to scale a uniform embedding without changing its moduli unless, of course, the moduli α and β are both homogeneous of the same order). The geometric consequences of Theorem.3 We begin with the following well known fact, which relates edge expansion to certain Poincaré inequalities. Fact.. Let G =V, E) be a graph. Then for every function f : V L, E u,v} E fu) fv) αg) V fu) fv). 6

7 Proof. We include the standard proof for the sake of completeness. Note that for every S V, E u,v} E S u) S v) = es, V \ S) E S V S ) αg) V = αg) V S u) S v). Fix f : V L. By the the cut-cone representation of the L metric fv ) [5], for every S V there is t S 0 such that for every u, v V, fu) fv) = S V t S S u) S v). Hence E u,v} E fu) fv) = S V S V t S E = αg) V u,v} E t S αg) V S u) S v) S u) S v) fu) fv). We are now in position to prove the second assertion in Theorem.3, i.e., the fact that the result is existentially optimal. Fix an integer d and consider the discrete cube V = 0, } d, equipped with the Hamming metric ρx, y) = i : x i y i }. The vertex isoperimetric inequality for the counting measure on V see []) implies that for every S V with S d, x V : 0 <ρx, S) 0 d} S. It follows that if we define a graph G =V, E) by E = ) } : ρu, v) [, 0 d] then hg). Let E be a set of edges on V for which u, v} V αv, E ) c. By Fact. applied to the identity mapping from V into l d E u,v} E ρu, v) c V ρu, v) = c d d ) d k = cd k. k=0 we get that It follows that there is an edge u, v} E with ρu, v) cd. Since ρu, v) 0 d d G u, v) we deduce that d G u, v) c d c 0 log V, as required. The following result is an immediate consequence of Theorem.3 and Fact. using the fact that l is isometric to a subset of l ): 7

8 Corollary.. Let G =V, E) be an n-vertex graph with hg). Assume that f : V l satisfies n fu) fv). Then there are u, v V with d G u, v) C log n such that fu) fv) c. Here C, c are as in Theorem.3. In order to deduce various geometric results from Corollary. we require the following simple combinatorial fact. Here, and in what follows, given a graph G =V, E) and a subset of the vertices U V, we denote the graph induced by G on U by G[U], i.e., G[U] = U, E U ) ). Lemma.3. Fix 0 <ε 0 and let G =V, E) be a graph such that for every X, Y V satisfying X, Y ε V, d G X, Y ). Then there is U V with U ε) V such that hg[u]). Proof. Construct graphs G = G 0, G =V,E ),..., G k =V k,e k ) as follows. If there exists a set W i V i such that W i V i and N Gi W i ) W i, put G i+ = G i [V i \ W i ]. By construction, when this process terminates hg k ). Define W = k i=0 W i. If W ε V we are done. Otherwise let j be the minimal integer such that W W j > ε V. For X = j i= W i we have that d G X, V \ [X N G X)]) >. By our assumption it follows that V X N G X) <ε V, or N G X) > ε) V X. But k ε) V X N G X) N Gi W i ) < i=0 k i=0 W i = W, or X > ε) 3 V. By the minimality of j, W W j ε V, so that V W j X ε V > ε) V ε V. 3 It follows that > ε) 3 ε, contradicting the fact that ε 0. We now prove a generalization of Theorem.4, which was stated for negative type metrics in Section.. The generalization applies to all metrics that uniformly embed into l. We show that every such metric has two large subsets that are far apart. The main idea of the proof is that if every pair of sufficiently large subsets are close, then the graph connecting pairs of close points contains a large vertex expander by Lemma.3), but then the embedded edge expander constructed by the edge replacement theorem Theorem.3) violates the Poincaré inequality proved in Fact.. Since negative type metrics embed uniformly with moduli αt) =βt) = t, readers who are chiefly interested in the application to sparsest cut may simplify the proof below by specializing to this case. Theorem.4. Let X, d) be an n-point metric space with diameter. Fix > 0 and a uniform embedding f : X l with moduli α and β. Assume that n dx, y). x,y X Then there are A, B X with A, B κ 6n and da, B) log n, where κ = κ, α, β) depends only on, α and β. 8

