Prove the following inequalities Gk−diam((X,d), (ˆC1, . . ., ˆCk )) ≤ 2r Gk−diam((X,d), (C∗ 1 , . . .,C∗ k )) ≥r .

1. Given a metric space (X,d), where |X<>∞, and ∈ N, we would like to find a partition of into C1, . . .,Ck which minimizes the expression Gk−diam((X,d)(C1, . . .,Ck )) = max j∈[d] diam(Cj )where diam(Cj ) = maxx,xCj d(xx_) (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 ∈ and set μ1 = x. Then, the algorithm iteratively sets ∀ ∈ {2,

Hint: Consider the point μk+1 (in other words, the next center we would have chosen, if we wanted +1 clusters). Let = minj∈[kd(μjk+1). Prove the following inequalities Gk−diam((X,d)C1, . . ., ˆCk )) ≤ 2r Gk−diam((X,d)(C∗ 1 , . . .,C∗ )) ≥.

 

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