Online Clustering with Variable Sized Clusters

Csirik, János; Epstein, Leah; Imreh, Csanád; Levin, Asaf

doi:10.1007/978-3-642-15155-2_26

Cited by 6 publications

(16 citation statements)

References 14 publications

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“…In [15] the one-dimensional variant of our problem is examined (with linear cost), where there is no restriction on the length of a cluster, and the cost of a cluster is the sum of a fixed setup cost and its diameter. Both the strict and the flexible model have been investigated.…”

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

confidence: 99%

“…An intermediate model, where the diameter is fixed in advance but the exact location can be modified is also studied. In [15] tight bounds are given on the competitive ratio of any online algorithm belonging to any of these variants. In the strict model tight bounds are given of 1 + √ 2 ≈ 2.414 on the competitive ratio for the online problem, and tight bounds of 2 in the semi-online version (points are presented sorted by their location).…”

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

confidence: 99%

“…Using the flexible model, the best competitive ratio dropped to Φ = 1+ √ 5 2 ≈ 1.618. The semi-online version of this model is solved optimally using a trivial algorithm which is discussed as well in [15].…”

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

confidence: 99%

“…This is an interesting transition: our offline clustering problem on the line with linear objective function can be solved with a simple greedy algorithm with O(n · logn) time complexity (see [15]), the problem on the line with square cost can be solved by a standard O(n 3 ) time dynamic programming algorithm. On the other hand the 2-dimensional case seems to be much harder, we conjecture that it is NP-hard.…”

Section: Lemmamentioning

confidence: 99%

“…In the case of 1 dimension with the linear cost the ECC (extend closed cluster) algorithm (see [15]) has competitive ratio Proof. Consider the following sequence of requests which contains n + 1 points.…”

Section: The Online Problemmentioning

confidence: 99%

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Online algorithms for clustering problems

Divéki¹

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

confidence: 99%

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

confidence: 99%

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

confidence: 99%

Section: Lemmamentioning

confidence: 99%

Section: The Online Problemmentioning

confidence: 99%

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Online algorithms for clustering problems

Divéki¹

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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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Online Sum-Radii Clustering

Fotakis

Koutris

2012

Mathematical Foundations of Computer Science 2012

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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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Online Clustering with Variable Sized Clusters

Cited by 6 publications

References 14 publications

Online algorithms for clustering problems

Online algorithms for clustering problems

Online Unit Clustering in Higher Dimensions

Online Sum-Radii Clustering

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