9 ) Proof. Let c, C be as in Corollary.. We will show that κ = cα/) C β 4 works. Assume the contrary. By translation, without loss of generality fx) β)b n, where B n is the unit Euclidean ball in R n. Define a graph G =X, E) by setting E = x, y} X ) : dx, y) <β cα/) 4 ) } C. log n By the contrapositive assumption for every A, B X with A, B 6 n, d GA, B). By Lemma.3 there is a subset X X with X /6)n such that hg[x ]). Denoting D = x, y) X X : dx, y) } we have that D n. It follows that D X X ) 4 n. So, X fx) fy) n D X X ) α/) α/). 4 x,y X By Corollary. there are x, y X such that fx) fy) cα/) 4, and d G x, y) C log n. It follows that there is k C ) log n and x = x 0,x,..., x k,x k = y} X such that for all i, dx i,x i ) <β cα/) 4 C log n. But, ) cα/) β dx, y) 4 a contradiction. k i= dx i,x i ) <C ) cα/) log n β 4 C log n, Remark.. We note the assumption of unit diameter in Theorem.4. It is easy to see that the proof generalizes to the case of arbitrary diameter, but then the constant κ would depend nontrivially on diamx). This is a manifestation of the fact that uniform embeddings do not in general scale well, and is also the reason we focus mainly on quasisymmetric embeddings see below). For the specific application to the sparsest cut algorithm in [], Theorem.4 is sufficient because the argument makes use only of ratios between distances, and thus is scale-free see Section.). We now present an analog of Theorem.4 which applies to any metric that is quasisymmetrically embeddable in l. As discussed in the remark above, this version has the advantage of being scalefree in the sense that the result holds uniformly in the diameter). Theorem.5. Let X, d) be an n-point metric space and f : X l a quasisymmetric embedding with modulus η. Assume that n dx, y) diamx) x,y X for some > 0. Then there are A, B X with A, B γ diamx) 6n and da, B) log n, where γ = γ, η) depends only on and η. Proof. The proof is similar to the proof of Theorem.4. For c, C as in Corollary. denote ) η c 8η/)+4 γ = ) η c + C, 8η/)+4 9

10 and define a graph G =X, E) by setting E = ) : dx, y) <γ diamx)/ log n }. We x, y} X assume for the sake of contradiction that there are no A, B X with da, B) γ diamx)/ log n and A, B n 6, i.e., for all A, B of this size we have d GA, B). By Lemma.3 there is a subset X X with X /6)n such that hg[x ]). Denoting D = x, y) X X : dx, y) diamx)} we have that D n. It follows that D X X ) 4 n. Fix x 0,y 0 ) D X X ) and x, y X. Since dx 0,x) diamx) dx 0,y 0 ) we have that fx 0 ) fx) η/) fx 0 ) fy 0 ). Similarly, fy 0 ) fy) η/) fx 0 ) fy 0 ), so fx) fy) [η/) + ] fx 0 ) fy 0 ). This shows that fx 0 ) fy 0 ) diamfx)) η/)+ whenever x 0,y 0 ) D X X ). Hence X fx) fy) D X X ) n x,y X diamfx)) η/)+ 8η/) + 4 diamfx)). By Corollary. there are x, y X c such that fx) fy) 8η/)+4 diamfx)), and d G x, y) C log n. It follows that there is k C log n and x = x 0,x,..., x k,x k = y} X such that for all i, dx i,x i ) <γdiamx)/ log n. Consider now an arbitrary pair x,y X. We have fx) fx ) diamfx)) 8η/)+4 fx) fy), so since f is a quasisymmetry with modulus η, or dx,x) dx,y) η c 8η/)+4 Since this is true for all x,y X, But ) dx, y) η dx fx) fy),x) fx ) fx) c c 8η/) + 4, ). Similarly dy,y) can be bounded by the same quantity, so that dx,y ) η η dx, y) η c 8η/)+4 c 8η/)+4 c 8η/)+4 ) ) + ) + dx, y). diamx) =Cγ diamx). Cγ diamx) dx, y) k dx i,x i ) <C log n γ diamx)/ log n, i= a contradiction. Corollary.6. Let X, d) be a finite metric space and N, > 0. Assume that µ is a probability measure on X such that for every x X, µx) N and X X dx, y)dµx)dµy) diamx). Let f : X l be a quasisymmetric embedding with modulus η. Then there are A, B X with µa),µb) γ diamx) 6 and da, B) log N, where γ = γ, η) depends only on and η. 0

