1. 13 th Nov 2006 1 Geometry of Graphs and It’s Applications Suijt P Gujar. Topics in Approximation Algorithms Instructor : T Kavitha

2. 13 th Nov 2006 2 Agenda Introduction Definitions Some Important Results Embedding Finite metric space into (R d, L p ) Multi Commodity Flow via Low Distortion Embeddings Applications.

3. 13 th Nov 2006 3 Geometry Graphs Geometry of Graphs simply viewing Graphs from Geometric perspective Topological Models Adjacency Models Metric Models In this talk we will be discussing Paper “The Geometry of Grpahs and some of its algorithmic applications” by London, Linial, Rabinovich (LLR’94)

4. 13 th Nov 2006 4 What is Metric Space? Metric Space: A pair (X,d ) where X is a set and d is a distance function such that for x,y in X : Banach Space: A vector space and a norm |v |, which defines a metric d (u,v)=|v-u|. Hilbert Space :A vector space with inner product along with induced norm |v |, which defines a metric d (u,v)=|v-u|. E.g. (R d, L p )

5. 13 th Nov 2006 5 Examples of Metrics Minkowski L p Metric: Let X = R d. L inf (Chessboard): Hamming Distance: Let X = {0,1} k. Number of 1-bits in the exclusive-or L 1 : Manhattan Distance, L 2 : Euclidian Distance Cut Metric : X = A U B where A,B is partition of X d (x,y) = 0 iff x,y both Є A or both Є B = 1 otherwise.

6. 13 th Nov 2006 6 Embedding We will be considering Embedding of Metric Spaces to Banach Spaces esp. (R d,L p ) Metric embedding is a function f : (X,d x ) (Y,d y ) Distortion : The embedding is said to have distortion C if for any x 1,x 2 in X

7. 13 th Nov 2006 7 Example Consider Graph G with 4 vertices with unit distance between any pair of Vertices. Embed this in (R 2,L 2 ) with 4 vertices as vertices as square with diagonal length ‘1’. AB CD d (A,B) G1G1 R 2 0.7071 d (A,C)10.7071 d (A,D)11 d (B,C)11 d (B,D)10.7071 d (C,D)10.7071

8. 13 th Nov 2006 8 Isometrics The isometry is mapping f from Metric space (X,d x ) to metric space (Y,d y ) which preserves distance. i.e. Distortion C = 1. Isometric Dimension of Metric space (X,d x ) is the least dimension for which there exists embedding of X into any real normed space.

9. 13 th Nov 2006 9 dim (X) ≤ n for ‘n’ point metric space. Let X = {x 1, x 2, …, x n } with d ij = d(x i,x j ). Map each point x i to z i Є R n whose k th coordinate is z i k = d ik. || z i – z j || inf = max k | z i k - z j k | ≥ | z i j - z j j | = |d ij - d jj | = d ij On other hand, | z i k - z j k | = |d ik - d jk | ≤ d ij (Triangular inequality) so, || zi – zj || inf = d ij.

10. 13 th Nov 2006 10 Johnson – Lindenstrauss Theorem (84) Any set of n points in a n - dimesional Euclidian space can be mapped to R d where d = O(ε -2 log n) with distortion ≤ 1 + ε. Such mapping may be found in random polynomial time. Idea is to project n dimensional space orthogonally to d dimensional subspace

11. 13 th Nov 2006 11 JL Theorem contd… Take A 1, A 2, …, A d set of orthonormal Vectors randomly chosen in R n. A = [A 1 A 2 … A d ] t For any x in X, x’ = Ax. Consider x Є R n st || x || 2 = 1. So, E[x i 2 ] = 1/n. E[x’.x’] = d/n. E[||x’||] = √(d/n) = m.

12. 13 th Nov 2006 12 Let x,y be two vectors in R n. And x’, y’ be corresponding embedding in R d. X’ = Ax, y’ = Ay. ||x’-y’|| = A(x-y). Pr ( | ||x’-y’|| - m||x-y|| | > εm||x-y|| ) ≤ e Ω(-d/ ε* ε) When d O(ln n / ε* ε ), this Probability of failure < 1/n 2. Best known bound is d = 16*ln n / ε 2

13. 13 th Nov 2006 13 Some results for embeddings We will define L d p = (R d,L p ). C p (X) = minimum distortion with which X may be embedded in L p.

