Coresets for $$(k, \ell )$$-Median Clustering Under the Fréchet Distance

Buchin, Maike; Rohde, Dennis

doi:10.1007/978-3-030-95018-7_14

Cited by 6 publications

(5 citation statements)

References 17 publications

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“…Thus combing with our Theorem 3.2 and Theorem 5.2, we obtain an ǫ-coreset for k-Median on P with size Õ ǫ −3 k 3 d 2 ℓ 2 log(m) . Compared with recent results, our coreset has size independent of n which improves over the O(log n) dependence in[BR22], and has only logarithmic dependence in m instead of poly(m) as in[Nat21].Finally, similar results can also be obtained for clustering under Hausdorff distance, since a similar shattering dimension bound for Hausdorff distance is obtained in [DNPP21, Theorem 7.7] as well.Similarly, we have that cost z (P, C, Γ) − cost z (S, C, Γ) ≤ ǫ(α + 1) cost z (P, C, Γ).and conclude that | cost z (P, C, Γ) − cost z (S, C, Γ)| ≤ ǫ(α + 1) • cost z (P, C, Γ).…”

supporting

confidence: 74%

“…In this problem, also known as (k, ℓ)-Median, the data set comprises of polygonal curves in R d , each with at most m line segments, and the center curves are restricted to at most ℓ line segments. This is the first coreset whose size is independent of the number of input curves, improving over [BR22].…”

Section: Our Resultsmentioning

confidence: 99%

“…We present two more results that can be obtained by plugging in known uniform shattering dimension bound into our framework. Interestingly, whether the weighted shattering dimensions of both metric spaces are bounded is still open and in previous works [BJKW21a,BR22] much effort has been put in bypassing it.…”

Section: More Coresets For K-median Via Uniform Shattering Dimensionmentioning

confidence: 99%

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The Power of Uniform Sampling for Coresets

Braverman¹,

Cohen-Addad²,

-C.³

et al. 2022

Preprint

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Motivated by practical generalizations of the classic k-median and k-means objectives, such as clustering with size constraints, fair clustering, and Wasserstein barycenter, we introduce a meta-theorem for designing coresets for constrained-clustering problems. The meta-theorem reduces the task of coreset construction to one on a bounded number of ring instances with a much-relaxed additive error. This reduction enables us to construct coresets using uniform sampling, in contrast to the widely-used importance sampling, and consequently we can easily handle constrained objectives. Notably and perhaps surprisingly, this simpler sampling scheme can yield coresets whose size is independent of n, the number of input points.Our technique yields smaller coresets, and sometimes the first coresets, for a large number of constrained clustering problems, including capacitated clustering, fair clustering, Euclidean Wasserstein barycenter, clustering in minor-excluded graph, and polygon clustering under Fréchet and Hausdorff distance. Finally, our technique yields also smaller coresets for 1-median in lowdimensional Euclidean spaces, specifically of size Õ(ε −1.5 ) in R 2 and Õ(ε −1.6 ) in R 3 .

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supporting

confidence: 74%

Section: Our Resultsmentioning

confidence: 99%

Section: More Coresets For K-median Via Uniform Shattering Dimensionmentioning

confidence: 99%

See 1 more Smart Citation

The Power of Uniform Sampling for Coresets

Braverman¹,

Cohen-Addad²,

-C.³

et al. 2022

Preprint

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“…. This improves over [BR21] and [BCJ + 22], although our construction is not polynomial time in that case.…”