11 T i x i Figure : The net N and the partition T,..., T n }. Proof. The proof is a simple duplication of points argument. Without loss of generality assume that µx) is rational for all x X, and write µx) = mx M, where x X m x = M; by our assumption on µ, M = ON). For every x X let xi)} mx i= be copies of x, and consider the semi-metric space X = x X xi)}m x i=, where d Xxi),yj)) = d X x, y) if x y and d Xxi),xj)) = 0. Clearly X = M = ON) and M a,b X d Xa, b) diam X). By Theorem.5 there are Ã, B X with Ã, B M 6 and d XÃ, B) ) = Ω diam X),η log M =Ω diamx),η log N ). One has to observe here that the proof of Theorem.5 works for semi-metrics as well, i.e., the condition dx, y) > 0 for x y was never used.) Denote A = x X : i, xi) Ã} and B = x X : i, xi) B}. Then da, B) = Ω diamx),η log N ). Additionally µa) = x A m x M M x A i m x : xi) Ã} = Ã M 6, and similarly µb) 6, as required. We are now in position to prove Theorem.7 which was stated in the introduction. Proof of Theorem.7. Let k be a large) integer which will be determined later; for now assume that k 4. Recall that a subset S X is called ε-separated if for every distinct x, y S, dx, y) ε. Let N be a maximal k diamx) separated set in X. Since the balls Bx, k diamx))} x N are disjoint and for every x X the doubling condition implies that µbx, k diamx))) λ k µbx, diamx)) = λ k, we have N n λ k+. Write N = x,..., x n }, and define inductively T = x X : dx, x )=dx, N )}, T i+ = x X : dx, x i )=dx, N )} \ i j= T j. Then T,..., T n } is a partition of X which is described schematically in Figure ), and for every x T i, dx, x i ) k diamx). Define a probability measure ν on N by νx i )=µt i ). Since T i Bx i, k diamx)), we

12 have that νx i ) λ k. Observe that diamx) dx, y)dµx)dµy) = = X X n n i= j= n i= j= n i= j= n i= j= dx, y)dµx)dµy) T i T j n [dx, x i )+dx i,x j )+dx j,y)]dµx)dµy) T i T j n [dx i,x j )+ k+ diamx)]dµx)dµy) T i T j n dx i,x j )νx i )νx j ) + k+ diamx). Since we assume that k 4, this implies N N dx, y)dνx)dνy) diamn ). Hence by c diamn ) Corollary.6 there are A, B N with νa),νb) and da, B) k log λ, where c is a constant depending on η and. Define A = x i A T i and B = x i B T i. Then µa )=νa) 3 and µb )=νb) 3. Observe that since N is a k diamx) net in X, diamn ) k+ ) diamx). Fix a A and b B. There are x i A and x j B such that dx i,a) k diamx) and dx j,b) k diamx). Hence, da, b) dx i,x j ) dx i,a) dx j,b) c k+ ) diamx) k log λ So, for k log log λ we get that da,b ) > 3 c diamx) log λ)log log λ), k+ diamx). where c is a constant depending only on η and. Denote ζ = I µ ) µa )/). We claim that µa ζ ). Indeed otherwise, the fact that X \ A ζ ) ζ A = implies that µa ) µ X \ A ζ ) ζ which is a contradiction. Denote ε = ) Iµ ζ) = µa ), c diamx) log λ)log log λ). If ε ζ then since A ζ ) ε/ A ε we have that µx \ A ε) I µ ε/). But B X \ A ε, so that I µ ε/) µb ) 3. On the other hand, if ε< then 3 µb ) µx \ A ε) I µε/) = I µε/) I µ ζ) µa )/ 64I µε/). In both cases we obtain the lower bound I µ ε/) 500.

13 Remark.. A statement analogous to Theorem.7 in the case of uniform embeddings can be proved along the same lines using Theorem.4. In this case, the implicit constants must also depend on diamx), since uniform embeddings do not scale well. This is why we preferred to deal with quasisymmetric embeddings we feel that mappings which preserve shape are more naturally compatible with isoperimetric problems. Remark.3. Consider the discrete cube X = 0, } d, equipped with the Hamming metric ρx, y) = i : x i y i }. Since X, ρ) is isometric to a subset of l, it is also quasisymmetrically equivalent to a subset of l with modulus ηs) = s). The concentration inequality for the uniform measure on X shows that Theorem.5 is optimal. Remark.4. We do not know whether Theorem.5 is optimal when restricted to subsets of Hilbert space i.e., the case ηs) =s). We can, however, show that it is optimal up to a double logarithmic term. Indeed, by Lemma in [6] there is a constant c such that for every < γ < π/ there are disjoint subsets S,..., S N S d of equal surface area, diameter at most γ and N c/γ) d. For each i =,..., N pick an arbitrary point x i S i. Assume that A, B x,..., x N } satisfy A, B N. Then, denoting A = x i A S i and B = x i B S i, we have that σa ),σb ) here σ is the normalized surface area measure on S d, and we have used the fact that all of the ) S i have the same surface area). By the concentration inequality for σ, da,b log/) )=O d. ) Since each of the sets S i has diameter at most γ, we deduce that da, B) O Choosing γ log/) d log/) d we get that N [c)d] d, implying that da, B) =O log log N log N ). +γ. Theorem.5 implies that certain spaces do not quasisymetrically embed into Hilbert space, namely spaces for which the observable diameter is much smaller than the diameter. Examples of such spaces are bounded degree expanders, i.e., regular graphs of bounded degree whose edge expansion is large. However, in this particular case it is easy to deduce an even stronger restriction on their quasisymmetric embeddability into L p, p< : Proposition.7. Let G =V, E) be an n-vertex d-regular graph. Fix p< and assume that f : V L p is a quasisymmetric embedding with modulus η. Then ) η log d n/4) αg) 4p. Proof. In [] see also [3]) it is shown that for every f : V l p, n fu) fv) p p ) p p α E u,v} E fu) fv) p p. 3) Since the number of vertices at distance at most t from a fixed vertex u V is at most +d + d + + d t d t, it follows that for every u V, v V : d G u, v) log d n/4)} n/. 3

