Fully dynamic hierarchical diameter k-clustering and k-center

TL;DR

This paper introduces dynamic data structures for hierarchical k-center clustering in discrete spaces, efficiently handling insertions, deletions, and queries in low and high dimensions with approximation guarantees.

Contribution

It presents novel dynamic data structures for hierarchical k-center clustering that operate efficiently in both low and high dimensions, with provable approximation bounds.

Findings

Low-dimensional data structure processes updates in polylogarithmic time.

High-dimensional structure achieves an O(d ll)-approximation with polylogarithmic update times.

Queries for cluster representatives are answered efficiently with high probability.

Abstract

We develop dynamic data structures for maintaining a hierarchical k-center clustering when the points come from a discrete space $\{1,\ldots,\Delta\}^d$. 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 $\log^{O(1)} (\Delta n)$ time, where $n$ is the current size of the point set. For the high dimensional case and an integer parameter $\ell > 1$, we provide a randomized data structure that maintains an $O(d \ell)$-approximation. The amortized expected insertion time is $O(d^2 \ell \log n \log \Delta)$. The amortized expected deletion time is $O(d^2 n^{1/\ell} \log^2 n \log \Delta)$. At any point of time, with probability at least $1-1/n$, the data structure can correctly answer all queries for cluster representatives in $O(d \ell \log n \log \Delta)$ time per query.