We revisit the online Unit Clustering problem in higher dimensions: Given a set of n points in \mathbb{R , 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 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 Epstein and van Stee (WAOA 2008). We also give a randomized online algorithm with competitive ratio O(d^2) for Unit Clustering of integer points (i.e., points in Z^d, d\in N, under L_{\infty} norm)  . We complement these results with some additional lower bounds for related problems in higher dimensions.
We revisit the online Unit Clustering and Unit Covering problems in higher dimensions: Given a set of n points in a metric space , that arrive one by one, Unit Clustering asks to partition the points into the minimum number of clusters (subsets) of diameter at most one ; while Unit Covering asks to cover all points by the minimum number of balls of unit radius . In this paper, we work in R^d using the L_\infty norm. We show that the competitive ratio of any online algorithm (deterministic or randomized) for Unit Clustering must depend on the dimension d. We also give a randomized online algorithm with competitive ratio O(d^2) for Unit Clustering of integer points (i.e., points in Z^d, d\in N, under L_{\infty} norm) . We show that the competitive ratio of any deterministic online algorithm for Unit Covering is at least 2^d. This ratio is the best possible, as it can be attained by a simple deterministic algorithm that assigns points to a predefined set of unit cubes . We complement these results with some additional lower bounds for related problems in higher dimensions.
R
problem
and Unit Covering problems
R
\mathbb{R
a metric space
A
Unit Clustering asks to
A
the minimum number of
R
, so as to minimize the number of clusters used
; while Unit Covering asks to cover all points by the minimum number of balls of unit radius
A
online
R
this problem
Unit Clustering
D
This resolves an open problem raised by Epstein and van Stee (WAOA 2008).
A
. We show that the competitive ratio of any deterministic online algorithm for Unit Covering is at least 2^d. This ratio is the best possible, as it can be attained by a simple deterministic algorithm that assigns points to a predefined set of unit cubes