Online unit clustering in higher dimensions

@article{Dumitrescu2017OnlineUC,
  title={Online unit clustering in higher dimensions},
  author={Adrian Dumitrescu and Csaba D. T{\'o}th},
  journal={ArXiv},
  year={2017},
  volume={abs/1708.02662}
}
We revisit the online Unit Clustering problem in higher dimensions: Given a set of n points in \(\mathbb {R}^d\), that arrive one by one, partition the points into clusters (subsets) of diameter at most one, so as to minimize the number of clusters used. In this paper, we work in \(\mathbb {R}^d\) using the \(L_\infty \) norm. We show that the competitive ratio of any algorithm (deterministic or randomized) for this problem must depend on the dimension d. This resolves an open problem raised by… 
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References

SHOWING 1-10 OF 44 REFERENCES
Online Unit Covering in Euclidean Space
TLDR
The online Unit Covering problem in higher dimensions is revisited, this time using Euclidean distance and the current best competitive ratio of an online algorithm is O(2^d d \log {d})\), due to Charikar et al. (2004).
An improved lower bound for one-dimensional online unit clustering
An online 2-dimensional clustering problem with variable sized clusters
TLDR
Two variants of the online clustering problems, both maintaining the properties that a point which was assigned to a given cluster must remain assigned to this cluster, and clusters cannot be merged are studied.
Better bounds on online unit clustering
Optimal algorithms for approximate clustering
TLDR
This work gives a polynomial time approximation scheme that estimates the optimal number of clusters under the second measure of cluster size within factors arbitrarily close to 1 for a fixed cluster size.
Online Clustering with Variable Sized Clusters
TLDR
Tight bounds on the competitive ratio of any online algorithm in each of the variants having the two essential properties that a point which has been assigned to a given cluster must remain assigned to that cluster and no pair of clusters can be merged are given.
On the online unit clustering problem
TLDR
This work designs a deterministic algorithm with a competitive ratio of 7/4 for the one-dimensional case, and provides the first non-trivial deterministic lower bound, improve the randomized lower Bound, and prove the first lower bounds for higher dimensions.
Grid based online algorithms for clustering problems
  • G. Divéki, C. Imreh
  • Computer Science, Mathematics
    2014 IEEE 15th International Symposium on Computational Intelligence and Informatics (CINTI)
  • 2014
TLDR
This paper considers the online variable sized clustering problem in d-dimensional Euclidean spaces where the cost of a cluster depends on the p-th power of its side and determines the competitive ratio of the algorithm GRID in the general case.
A Randomized Algorithm for Online Unit Clustering
TLDR
It is shown that surprisingly the naïve upper bound of 2 on the competitive ratio can be beaten and a new randomized 15/8-competitive online algorithm is presented, which provides some lower bounds and an extension to higher dimensions.
An Improved Algorithm for Online Unit Clustering
TLDR
A new randomized online algorithm is presented that achieves expected competitive ratio 11/6 against oblivious adversaries, improving the previous ratio of 15/8 and leads to improved upper bounds for the problem in two and higher dimensions as well.
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