14 Now n fu) fv) p p = dn dn dn = w N G u) w N G u) u V w N G u) fu) fv) p p fw) fu) p p [η/d G u, v))] p fw) fu) p p [η/ log d n/4))] p v V : d Gu, v) log d n/4)} dn[η/ log d n/4))] p [η/ log d n/4))] p E u,w V u,w} E u,w V u,w} E This lower bound, combined with 3), implies the required result. fu) fw) p p fu) fw) p p. 3 Proof of Theorem.3 In Section 3. we show how the geometric ideas of [] can be used to obtain the following statement, which is weaker than Corollary.: For every > 0 there are constants c),c) > 0 such that, if G = V, E) is an n-vertex graph satisfying hg) and f : V l is a Hilbert space valued function for which n fu) fv), then there are u, v V with d G u, v) C) log n and fu) fv) c). The same statement with C) uniformly bounded in and c) proportional to would suffice to prove Theorem.3. Unfortunately, we are unable to prove this fact directly. Therefore in Section 3. we augment the above statement with a combinatorial argument which yields Theorem.3. Remark 3.. A natural approach for proving Theorem.3 is to take E = u, v} : u, v V and d G u, v) =C } log n. This idea is easily discarded through the following example: Let G =V V,E E E3 ), where V,E ) and V,E ) are disjoint isomorphic log n)-regular expander graphs with girth log n, and E 3 is a perfect matching between V and V. Consider a node u V. The number of nodes v V such that d G u, v) =C log n is at least log n) C log n. On the other hand, the number of nodes v V such that d G u, v) =C log n is at most C log n log n) C log n. Therefore, αv, E ) C log n. Similar arguments show that other uniform constructions fail, such as taking pairs u, v} with d G u, v) C log n or taking random walks of length at most C log n. It therefore seems that the edges in E have to be chosen judiciously. 4

15 3. Combinatorial preliminaries As stated above, in Section 3. we prove a result which is weaker than Corollary.. In this section we show that nevertheless, in the present setting such a weaker statement suffices to yield the full force of Theorem.3. Informally, the weaker result replaces edges and takes care of the expansion of large sets, but we may be left with a small set of vertices that has poor edge expansion. In this section we fix this poorly expanding set. Lemma 3.6 isolates the poorly expanding set. To fix it, we first reduce its size considerably by adding a large matching across the bad cut. The matching is constructed in Corollary 3.3. It may reduce the expansion by a constant factor, as shown in Lemma 3.5. For this reason, we cannot apply the matching argument iteratively on the remaining set. We therefore connect the remaining vertices iteratively to several vertices on the good side Lemma 3.), thus reducing the expansion by less than a constant factor in each iteration Lemma 3.4). The cumulative effect of all the iterations reduces the expansion by another constant factor. Lemma 3.. Let G =V, E) be a graph with hg) and t N. Fix a subset S V with S V 3/). Then there exists a bipartite graph H =S, V \ S, F ) and a subset S S such t that the following conditions hold:. u, v} F, d G u, v) t.. S \ S S 3/) t/. 3. v V \ S, deg H v). 4. u S, deg H u) 3/) t/. Proof. Fix R S of cardinality R > S /3/) t/. As hg), we have that for every t N, u V : d G R, u) t} 3/) t R > S 3/) t/. Consider the following iteration, starting with S, T, F = : Find a pair of vertices u, v V such that u S \S, v V \S T ), and d G u, v) t. Place u, v} in F, place v in T, and if currently deg H u) = 3/) t/, place u in S. Terminate when no such pair of nodes exists. Notice that while R = S \ S satisfies R > S /3/) t/, then u V : d G R, u) t} > S 3/) t/ = S + S 3/) t/ ) S + T. Therefore, the iteration terminates with R S /3/) t/, and at that point the bipartite graph H =S, V \S, F ) satisfies all the required conditions. Lemma 3.. Let G =V, E) be a connected graph with hg), s,s N, and X, Y V satisfying X V /3/) s, Y V /3/) s. Then d G X, Y ) s + s. Proof. Since hg), the sets X = v V : d G v, X) s } and Y = v V : d G v, Y ) s } satisfy X, Y V /. As G is connected, d G X,Y ). Corollary 3.3. Let } G =V, E) be a connected graph with hg), r N and X, Y V satisfying X min Y, V. Then there is Z X and a one to one mapping M : Z Y such that. For every v Z, d G v, Mv)) < r.. X \ Z V /3/) r. 5

