Online Unit Clustering in Higher Dimensions [chapter]

Adrian Dumitrescu, Csaba D. Tóth
2018 Lecture Notes in Computer Science  
We revisit the online Unit Clustering problem in higher dimensions: Given a set of n points in 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 R d using the L ∞ 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
more » ... 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 ∈ N, under L ∞ norm). We complement these results with some additional lower bounds for related problems in higher dimensions. Problem 3. Unit Clustering. Given a set of n points in R d , partition the set into clusters of diameter at most one so that number of clusters is minimized. Problems 1 and 2 are easily solved in polynomial time for points on the line, i.e., for d = 1; however, both problems become NP-hard already in the Euclidean plane [17, 22] . Factor 2 approximations are known for k-Center in any metric space (and so for any dimension) [16, 18] ;
doi:10.1007/978-3-319-89441-6_18 fatcat:bsw2tmso2bdtzojthumahiaxse