-4princeton u. sp’02 cos 598B: algorithms and complexityLecture 3-4: Embedding metrics into 1and applications tosparsest cutLecturer: Sanjeev Arora Scribe:Elad Hazan1 Bourgain’s theorem (1version)The main goal of these lectures is to prove the following theorem, which is a special case of Bour-gain’s theorem from the mid-eighties.Theorem 1 (Bourgain, 1985)Ev ery metric space embeds into 1with distortion O(log n).Bourgain’s theorem is actually more general, and holds for any lp. (We present a proof forthe 1case due to Fakcharoenphol, Rao and Talwar (2003) since it has been useful in subsequentdevelopments, as we will see.) Furthermore, examining the proof of the theorem one can derivean efficient algorithm to produce such a low-distortion embedding. (Aside, this is not always thecase with mathematical proofs. Later in the course we shall encounter proofs of existence forcombinatorial objects that do not entail efficient algorithms to construct these objects.)We start with some notation we will use throughout this scribe. Denote the original metric by(X, d), where X is a set of |X| = n elements and d : X × X →+a distance function. Denote byBall(x, R) ⊆ X the set of all elements y ∈ X such that the distance to x is at most R.We now describe a (fairly efficient) procedure to partition the elements of X. This procedurelies at the heart of the embedding.Procedure Partition(A, B)1. Pick uniformly at random a number R ∈ [A, B].2. Pick uniformly at random an order σ on the elements of X.3. Partition the items of X into at most n = |X| blocks as follows.(a) Proceed with the elements of X according to the order σ.(b) For each element x ∈ X, pick all non-assigned elements within distance R fromit, and form a new block.We call Pσ,Rthe partition created above, and denote by Pσ,R(x) the block in whichx was placed.remarks: (i) Some blocks may be empty. (ii) Pσ,R(x) may not be the same as the block createdusing x in part 3(b) of the above procedure. The reason is that x may already be assigned to someother block.The crucial property of this partitioning procedure is the following:12xCurrently assigned verticesBall(x, R)Figure 1: At each step, Partition takes the next element in order σ,sayx, and creates a blockconsisting of all nonassigned elements in Ball(x, R).Theorem 2 (Padded decomposition property)For every τ>0,x∈ X we have:Prσ,R[Ball(x, τ)  Pσ,R(x)] ≤4τB − Alog|Ball(x, B + τ)||Ball(x, A − τ)|Proof: Denote by Ezthe event that z is the first element by the order σ such that d(x, z) ≤ R+ τ.Obviously,Pr[Ez]=1|Ball(x, R + τ)|.Notice,z∈Ball(x,R+τ )Ezalways happens.We claim if Ball(x, τ)  Pσ,R(x), then the eventz:d(x,z)∈[R−τ,R+τ ]Ezmust have happened.(Indeed, consider the z which is the first element in Ball(x, R + τ) in the order σ.Ifd(x, z)were≤ R − τ then the block created using z would swallow all of Ball(x, τ).) We therefore bound theprobability that Ball(x, τ)  Pσ,R(x)by:PrR,σ[Ball(x, τ)  Pσ,R(x)] ≤ zPrR,σ[EzR ∈ [d(x, z) ±τ ]]= zPr [R ∈ [d(x, z) ± τ]] · Pr[Ez| R ∈ [d(x, z) ± τ]]≤2τB−A zPr [Ez| R ∈ [d(x, z) ± τ]] R is chosen uniformly≤2τB−A|Ball(x,B+τ )|r=|Ball(x,A−τ )|1rdr≤4τB−Alog|Ball(x,B+τ )||Ball(x,A−τ )|2Using the above procedure, we can now define the embedding into 1. We assume w.l.o.g (byscaling) that the given metric (X, d) has shortest distance 4, and largest distance Δ. The embeddingwe describe is probabilistic.The final embedding of (X, d)into1is a composition of several 1pseudo-metrics, one for eachdistance scale.3Procedure Embed(X, d)For every t ≥ 1 such that 2t< Δ do:Invoke Partition(2t, 2t+1) to create a partition Pσ,R. Then define an embeddingρt: X →Kwhere K is the number of blocks in Pσ,Rand|ρt(x) − ρt(y)|1=2tif x, y are in different blocks of Pσ,R0otherwise(1)(Notice that embedding ρtconsists of placing each block of Pσ,Ron a coordinate axis inKat a distance 2s−1from the origin.)The final embedding f is the trivial composition of ρtfor all scales. (Namely, use freshcoordinates to accomodate each ρtand never reuse them for any other scale.)Now we prove that the expected distortion of the embedding f is O(log n). We give a trivial(deterministic) lowerbound on |f(x) − f(y)|1, and a probabilistic (expectation) upperound.Lemma 3For any x, y ∈ X it holds that f(x) − f(y)1≥14d(x, y).Proof: Let Pσ,Rbe the partition created at scale t.Noticethatift is such that 2t+2<d(x, y),then since R ≤ 2t+1, any ball of radius R cannot contain both x and y,anditmustbethatPσ,R(x) = Pσ,R(y) and therefore ρt(x, y)=2t. Hence,f(x) − f(y)1= log Δt=0ρt(x, y) ≥t|2t+2<d(x,y)2t≥14d(x, y)2Lemma 4For any x, y ∈ X it holds that Eσ,R[f(x) − f(y)1] ≤ O(d(x, y) · log n).Proof: Using the definitions:E[f(x) − f(y)1]= log Δt=0Eσ,R[ρt(x, y)]≤ log d(x,y)t=0Eσ,R[ρt(x, y)] + log Δt>log d(x,y)Eσ,R[ρt(x, y)]≤ log d(x,y)t=02t+ log Δt>log d(x,y)Eσ,R[ρt(x, y)]≤ 4d(x, y)+ log Δt>log d(x,y)Eσ,R[ρt(x, y)]For the larger values of t we have:4 log Δt>log d(x,y)Eσ,R[ρt(x, y)] = log Δt>log d(x,y)2t· Pr[Pσ,R(x) = Pσ,R(y)]≤ log Δt>log d(x,y)2t· Pr[Ball(x, d(x, y))  Pσ,R(x)]≤ log Δt>log d(x,y)2t·2d(x,y)2tlog|Ball(x,2t+1+d(x,y))||Ball(x,2t−d(x,y))|by theorem 2≤ 2d(x, y) log Δt>log d(x,y)log|Ball(x,2t+2)||Ball(x,2t−1)|≤ 2d(x, y) · 3log|Ball(x, Δ)| =6d(x, y)lognCombining both previous equations we get Eσ,R[f(x) − f(y)1]=O(d(x, y) · log n). 2From lemmas 3 and 4 we derive that the embedding described into 1has expected distortionO(log n). In order to prove theorem 1, we need an embedding that has a worst case distortionguaranty. Here, again, we use the fact that the set of n-point 1metrics is a convex cone. Therandomized embedding that we have described thus far can be viewed as a distribution over deter-ministic embeddings into 1. The convex combination of all these 1metrics is itself a an 1metricwith worst case distortion which is precisely equal to the expected distortion of the randomizedembedding, i.e. O(log n).Efficiency issues: Note that even though we have described a polynomial time procedurefor embedding into 1with low expected distortion, the last argument does not directly implyan