16 Proof. The proof is by induction on X. If X V /3/) r, then there is nothing to prove. Otherwise, by Lemma 3., there are v X and Mv) Y such that d G v, Mv)) r. Apply the induction hypothesis to X \v} and Y \Mv)}. Lemma 3.4. Fix > 0 and let G =V, E) be a graph with V E V and set α = minαg), / }. Let k, m N satisfy m V. Let V k,..., V m be disjoint k-vertex subsets of V. Construct a graph H =W, F ) by setting W = V s,..., s m }, where s,..., s m are m new vertices, and F = E m i= v V i s i,v}). Then αh) α 3 ). k Proof. Notice that F E + mk = W V + m E + mk V E + V /k V + ) E k V. Fix S W with S W / and write A = S s,..., s m } and B = S V. If A k S then the number of edges leaving S is at least k A B S S = S α E V S α ) F k W S. Otherwise, B ) k S, so if B V / then the number of edges leaving S is at least αg) E B α E ) S α 3 ) F V V k k W S. If, on the other hand B > V /, then by the definition of αg) applied to V \ B, the number of edges leaving S is at least αg) E V \ B ) α F V S ) V k W ) α F V m k W V + m S ) α F k W /k +/k S, implying the required result. Lemma 3.5. Fix > 0 and let G =V, E) be a graph with V E V. Fix a set U with U V 4 and U V = and let M : U V be a one to one function. Construct a graph H =W, F ) by setting W = U V and F = E } u U u, Mu)}}). Then αh) min αg) 3, 3. Proof. Notice that F W = E + U V + U E V. Consider a set S W with S W. If S U 3 S, then u S U : Mu) S} 3 S. Therefore, at least F 3 S > 3 W S edges leave S. Otherwise, S V 3 S, implying that at least αg) E S αg) F S V 3 3 W edges in E leave S. 6

17 Lemma 3.6. Fix α, ε > 0 and let V, F ) be a graph with the property that for every S V satisfying ε V S F V, e F : e S =} α V S. Then there is U V such that V \ U ε V and the graph G =U, F ), where F = e F : e U =}, has αg ) ε)α. Proof. Define a sequence of graphs V 0,F 0 ), V,F ),...,V k,f k ) and a sequence of disjoint subsets S,..., S k V as follows. Put V 0 = V and F 0 = F. If αv i,f i ) ε)α set k = i. Otherwise, there is S i+ V i for which e F i : e S i+ =} < ε)α F i V i S i+. Set V i+ = V i \ S i+ and put F i+ = e F i : e S i+ = }. We now show that V k ε) V. For contradiction, let j be the smallest index in,,..., k} for which V j < ε) V. Put S = j i= S j, so our assumption is that S >ε V. Then e F : e S =} < j e F i : e S i+ =} i=0 j i=0 j ε)α F i V i S i+ F ε)α ε) V S i+ i=0 = α F V S, in contradiction to the conditions stated in the lemma. Now αv k,f k ) ε)α and V k ε) V, so we can set U = V k and F = F k. Lemma 3.7. Fix, α, ε > 0. Let G =V, E) be a graph with hg) and let F V ) be a set of edges such that V F V and V, F ) satisfies the conditions of Lemma 3.6. } Then there exists F V ) such that the graph H =V, F ) satisfies αh) min ε)α 300, 000 max d G u, v) : u, v} F } max log log V, max d G u, v) : u, v} F }}. Proof. We will construct the graph H gradually. Initially, set H to be the graph G =U, F ) from Lemma 3.6. So, U ε) V, F F, and αg ) ε)α. Fix an integer r 4, which will be determined later. By Corollary 3.3, there is a set Z V \ U and a one to one mapping M : Z U such that V \ Z V /3/) r and for every u Z, d G u, Mu)) r. By Lemma 3.5, the graph H 0 =V 0,F 0 ) with V 0 = U Z and F 0 = F u, Mu)} : u Z} has αh 0 ) min and ε)α 3, 3 Put Z 0 = V \ V 0. Now iterate the following step, starting with i = 0. Use Lemma 3. to generate a bipartite graph M i =Z i,v i,e i+ ) and a set Z i+ Z i that satisfy the following conditions:. u, v} E i+, d G u, v) r.. Z i+ Z i /3/) r. 3. v V i, deg Mi v). }. 7

