In the following, RA refers to [MR95]. 1. JL also preserves angles.

In the following, RA refers to [MR95]. 1. JL also preserves angles. Show that the Johnson-Lindenstrauss lemma also (1±ε)-preserves angles among triples of points of P (you might need to increase the target dimension however by a constant factor). [Hint: For every angle, construct a equilateral triangle that its edges are being preserved by the projection (add the vertices of those triangles [conceptually] to the point set being embedded). Argue, that this implies that the angle is being preserved.] 2. Optimal k-median clustering for HST. [10 Points] Let (X, d) be a HST defined over n points, and let k > 0 be an integer. Provide an algorithm that computes the optimal k-median clustering of X in O(k 2 n) time. [Hint: Transform the HST into a tree where every node has only two children. Next, run a dynamic programming algorithm on this tree.] 3. Some metric space stuff. [10 Points] (A) Give a counter example to the following claim: Let (X, d) be a metric space, and let P be a partition of X. Prove that |U | = O(n). Namely, there are only n different resolutions that " matter " for a finite metric space.