princeton u. sp’02 cos 598B: algorithms and complexityLecture 1: Introductory LectureLecturer: Sanjeev Arora Scribe:Michael Dinitz1 Course OutlineThis course will cover three main areas:1. Geometric embeddings of metric spaces2. Nonconstructive methods in combinatorics and theoretical computer science3. Efficient algorithms for some geometric problems, especially convex optimizationIn this lecture we begin our study of geometric embeddings of metric spaces.2 Basic DefinitionsWe start with the definitions of metric spaces and normed spaces.Definition 1 Let X be a set, and let d : X × X → R+∪{0}.Thepair(X, d) is a metric spaceif for all x, y, z ∈ X,1. d(x, y)=d(y,x)2. d(x, y)=0⇔ x = y3. d(x, y)+d(y,z) ≥ d(x, z) (triangle inequality)Definition 2 A normed space is Rkfor some finite k together with an associated mapping a →afrom Rkto R+∪{0} such that:1. For all λ ∈ R, λa = |λ|a2. u + v≤u + v3. u =0if and only if a =0(the zero vector)Note that (Rk,d)whered(u, v)=u − v is a metric space. The following are some examplesof norms:1. l2is the normal Euclidean norm. If u =(u1,u2,...,uk)thenu2=(|ui|2)1/2.2. In general, lpfor p ≥ 1 is the norm up=(|ui|p)1/p.3. l1is sometimes referred to as the Manhattan norm since it is simply the sum of absolutevalues of the coordinates.124. As p approaches infinity the value is dominated by the largest coordinate, so l∞is the maxnorm and is the largest coordinate (in absolute value).Metric and normed spaces arise naturally in many practical applications (e.g., finding similarimages in a database) but we will not talk about practical applications here. Rather, we will beinterested in theoretical issues (which of course have a bearing on design of practical algorithms)such as how well metrics can be embedded into each other. We need a notion of distortion thatmeasures how good the embedding is.Definition 3 Let (X1,d1) and (X2,d2) be metric spaces. An embedding f : X1→ X2has distor-tion C if there is an r>0 such that ∀x, y ∈ X1,r · d1(x, y) ≤ d2(f(x),f(y)) ≤ Cr ·d1(x, y).3 Three types of problems3.1 TypesThere are three main flavors of problems:1. (Upperbound) Can metrics of class I embed into class II with distortion at most C?2. (Lowerbound) Show that there is a metric in class I that does not embed into any metric ofclass II with distortion at most C.3. (Algorithmic version) Given particular metrics (X1,d1)and(X2,d2), determine the minimumdistortion required to embed X1into X2.Typically solutions to problems of type 1 also give rise to an algorithm for embedding all metricsof class I into a metric of class II.3.2 Examples and ResultsThe following three theorems are examples of the first type of problem:Theorem 1 (Bourgain ’85)For p>1,everyn-point metric embeds into lpwith distortion O(log n).We will prove the above theorem later in the course.Theorem 2 (Frech´et)Every n-point metric embeds into l∞isometrically (distortion 1).Proof: Let (X, d)bean-point metric, and let f : X → R|X|map each element x ∈ X to(d(x, z))z∈X(i.e., vector in R|X|whose ith coordinate is the distance between x and the ith elementof X). We need to verify that f (x) − f(y)∞= d(x, y) for all x, y ∈ X.Notethatf(x) − f(y)=(d(x, z) − d(y, z))z∈X. By the triangle inequality, |d(x, z) − d(y, z)|≤d(x, y) for all z, and thusf(x) − f (y)∞≤ d(x, y). On the other hand, if z = x then |d(x, z) − d(y,z)| = |d(y,x)|,sof(x) − f(y)∞≥ d(x, y). Thus f(x) −f (y)∞= d(x, y). 2The following theorem will also be proved later in the course.3Theorem 3 (Arora, Lee, Naor ’05)Every n-point l1metric embeds in to l2with distortion O(√log n log log n).Examples of lowerbounds include the result that the Theorems 1 and 2 are tight up to a constantfactor, and that Theorem 3 is tight up to a log log n factor. A result of the third type is the following:Theorem 4For any metric space (X, d) we can determine in polynomial time the minimum C such that (X, d)embeds in to l2with distortion C.Proof: Note that the best C could be a transcendental number so “computing” it just meansbeing able to compute it upto additive error in time poly(log(1/ )). Such algorithms are calledfully polynomial-time approximation scheme (FPAS). For any C we can determine in polynomialtime if there exist vectors u1,u2,...,unsuch that ∀i, j ∈ X, d(i, j)2≤ui− uj22≤ C2d(i, j)2byusing semidefinite programming. The FPAS uses this algorithm and does a binary search on C. 23.3 Lower Bound ExampleWhat is an example of a metric that does not embed isometrically into l2? The three point metricwith all distances equal to 1 embeds as a triangle, and the four point metric with all distances equalto 1 embeds as a simplex. What about the four point metric where the outside square has distancesof 1 and the diagonals have distance 2? In other words, the shortest-path metric for the graph ofFigure 1 whose edges AB,BC,CD,DA all have unit length. (Thus d(A, C)=d(B,D)=2.) Thisobviously does not embed isometrically into l2, since any three of the points determine a uniqueembedding (in fact a line) that makes the fourth point impossible to embed. This argument doesnot lead immediately to a lower bound on the distortion, but there is a different argument thatdoes give a bound.)*+,Figure 1: A metric that does not embed isometrically into l2Theorem 5Every embedding of (X, d) into l2has distortion at least√2.Proof: This is an example of a popular methodology for proving lower bounds. We first come upwith two weightings w1: X ×X → R+∪{0} and w2: X ×X → R+∪{0}. Then we show that forall embeddings f,w1(x, y)d(x, y)2w2(x, y)d(x, y)2(1)4is different by a factor of k fromw1(x, y)f(x) − f(y)2w2(x, y)f(x) − f(y)2(2)This obviously implies that the required distortion is at least√k, since we can just use theembedding with the smallest distortion for f . In our specific case, we will let w2be the weightingthat puts equal weight on AB,BC,CD,DA and w1be the weighting that puts equal weight onAC, BD. Then equation (1) equals|AC|2+|BD|2|AB|2+|BC |2+|CD|2+|DA|2=4+41+1+1+1= 2. Equation (2) becomes|AC|22+|BD|22|AB|22+|BC |22+|CD|22+|AD|22which, by the quadrilateral inequality from high school geometry, is lessthan or equal to 1 for any four points in Rk. Thus the required distortion is at least√2. 2The Boolean hypercube