Fully dynamic hierarchical diameter k-clustering and k-center
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We develop dynamic data structures for maintaining a hierarchical k-center clustering when the points come from a discrete space {1,...,Δ}^d. Our first data structure is for the low dimensional setting, i.e., d is a constant, and processes insertions, deletions and cluster representative queries in ^O(1) (Δ n) time, where n is the current size of the point set. For the high dimensional case and an integer parameter ℓ > 1, we provide a randomized data structure that maintains an O(d ℓ)-approximation. The amortized expected insertion time is O(d^2 ℓ n Δ). The amortized expected deletion time is O(d^2 n^1/ℓ^2 n Δ). At any point of time, with probability at least 1-1/n, the data structure can correctly answer all queries for cluster representatives in O(d ℓ n Δ) time per query.
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