Growing Balls in ℝ<sup><i>d</i></sup>

Bahrdt, Daniel; Becher, Michael; Funke, Stefan; Krumpe, Filip; Nusser, André; Seybold, Martin P.; Storandt, Sabine

doi:10.1137/1.9781611974768.20

Cited by 6 publications

(9 citation statements)

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“…The goal is to compute the elimination sequence efficiently. Bahrdt et al [4] and Funke and Storandt [7] improved upon the initial results and presented bounds which depend on the ratio ∆ of the largest to the smallest radius. Specifically, Funke and Storandt [7] show how to compute an elimination sequence for n balls in O(n log ∆(log +∆ d−1 )) time in arbitrary dimensions and in O(Cn polylog n) time for d = 2, where C denotes the number of different radii.…”

Section: Introductionmentioning

confidence: 92%

Agglomerative Clustering of Growing Squares

Castermans

Speckmann

Staals

et al. 2018

LATIN 2018: Theoretical Informatics

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We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs.We present a fully dynamic kinetic data structure that maintains a set of n disjoint growing squares. Our data structure uses O(n(log n log log n) 2 ) space, supports queries in worst case O(log 3 n) time, and updates in O(log 7 n) amortized time. This leads to an O(nα(n) log 7 n) time algorithm to solve the agglomerative clustering problem. This is a significant improvement over the current best O(n 2 ) time algorithms. ACM Subject Classification I.3.5 Computational Geometry and Object Modeling

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Section: Introductionmentioning

confidence: 92%

Agglomerative Clustering of Growing Squares

Castermans

Speckmann

Staals

et al. 2018

LATIN 2018: Theoretical Informatics

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“…Whenever two growing balls touch, the ball with the lower priority is eliminated. Bahrdt et al [6] and following up on their work Funke and Storandt [19] show how to compute an elimination sequence for n balls in O(n log ∆(log n + ∆ d−1 )) time in arbitrary dimensions and in O(Cn polylog n) time for d = 2, where C denotes the number of different radii and ∆ the ratio of the largest to the smallest radius. Quite recently Ahn et al [3] showed how to compute elimination orders for ball tournaments in sub-quadratic time, for balls and boxes in two or higher dimensions.…”

Section: Related Workmentioning

confidence: 99%

A Practical Algorithm for Spatial Agglomerative Clustering

Castermans

Speckmann

Verbeek

2019

2019 Proceedings of the Twenty-First Workshop on Algorithm Engineering and Experiments (ALENEX)

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“…The goal is to compute the elimination sequence efficiently. Bahrdt et al [4] and Funke and Storandt [8] improved upon the initial results and presented bounds which depend on the ratio Δ of the largest to the smallest radius. Specifically, Funke and Storandt [8] show how to compute an elimination sequence for n balls in O n log Δ(log +Δ d−1 ) time in arbitrary dimensions and in O Cn polylog n time for d = 2, where C denotes the number of different radii.…”

mentioning

confidence: 92%

“…In Sect. 4 we analyze the relation between canonical subsets in dominance queries. We show that for two range trees T R and T B in R d , the number of pairs of nodes r ∈ T R and b ∈ T B for which r occurs in the canonical subset of a dominance query defined by b and vice versa is only O n(log n log log n) d−1 , where n is the total size of T R and T B .…”

mentioning

confidence: 99%

Agglomerative Clustering of Growing Squares

et al. 2021

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We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of n disjoint growing squares. Our data structure uses $$O\bigl (n \log n \log \log n\bigr )$$ O ( n log n log log n ) space, supports queries in worst case $$O\bigl (\log ^2 n\bigr )$$ O ( log 2 n ) time, and updates in $$O\bigl (\log ^5 n\bigr )$$ O ( log 5 n ) amortized time. This leads to an $$O\bigl (n\,\alpha (n)\log ^5 n\bigr )$$ O ( n α ( n ) log 5 n ) time algorithm to solve the agglomerative clustering problem. This is a significant improvement over the current best $$O\bigl (n^2\bigr )$$ O ( n 2 ) time algorithms.

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Growing Balls in ℝ^d

Cited by 6 publications

References 0 publications

Agglomerative Clustering of Growing Squares

Agglomerative Clustering of Growing Squares

A Practical Algorithm for Spatial Agglomerative Clustering

Agglomerative Clustering of Growing Squares

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Growing Balls in ℝd

Cited by 6 publications

References 0 publications

Agglomerative Clustering of Growing Squares

Agglomerative Clustering of Growing Squares

A Practical Algorithm for Spatial Agglomerative Clustering

Agglomerative Clustering of Growing Squares

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Growing Balls in ℝ^d