18 Z Z 0 Z U U ε )V Figure : The iterative construction of the graphs H i. 4. u Z i \ Z i+, deg Mi u) = 3/) r. This construction is described schematically in Figure Set H i+ =V i Zi,F i Ei+ ). Notice that after k = The number of edges in H is at most V +ε V + Thus, by Lemma 3.4, for every i =,,..., k, } min αh i ), min 0 Therefore, for r = 0 log log V, as required. αh k ) min αh 0 ), ε)α min, 3 ε)α min, log V r steps, Z k =. Set H = H k. 6 V log V 3/) r 0 V, provided r 0 log log V. αh i ), } } 6 ) 0 3/) r. 3/) r } 6 }, ) k 3/) r ) 3 log V r Lemma 3.8. Let G =V, E) be a graph and let π be a probability distribution on E. Assume that for every S V such that V 4 S V, πe E : e S =}) p. Then there exists a graph H =V, F ) such that F E, F 0 V p V, and for every S V with 4 S V, e F : e S =} 5 V. Proof. Let k = 0 V /p. Consider a sample F = e,e,..., e k } of E, where the e i are independent, identically distributed random variables with Pr[e i = e] =πe). Consider a set of vertices S V with V V 4 S. Let X i denote the indicator variable for the event e i S =. Then Pr[X i = ] p. Trivially, k e F : e S =} = X i. 8 i=

19 Put X = i X i. Then E[X] 0 V and Pr[X <5 V ] Pr[Y < 5 V ], where Y is distributed as the sum of k Bernoulli trials with success probability p. Using standard bounds on the deviation of Y, we get On the other hand, Pr[X <5 V ] Pr[Y <5 V ] = Pr[Y <E[Y ]/] <e E[Y ]/8 <e V. S V : V 4 } V S V. Therefore, with probability approaching, F satisfies the required property. In what follows we denote by B the unit ball of l. Lemma 3.9. Let G =V, E) be a graph and, p, k > 0. Assume that for every f : V B with n fu) fv), there are u, v V such that d G u, v) k and fu) fv) p. Then there is a probability distribution π on u, v} : u, v V d G u, v) k} such that for every S V with 4 /3 V S V we have that πu, v} : u, v} S =}) p. Proof. Let F be the set of all S V with 4 /3 V S V. If S F then we have that V [ Su) S v)] /3. Denote by D F the set of all probability distributions on F, i.e., the set of all t S ) S F such that t S 0 and S F t S =. For t D F define f t : V R F by f t v) S = t S S v). Then f t v) for all v V and V f t u) f t v) /3. Denoting D = x, y) X X : fu) fv) /3 } we have that D /3 V. Hence V f t u) f t v). By our assumption it follows that there are u, v G with d G u, v) k and fu) fv) p. Now let Π be the set of all probability distributions π on u, v) V V : d G u, v) k}. The Min-Max Theorem implies that max min πu, v) : u, v} S =}) min max πu, v) f t u) f t v) p, π Π S F t D F π Π as required. d G u,v) k Corollary 3.0. Let G =V, E) be an n-vertex graph satisfying hg). Assume that there exist constants c, C > 0 such that for every f : V B satisfying n fu) fv) 64 there are u, v V with d G u, v) C log n and fu) fv) c. Then there are edges E on V such that for every u, v} E, d G u, v) =OC log n) and αv, E ) = Ωc ). Proof. By Lemma 3.9 there is a probability distribution π on u, v} : u, v V d G u, v) C log n} such that for every S V with n 4 S n we have that πu, v} : u, v} S = }) c. By Lemma 3.8 there are edges F on V such that F 0n/c, for every u, v} F, d G u, v) C log n and for every S V with n 4 S n, e F : e S =} 5 V. Now, the graph V, F ) satisfies the conditions of Lemma 3.6 with ε = 4 and α = c 4. So, Lemma 3.7 yields new edges F on V for which αv, F ) = Ωc ) and for every u, v} F, d G u, v) =OC log n). 9

