An online 2-dimensional clustering problem with variable sized clusters

Divéki, Gabriella; Imreh, Csanád

doi:10.1007/s11081-013-9231-9

Cited by 6 publications

(10 citation statements)

References 14 publications

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“…This result is particularly interesting because it demonstrates that the deterministic competitive ratio of OnlSumRad (and of CSDF) changes abruptly, from constant to logarithmic, when we move from the line to the plane. Interestingly, this does not happen when the cost of each cluster is proportional to its area [9].…”

Section: Contributionmentioning

confidence: 99%

“…only studied CSDF on the line and proved that its deterministic competitive ratio is 1 + √ 2 for the strict and the intermediate model and (1 + √ 5)/2 for the flexible model. Recently, Divéki and Imreh [9] studied online clustering in two dimensions to minimize the sum of the setup costs and the area of the clusters. They proved that the competitive ratio of this problem lies in (2.22, 9] for the strict model and in (1.56, 7] for the flexible model.…”

Section: Introductionmentioning

confidence: 99%

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Online Sum-Radii Clustering

Fotakis

Koutris

2014

Theoretical Computer Science

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Abstract. In Online Sum-Radii Clustering, n demand points arrive online and must be irrevocably assigned to a cluster upon arrival. The cost of each cluster is the sum of a fixed opening cost and its radius, and the objective is to minimize the total cost of the clusters opened by the algorithm. We show that the deterministic competitive ratio of Online Sum-Radii Clustering for general metric spaces is Θ(log n), where the upper bound follows from a primal-dual algorithm and holds for general metric spaces, and the lower bound is valid for ternary Hierarchically Well-Separated Trees (HSTs) and for the Euclidean plane. Combined with the results of (Csirik et al., MFCS 2010), this result demonstrates that the deterministic competitive ratio of Online Sum-Radii Clustering changes abruptly, from constant to logarithmic, when we move from the line to the plane. We also show that Online Sum-Radii Clustering in metric spaces induced by HSTs is closely related to the Parking Permit problem introduced by (Meyerson, FOCS 2005). Exploiting the relation to Parking Permit, we obtain a lower bound of Ω(log log n) on the randomized competitive ratio of Online Sum-Radii Clustering in tree metrics. Moreover, we present a simple randomized O(log n)-competitive algorithm, and a deterministic O(log log n)-competitive algorithm for the fractional version of the problem.

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Section: Contributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Online Sum-Radii Clustering

Fotakis

Koutris

2014

Theoretical Computer Science

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“…A survey of algorithms for Unit Clustering in the context of online algorithms appears in [11]; see also [15] for a review overview. Clustering with variable sized clusters has been studied in [12,13]. Gridbased online algorithms for clustering problems have been developed by the same authors [14].…”

Section: Introductionmentioning

confidence: 99%

Online Unit Clustering in Higher Dimensions

Dumitrescu

Tóth

2018

Approximation and Online Algorithms

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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 ∞ 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 inWe 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.

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“…The same example which was used to prove the tightness in [19] in case of p = 2 also works for arbitrary p.…”

Section: Two Dimensional Versionmentioning

confidence: 94%

“…In this thesis the 1-dimensional and the 2-dimensional variants of the online clustering with variable sized clusters problem are considered which are presented in [17], [19] and [20]. In our model points of the 1-dimensional and 2-dimensional Euclidean space arrive one by one.…”

Section: Clustering Problems With the Cost Depending On The Diameter mentioning

An online 2-dimensional clustering problem with variable sized clusters

Cited by 6 publications

References 14 publications

Online Sum-Radii Clustering

Online Sum-Radii Clustering

Online Unit Clustering in Higher Dimensions

Online algorithms for clustering problems

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