14. 13 th Nov 2006 14 XYC L 2 n, n pointsL 2 O(log n) 1 + ε i.e. O(1) RJL(84) n point metric space L 2 2^n O (log n)DBourgain(85) L2nL2n C 2 (X) + εD L 2 O(logn) O (log n)R L p O(logn) 1 ≤ p ≤ 2 O (log n)RLLR(94) L p O(n^2) 1 ≤ p ≤ 2 O ( C 2 (X) )DLLR(94) L p O(logn^2) 1 ≤ p O (log n)RLLR(94)

15. 13 th Nov 2006 15 LLR Algorithm for Embedding Let q = O(log n). [Constant affects the constant in distortion.] For i = 1,2,…,log n do For j = 1 to q do A i,j = random subset of X of size 2 i. Map x to the vector {d i,j }/Q 1/p d i,j is the distance from x to the closest point in A i,j and Q is the total number of subsets.

16. 13 th Nov 2006 16 Theorem 1 Let (X,d) be a finite metric space and {(s i,t i ) | i = 1,2,…,k} Є X x X. There exists a deterministic algorithm that finds an embedding f : X → l 1 O(n^2), so that d (x,y) ≥ ||f(x) – f(y)|| 1 for every x,y in X and ||f(s i ) – f(t i )|| 1 ≥ Ω(1/log k)*d (s i,t i ) for every i = 1,2,…,k.

17. 13 th Nov 2006 17 Multi commodity flows via low distortion embeddings Problem : Given an undirected Graph G(V,E) with n vertices, Capacity Ce associated with every edge in E. There are k source-sink pairs (s i,t i ) and Demand D i associated with it. Flow conservation law should hold true. Total flow through each edge should not exceed the capacity. Find the maxflow, largest f such that, it is possible to simultaneously flow f*D i, between (s i,t i ) for all i.

18. 13 th Nov 2006 18 Max flow – Min Cut gap f* be the maxflow. Trivial upper bound f* ≤ a* Are these two equal? No. Leigthon-Rao (‘87) showed in some cases this gap ≤ O (log n) Garg, Vazerani (‘93) showed in case of unit demand among all source-sink pairs, this gap ≤ O (log k) LLR (‘94) : This gap is always ≤ O (log k) using, least distortion embedding of graph in L 1.

19. 13 th Nov 2006 19 LP for Max flow multi commodity Garg, Vazerani : Where minimum is over all metrics over G

20. 13 th Nov 2006 20 Let d be optimizing metric. Apply theorem 1 to embed (V,d) into L 1 m. say {x 1,x 2,…,x n }. ||x i -x j || 1 ≤ d i,j for all i,j. And, ||x_s i – x_t i )|| 1 ≥ Ω(1/log k)*d (s i,t i ) for every i = 1,2,…,k. Lets denote, x i,j = ||x i - x j || 1

21. 13 th Nov 2006 21 Lemma

22. 13 th Nov 2006 22 Max Flow Min Cut gap Suppose the for minimizing r, all x i,r in {0,1}. Then, for that r, a* ≤ Cap(S)/Dem(S) ≤ f* O (log k) So, Max-flow min cut gap is bounded by O (log k)

23. 13 th Nov 2006 23 Variational Argument Consider the expression 1.If all x’s take only two values, the valuation can be replaced by 0,1 2.Suppose x’s take three values, s > t > u. Then Consider the x’s which take value t. Fixing all other values let t varies over [u,s], 3.The expression is linear function in t. So changing t to u or s, the value of expression won’t increase. 4.Repeat this procedure till all variables take only two values.

24. 13 th Nov 2006 24 Algorithm Solve LP to find f*. Embed Graph with optimizing metric, into L 1 m. Find r which minimizes, Using Variational Argument, get near Optimal Cut

25. 13 th Nov 2006 25 Limitations Limitations of the LLR embedding: O(log 2 n) dimension: This is a real problem. O(n 2 ) distance computations must be performed in the process of embedding and embedding a query point requires O(n) distance computations: Too high if distance function is complex. O(log n) distortion: Experiments show that the actual distortions may be much smaller.

26. 13 th Nov 2006 26 Questions???

27. 13 th Nov 2006 27 Thank You !!!