20 3. The Euclidean argument In what follows S d denotes the unit Euclidean sphere in R d and B d denotes the unit Euclidean ball in R d. The normalized Haar measure on S d is denoted by σ. By Corollary 3.0, Theorem.3 will be proved once we establish the following result, the proof of which is based on the chaining argument from []. Proposition 3.. Let G =V, E) be an n-vertex graph with hg) and 0, ]. Assume that f : V B d satisfies n Then there are u, v V such that fu) fv). 4) d G u, v) 0[log750/)] log n and fu) fv) 000 log4/). Before we proceed with the proof, we present an informal description of the chaining argument. The basic idea is that if there is a set A V ) of nearby pairs of vertices of G such that, for almost every direction y S d, there are pairs of vertices u, v} A with projections y, fu) and y, fv) far apart, then there is a pair of vertices u, v with far apart fu) and fv). In order to construct the set of pairs A, one begins with pairs that are very close in G but have projections that are not far enough Lemma 3.3). Then, these pairs are iteratively chained to create new pairs that are more distant in G and have better projections. In each iteration, the measure of good directions is boosted using measure concentration, and the number of pairs is boosted using the vertex expansion of G. We begin with the following simple numerical fact: Lemma 3.. Fix η 0, ) and let X R be an n-point subset of the the real line satisfying Then there exist A, B X with A, B x B and y A. x, y) X X : x y a} ηn. η ) n η 4 n and such that x y + a η for all Proof. Let m be a median of X. Write k = X m a/,m+ a/). Then k η)n, i.e., k ) ηn. It follows that either X,m a/] n or X [m + a/, ) η ) n. In the first case take A = X,m a/] and B = X [m, ). In the second case take A = X,m], B = X [m + a/, ). Lemma 3.3. Let Z be an n-point subset of B d such that n z w. z,w Z Then σ y S d : z, w) Z Z : z w, y 6 d } } n 4. 0

21 Proof. Let D = z, w) Z Z : z w }. Since diamz), D 4 n. If z, w) D then a simple calculation yields σ y S d : z w, y } 6 d. Hence n y S d : z w,y /6 d)} dσy), S d implying the required result. z,w Z Lemma 3.4. Let G and f be as in the statement of Proposition 3.. There exists a subset U V satisfying U n 30 with the following property. For every y Sd there is W y U and a one to one mapping M y : W y U \ W y such that, for every v W y, we have d G v, M y v)) 6 log6/) and fm y v)) fv),y 3 d, 5) and for every v U, σy S d : v W y } ) Proof. By Lemma 3.3 applied to fv ) B d, there exists a subset T Sd such that σt ) 4 and for every y T there are A y,b y V such that A y, B y n 8 and for every p B y, q A y, fp),y fq),y + 3 d. By Corollary 3.3 there is a subset A y A y with A y n 6 and a one to one mapping M y : A y B y such that for every v A y, d G v, M y v)) 6 log6/). Fix a subset T T for which σt ) 8 and T T )=. We will construct inductively sets V = V V V k as follows. Assuming V i has been defined, denote For every v V i write A i y = A y V i [M y ) MA i ) V i )]. S i v = y T : v A i y My A i y)}. If there is v V i for which σsv) i < 360, define V i+ = V \v}. This procedure ends when for every v V k we have σsk i ) 360. We choose U = V k and for all y T, W y = A k y. We symmetrize by setting, for all y T, W y = M y W y ) and M y =M y ). With these definitions, the construction implies that 5) and 6) are satisfied. It remains to bound U from below. Fix i < k and write u} = V i \ V i+ Observe that σsv i+ ) = σsv) i σsu) i v V i+ v V i [σy T : v A i y u = M y v)} + σy T : u A i y v = M y u)}] v V i+ = v V i σs i v) σs i u) v V i σs i v) 30.

22 By induction we have U v V k σs k v ) v V σs v) implying the required estimate. n U ) 30 = A y dσy) T n U ) 30 n 64 n U ), 30 Proof of Proposition 3.. Assume for the sake of contradiction that, for every u, v V with d G u, v) 0[log750/)] log n, we have fu) fv). Let U be as in 000 log4/) Lemma 3.4. We claim that this implies that for every i log n there is a subset Y i U with Y i U such that, for every v Y i, σ y S d : u U s.t. d G u, v) 0[log750/)] i and fu) fv),y i } 64 d 500 ). 7) Assuming this for the moment, we conclude the proof of Proposition 3. as follows. Set k = log n, take any v Y k and consider the ball B = B G v, 0[log750/)] k) =u V : d G u, v) 0[log750/)] k} in G. For every u B, our assumption implies that fu) fv) 000, so that σ y S d : fu) fv),y k } 64 < d n 3. By the union bound, this contradicts 7). It remains to prove 7). The proof is by induction on i. For i = 0 the claim is vacuous. Assuming the existence of Y i for some i log n, we will deduce the existence of Y i+. Fix v Y i and denote T v = y S d : u U s.t. d G u, v) 0[log750/)] i and fu) fv),y i 64 d so that by the inductive hypothesis σt v ) /500. It follows from the definition that there is a function N v : T v B G v, 0[log750/)] i) U such that for every y T v fn v y)) fv),y i 64 d. By 6) there is a subset T v T v with σt v) 000 such that, for every y T y, v W y. Observe that for y T v, d G M y v),n v y)) 0[log750/)] i + 6 log6/) and fn v y)) fm y v)),y = fn v y)) fv),y + fv) fm y v)),y For every u U consider the set K u = y S d : v Y i s.t. y T v and u = M y v)}, i 64 d + 3 d. }, and define Z = u U : σk u ) }. 4000