Section: Application To Minor-excluded Graphs Graphs With Bounded Tre...mentioning

confidence: 88%

A new coreset framework for clustering

Cohen-Addad

Saulpic

Schwiegelshohn

2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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In all state-of-the-art sketching and coreset techniques for clustering, as well as in the best known fixed-parameter tractable approximation algorithms, randomness plays a key role. For the classic k-median and k-means problems, there are no known deterministic dimensionality reduction procedure or coreset construction that avoid an exponential dependency on the input dimension d, the precision parameter ε −1 or k. Furthermore, there is no coreset construction that succeeds with probability 1 − 1/n and whose size does not depend on the number of input points, n. This has led researchers in the area to ask what is the power of randomness for clustering sketches [Feldman WIREs Data Mining Knowl. Discov'20].Similarly, the best approximation ratio achievable deterministically without a complexity exponential in the dimension are 1 + √ 2 for k-median [Cohen-Addad, Esfandiari, Mirrokni, Narayanan, STOC'22] and 6.12903 for k-means [Grandoni, Ostrovsky, Rabani, Schulman, Venkat, Inf. Process. Lett. '22]. Those are the best results, even when allowing a complexity FPT in the number of clusters k: this stands in sharp contrast with the (1 + ε)-approximation achievable in that case, when allowing randomization.In this paper, we provide deterministic sketches constructions for clustering, whose size bounds are close to the best-known randomized ones. We show how to compute a dimension reduction onto ε −O(1) log k dimensions in time k O(ε −O(1) +log log k) poly(nd), and how to build a coreset of size O k 2 log 3 kε −O(1) in time 2 ε O(1) k log 3 k + k O(ε −O(1) +log log k) poly(nd). In the case where k is small, this answers an open question of [Feldman WIDM'20] and [Munteanu and Schwiegelshohn, Künstliche Intell. '18] on whether it is possible to efficiently compute coresets deterministically.We also construct a deterministic algorithm for computing (1+ε)-approximation to k-median and k-means in high dimensional Euclidean spaces in time 2 k 2 log 3 k/ε O(1) poly(nd), close to the best randomized complexity of 2 (k/ε) O(1) nd (see [Kumar, Sabharwal, Sen, JACM 10] and [Bhattacharya, Jaiswal, Kumar, TCS'18]).Furthermore, our new insights on sketches also yield a randomized coreset construction that uses uniform sampling, that immediately improves over the recent results of [Braverman et al. FOCS '22] by a factor k.

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“…The running time is Õ n • m O(kℓ) • 2 O((k 3 /ε 2 ) log 2 (1/µ)) • (k/(µε)) O(dkℓ) . There are some results on coresets for (k, ℓ)-median clustering under Fréchet distance [7].…”

Section: Introductionmentioning

confidence: 99%

Curve Simplification and Clustering under Fréchet Distance

Cheng¹,

Huang²

2022

Preprint

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We present new approximation results on curve simplification and clustering under Fréchet distance. Let T = {τ i : i ∈ [n]} be polygonal curves in R d of m vertices each. Let ℓ be any integer from [m]. We study a generalized curve simplification problem: given error boundsWe present an algorithm that returns a null output or a curve σ of at most ℓ vertices such that O(dℓ) . This algorithm yields the first polynomial-time bicriteria approximation scheme to simplify a curve τ to another curve σ, where the vertices of σ can be anywhere in R d , so that d F (σ, τ ) ≤ (1+ε)δ and |σ| ≤ (1+α)•min{|c| : d F (c, τ ) ≤ δ} for any given δ > 0 and any fixed α, ε ∈ (0, 1). The running time is Õ m O(1/α) •(d/(αε)) O(d/α) . By combining our technique with some previous results in the literature, we obtain an approximation algorithm for (k, ℓ)-median clustering. Given T , it computes a set Σ of k curves, each of ℓ vertices, such that i∈[n] min σ∈Σ d F (σ, τ i ) is within a factor 1+ε of the optimum with probability at least 1−µ for any given µ, ε ∈ (0, 1). The running time is Õ n • m O(kℓ 2 ) • µ −O(kℓ) • (dkℓ/ε) O((dkℓ/ε) log(1/µ)) .

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Coresets for $$(k, \ell )$$-Median Clustering Under the Fréchet Distance

Cited by 6 publications

References 17 publications

The Power of Uniform Sampling for Coresets

The Power of Uniform Sampling for Coresets

A new coreset framework for clustering

Curve Simplification and Clustering under Fréchet Distance

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