Geometry of Graphs and It’s Applications

1. Geometry of Graphsand It’s Applications Suijt P Gujar. Topics in Approximation Algorithms Instructor : T Kavitha

2. Agenda • Introduction • Definitions • Some Important Results • Embedding Finite metric space into (Rd, Lp) • Multi Commodity Flow via Low Distortion Embeddings • Applications.

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. 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. (Rd, Lp)

5. Linf (Chessboard): Hamming Distance:Let X = {0,1}k. Number of 1-bits in the exclusive-or Examples of Metrics Minkowski Lp Metric:Let X = Rd. L1 : Manhattan Distance , L2 : Euclidian Distance CutMetric : 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. Embedding • We will be considering Embedding of Metric Spaces to Banach Spaces esp. (Rd,Lp) • Metric embedding is a function • f : (X,dx) (Y,dy) • Distortion : The embedding is said to have distortion C if for any x1,x2 in X

7. A B C D Example • Consider Graph G with 4 vertices with unit distance between any pair of Vertices. • Embed this in (R2,L2) with 4 vertices as vertices as square with diagonal length ‘1’.

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

9. dim (X) ≤ n for ‘n’ point metric space. • Let X = {x1, x2, …, xn} with dij = d(xi,xj). • Map each point xi to ziЄ Rn whose kth coordinate is zik = dik. • || zi – zj ||inf = maxk | zik - zjk | ≥ | zij - zjj | = |dij - djj | = dij • On other hand, | zik - zjk| = |dik - djk| ≤ dij (Triangular inequality) so, || zi – zj ||inf = dij.

10. Johnson – Lindenstrauss Theorem (84) • Any set of n points in a n - dimesional Euclidian space can be mapped to Rd where d = O(ε-2log 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. JL Theorem contd… • Take A1, A2, …, Ad set of orthonormal Vectors randomly chosen in Rn. • A = [A1 A2 … Ad]t • For any x in X, x’ = Ax. Consider x Є Rn st || x ||2 = 1. So, E[xi2] = 1/n. E[x’.x’] = d/n. E[||x’||] = √(d/n) = m.

12. Let x,y be two vectors in Rn. And x’, y’ be corresponding embedding in Rd. • 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/n2. Best known bound is d = 16*ln n / ε2

13. Some results for embeddings We will define • Ldp = (Rd,Lp). • Cp(X) = minimum distortion with which X may be embedded in Lp.

15. LLR Algorithm for Embedding • Let q = O(log n). [Constant affects the constant in distortion.] • For i = 1,2,…,log n doFor j = 1 to q doAi,j = random subset of X of size 2i. • Map x to the vector {di,j}/Q1/p • di,j is the distance from x to the closest point in Ai,j and Q is the total number of subsets.

16. Theorem 1 • Let (X,d) be a finite metric space and {(si,ti) | i = 1,2,…,k} Є X x X. There exists a deterministic algorithm that finds an embedding f : X → l1O(n^2), so that d (x,y) ≥ ||f(x) – f(y)||1 for every x,y in X and ||f(si) – f(ti)||1 ≥ Ω(1/log k)*d (si,ti) for every i = 1,2,…,k.

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 (si,ti) and Demand Di 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*Di, between (si,ti) for all i.

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 L1.

19. LP for Max flow multi commodity • Garg, Vazerani : Where minimum is over all metrics over G

20. Let d be optimizing metric. • Apply theorem 1 to embed (V,d) into L1m. say {x1,x2,…,xn}. • ||xi-xj||1≤ di,j for all i,j. And, ||x_si – x_ti)||1 ≥ Ω(1/log k)*d (si,ti) for every i = 1,2,…,k. Lets denote, xi,j = ||xi - xj||1

21. Lemma

22. Max Flow Min Cut gap • Suppose the for minimizing r, all xi,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. Variational Argument Consider the expression • If all x’s take only two values, the valuation can be replaced by 0,1 • 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], • The expression is linear function in t. So changing t to u or s, the value of expression won’t increase. • Repeat this procedure till all variables take only two values.

24. Algorithm • Solve LP to find f*. • Embed Graph with optimizing metric, into L1m. • Find r which minimizes, • Using Variational Argument, get near • Optimal Cut

25. Limitations Limitations of the LLR embedding: O(log2n) 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. Questions???

27. Thank You !!!