1. Given a metric space (X,d), where |X| <>∞, and k ∈ N, we would like to find a partition of X into C1, . . .,Ck which minimizes the expression Gk−diam((X,d), (C1, . . .,Ck )) = max j∈[d] diam(Cj ), where diam(Cj ) = maxx,x_ Cj d(x, x_) (we use the convention diam(Cj ) = 0 if |Cj | <>2).
Similarly to the k-means objective, it is NP-hard to minimize the k-diam objective. Fortunately, we have a very simple approximation algorithm: Initially, we pick some x ∈ X and set μ1 = x. Then, the algorithm iteratively sets ∀ j ∈ {2,
Hint: Consider the point μk+1 (in other words, the next center we would have chosen, if we wanted k +1 clusters). Let r = minj∈[k] d(μj,μk+1). Prove the following inequalities Gk−diam((X,d), (ˆC1, . . ., ˆCk )) ≤ 2r Gk−diam((X,d), (C∗ 1 , . . .,C∗ k )) ≥r .
