By Alan George

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In step i = 1, . . ,n where di (z) := min j≤i z − c j denotes the distance of z to the computed centers c j , j ≤ i. After the computation of c1 , . . , ck , assign each point zi , i = 1, . . , n, to its nearest center. 29. The farthest-point clustering algorithm computes a partition with maximum radius at most twice the optimum. Proof. Let ck+1 denote the next center chosen in the farthest-point clustering algorithm. Then the maximum radius of the farthest-point clustering solution equals dk (ck+1 ).

1, we consider the multiplication of such matrices by a vector in Sect. 2. It will be seen that this can be done with logarithmic-linear complexity. In Sect. 3 we describe how to perform this multiplication on a parallel computer. Matrix operations such as addition and multiplication and relations between local and global norm estimates were investigated in detail in [116]. We review the results and improve some of the estimates in Sect. 4, Sect. 5, and Sect. 7, since these results will be important for higher matrix operations.

17) give ν µ (Xt ) ≥ ∑i∈t µ (Xi ) ≥ |t| maxi∈I µ (Xi )/cU . Cardinality balanced clustering leads to geometrically balanced trees under suitable assumptions. 25. Assume that Xi , i ∈ I, are quasi-uniform. 22) is satisﬁed. ( ) Proof. We assume that |I| = 2 p for some p ∈ N. 24 that µ (Xt ) ≥ |t| max µ (Xi )/(cU ν ) = 2− |I| max µ (Xi )/(cU ν ) ≥ 2− µ (Ω )/(cU ν ), i∈I i∈I since µ (Ω ) ≤ ∑i∈I µ (Xi ) ≤ |I| maxi∈I µ (Xi ). On the other hand, µ (Xt ) ≤ |t| max µ (Xi ) = 2− |I| max µ (Xi ) ≤ cU ν 2− µ (Ω ), i∈I i∈I because ν µ (Ω ) ≥ ∑i∈I µ (Xi ) ≥ |I| mini∈I µ (Xi ).