Assume we have a set P = {p. ....p,} of n distinct points in plane, where p= (x(p), x(y))and x(p1) and y(p1) refers to x and y coordinates of p

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8. Assume we have a set P = {p. ....p,} of n distinct points in plane, where p= (x(p), x(y))and x(p1) and y(p1) refers to x and y coordinates of p, respectively. For any point z in plane. define the function (2. P) asx(z,P) =  11m — zII (1)

Note that 11m — zll refers to the square of the Euclidean distance between pj and z. Answer the following  x(p,)

(a) Let the point z =  ,  ). Show that for any arbitrary point z in plane. (zP) = (z4,P) + nIIz — zII

(b) Show that among all points in the plane. the point z minimizes Equation 

(c) Choose a point uniformly at random from P. Let the chosen point be p E P. Show that E1(p.P)]  2(z.P)

9. Show that any metric space (X,d) on n-points, can be embedded in O(Log2 n)-dimensional space with a distortion factor of O(log2 n). where the distances are measured with respect to Lj-metric (i.e. the Manhattan metric). [See Corollary 12.2.10. You need to provide missing

details in the proof

10. Let G = (V = L U R,E) be a bipartite graph, where ILl = RL Each edge e E E is also colored by one of the possible two colors {Green. Blue). Design a (randomized) polynomial-time algorithm that can determine whether there is a perfect matching in G that consists of exactly k-Green edges, where O  k  ILl is an integer. The input to the decision problem consists of the bipartite graph G and the niunber k.

(Hint: You should think of how we determined whether a bipartite graph has a perfectmatching via polynomial identity testing using Schwartz-Zippel Lemma and Lo1ation Lemma.You ma a.ssuine that the determinant of a matrix can he computed in polynomial time.)

How can you approximate Euclidean Minimum Spanning Tree for a set of n point.s P in Wusing the ideas from the Locality-Sensitive ordering paper/lecture? EMST is the minimumspanning tree of the complete graph on P. Veight of each edge e = (or) is the Euclidean

distance between the points u and r of P. What is the approximation factor? What is approximately the running time? [You may like to see the scribe notes b Dorsa on Locality- Sensitive Ordering

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