23 Now we have σt v) Y i 000 U 000. v Y i On the other hand, since M y is one to one, σt v) = v Y i : y T v} dσy) v Y S d i = u U : v Y i s.t. y T v u = M y v)} dσy) S d = u U σk u ) Z + U It follows that Z U Fix u Z and define L u = y S d : v U B G u, 0[log750/)] i + 6 log6/)) } fv) fu),y i 64 d + 50 d We claim that L u K u ) 7 [log4/)]/d recall that A r denotes the Euclidean r-neighborhood of a set A). Indeed, if y K u then, by the definition of K u, there is w U B G u, 0[log750/)] i + 6 log6/)) such that fw) fu),y i 64 + d 3. Observe that by our contrapositive d assumption), fw) fu). Fix z 000 log4/) Sd log4/) with z y 7 d. Then, fw) fu),z fw) fu),y fw) fu) z y i 64 d + 3 d 7 log4/) 000 log4/) d i 64 d + 50 d. By concentration of measure on S d, it follows that σl u ) 8000 e 49 log4/) > 000. We are almost done, except for the fact that Z is too small. This is where the lower bound on hg) is used again). Define Z = v V : d G v, Z) 6 log750/)}. We claim that Z U U. Otherwise, denote A = V \ Z. Since hg), 3 ) } 3 log750/) v V : d G v, Z) 3 log740/)} > min Z, n 3 ) } 3 log750/) U min 4000, n 750 ) } n min , n = n, 3 and

24 and similarly since A U / n/360), v V : d G v, A) 3 log740/)} > n. This implies that there is some v V for which d G v, Z) 3 log740/) and d G v, A) 3 log740/), which contradicts the fact that Z A =. We define Y i+ = Z U. It remains to show that 7) holds for every v Y i+. Indeed, there exists u Z such that d G u, v) 6 log740/) and σ y S d : w U s.t. d G w, u) 0[log750/)] i + 6 log6/) and fw) fu),y i 64 d + 50 d } 000. It follows that σ y S d : w U s.t. d G w, v) 0[log750/)] i + ) and } fw) fv),y i 64 d + 50 fu) fv),y d 000. But, since fu) fv), it follows that 000 log4/) σ y S d : fu) fv),y 50 ) } d < , implying 7). This completes the proof of Proposition 3., and hence of Theorem.3. References [] Alon, N. and Spencer, J. The Probabilistic Method, Wiley-Interscience, New York, second edition, 000. [] Arora, S., Rao, S. and Vazirani, U., Expander flows, geometric embeddings and graph partioning, In Proceedings of the 36th Annual ACM Symposium on the Theory of Computing, 3, 004. [3] Bartal, Y, Linial, N., Mendel, M. and Naor, A., On metric Ramsey-type phenomena. To appear in the Annals of Mathematics, 004. [4] Beurling A., and Ahlfors, L., The boundary correspondence under quasiconformal mappings. Acta Math. 96, 5 4, 956. [5] Deza, M. M. and Laurent, M., Geometry of cuts and metrics. Springer-Verlag, Berlin, 997. [6] Feige, U. and Schechtman, G., On the optimality of the random hyperplane rounding technique for MAX CUT, Random Structures and Algorithms 0, , 00. [7] Gromov, M., Metric structures for Riemannian and non-riemannian spaces, Birkhaüser 999. [8] Grötschel, M., Lovász, L. and Schrijver A., Geometric algorithms and combinatorial optimization. Springer-Verlag,

25 [9] Heinonen, J., Lectures on Analyis on Metric Spaces, Universitext, Springer-Verlag, New-York, 00. [0] Lee, J. R., On distance scales, embeddings, and efficient relaxations of the cut cone. In Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms, 005. [] Leighton, F.T. and Rao, S., Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms, Journal of the ACM 46, , 999. [] Matoušek, J., Embedding expanders into l p, Israel J. Math. 0, 89-97, 997. [3] Milman, V. and Schechtman, G., Assymptotic theory of finite dimensional normed spaces, Springer-Verlag, Berlin 986. [4] Tukia, P. and Väisälä, J., Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I. Math. 5, 97 4,