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On the Advice Complexity of Online Unit Clustering

Judit Nagy-György

Abstract

In online unit clustering, points of a metric space arriving one by one must be partitioned into clusters of diameter at most 1, where the cost is the number of clusters. This paper gives linear upper and lower bounds on the advice complexity of 1-competitive online unit clustering algorithms, in terms of the number of points in $\mathbb{R}^d$ and $\mathbb{Z}^d$.

On the Advice Complexity of Online Unit Clustering

Abstract

In online unit clustering, points of a metric space arriving one by one must be partitioned into clusters of diameter at most 1, where the cost is the number of clusters. This paper gives linear upper and lower bounds on the advice complexity of 1-competitive online unit clustering algorithms, in terms of the number of points in and .
Paper Structure (4 sections, 7 theorems, 13 equations, 1 table)

This paper contains 4 sections, 7 theorems, 13 equations, 1 table.

Table of Contents

  1. Introduction
  2. Lower bounds
  3. Upper bounds
  4. Acknowledgements

Key Result

Proposition 2

If for input sequences $I_1$ and $I_2$ there exists $k>1$ such that $I_1^{(k)}=I_2^{(k)}=I'$, where $I_j^{(k)}$ denotes the subsequence consisting of the first $k$ elements of $I_j$, furthermore $I'$ must have different clusterings in the optimal service of $I_1$ and $I_2$, then $\mathbf{A}$ needs d

Theorems & Definitions (15)

  • Definition 1
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 5 more