Nearly Optimal Dynamic $k$-Means Clustering for High-Dimensional Data

Hu, Wei; Song, Zhao; Yang, Lin F.; Zhong, Peilin

doi:10.48550/arxiv.1802.00459

Cited by 4 publications

(9 citation statements)

References 35 publications

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“…In this paper, we suppose all input and output points are in {1, 2, • • • , ∆} d for some ∆, d ∈ Z ≥1 . This assumption is without loss of generality since if the clustering cost is non-zero, we can always discretize the space by changing the cost by an arbitrary small multiplicative error [BFL + 17,HSYZ18]. Given a point set Q ∈ [∆] d , a strong (η, )-coreset of Q for capacitated k-clustering in r is a subset of points Q ⊆ Q with weights w : Q → R >0 such that for any capacity t ≥ |Q|/k and any set of…”

Section: Our Resultsmentioning

confidence: 99%

“…Let F denote the event that the total number of center cells is at most 2000kL. Lemma 3.2 (Lemma 14 of [HSYZ18]). F happens with probability at least 0.99.…”

Section: Points Partitioningmentioning

confidence: 99%

“…Lemma 4.2 (Lemma 19 in [HSYZ18]). For i ∈ {0, 1, • • • , L}, α, β ∈ Z ≥1 , δ ∈ (0, 0.5), there is a dynamic streaming algorithm Storing(G i , α, β, δ) which uses O(αβdL • log 2 (αβ/δ)) bits to process a stream of insertion and deletion of points such that 1. if the algorithm does not output FAIL, the algorithm will return…”

Section: Coreset Construction Over Dynamic Streammentioning

confidence: 99%

See 2 more Smart Citations

Streaming Balanced Clustering

Esfandiari¹,

Mirrokni²,

Zhong³

2019

Preprint

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Clustering of data points in metric space is among the most fundamental problems in computer science with plenty of applications in data mining, information retrieval and machine learning. Due to the necessity of clustering of large datasets, several streaming algorithms have been developed for different variants of clustering problems such as k-median and k-means problems. However, despite the importance of the context, the current understanding of balanced clustering (or more generally capacitated clustering) in the streaming setting is very limited. The only previously known streaming approximation algorithm for capacitated clustering requires three passes and only handles insertions.In this work, we develop the first single pass streaming algorithm for a general class of clustering problems that includes capacitated k-median and capacitated k-means in Euclidean space, using only poly(kd log ∆) space, where k is the number of clusters, d is the dimension and ∆ is the maximum relative range of a coordinate 1 . This algorithm only violates the capacity constraint by a 1 + factor. Interestingly, unlike the previous algorithm, our algorithm handles both insertions and deletions of points. To provide this result we define a decomposition of the space via some curved half-spaces. We used this decomposition to design a strong coreset of size poly(kd log ∆) for balanced clustering. Then, we show that this coreset is implementable in the streaming and distributed settings.1 Note that d log ∆ is the space required to represent one point.

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Section: Our Resultsmentioning

confidence: 99%

“…Let F denote the event that the total number of center cells is at most 2000kL. Lemma 3.2 (Lemma 14 of [HSYZ18]). F happens with probability at least 0.99.…”

Section: Points Partitioningmentioning

confidence: 99%

Section: Coreset Construction Over Dynamic Streammentioning

confidence: 99%

See 1 more Smart Citation

Streaming Balanced Clustering

Esfandiari¹,

Mirrokni²,

Zhong³

2019

Preprint

Self Cite

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“…The offline construction is almost the same as the algorithm proposed by [HSYZ18]. We put all analysis into Appendix B for completeness.…”

Section: Offline Coreset Constructionmentioning

confidence: 99%

“…The goal is to prove Theorem 5.2. The analysis is similar to [HSYZ18]. We include the analysis in this section for completeness.…”

Section: A1 Smooth Function and Smooth Histogrammentioning

confidence: 99%

Improved Sliding Window Algorithms for Clustering and Coverage via Bucketing-Based Sketches

Epasto¹,

Mahdian²,

Mirrokni³

et al. 2021

Preprint

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Streaming computation plays an important role in large-scale data analysis. The sliding window model is a model of streaming computation which also captures the recency of the data. In this model, data arrives one item at a time, but only the latest W data items are considered for a particular problem. The goal is to output a good solution at the end of the stream by maintaining a small summary during the stream.In this work, we propose a new algorithmic framework for designing efficient sliding window algorithms via bucketing-based sketches. Based on this new framework, we develop space-efficient sliding window algorithms for k-cover, k-clustering and diversity maximization problems. For each of the above problems, our algorithm achieves (1 ± )-approximation. Compared with the previous work, it improves both the approximation ratio and the space.

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Almost Tight Approximation Algorithms for Explainable Clustering

Esfandiari

Mirrokni

Narayanan

2022

Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

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In this paper, we design replicable algorithms in the context of statistical clustering under the recently introduced notion of replicability in (Impagliazzo et al., 2022). A clustering algorithm is replicable if, with high probability, it outputs the exact same clusters after two executions with datasets drawn from the same distribution when its internal randomness is shared across the executions. We propose such algorithms for the statistical kmedians, statistical k-means, and statistical k-centers problems by utilizing approximation routines for their combinatorial counterparts in a black-box manner. In particular, we demonstrate a replicable O(1)-approximation algorithm for statistical Euclidean k-medians (k-means) with poly(d) sample complexity. We also describe a O(1)-approximation algorithm with an additional O(1)-additive error for statistical Euclidean k-centers, albeit with exp(d) sample complexity.

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Nearly Optimal Dynamic $k$-Means Clustering for High-Dimensional Data

Cited by 4 publications

References 35 publications

Streaming Balanced Clustering

Streaming Balanced Clustering

Improved Sliding Window Algorithms for Clustering and Coverage via Bucketing-Based Sketches

Almost Tight Approximation Algorithms for Explainable Clustering

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