Book ChapterDOI
Small space representations for metric min-sum k-clustering and their applications
Artur Czumaj,Christian Sohler +1 more
- pp 536-548
TLDR
The first efficient construction of a coreset for a-preserving metric embeddings is shown, and a sublinear-time polylogarithmic-factor approximation algorithm for the min-sum k-clustering problem for arbitrary values of k is obtained.Abstract:
The min-sum k-clustering problem is to partition a metric space (P, d) into k clusters C1, . . . , Ck ⊆ P such that Σi=1k Σ p,q∈Ci d(p,q) is minimized. We show the first efficient construction of a coreset for this problem. Our coreset construction is based on a new adaptive sampling algorithm. Using our coresets we obtain three main algorithmic results.
The first result is a sublinear time (4+Ɛ)-approximation algorithm for the min-sum k-clustering problem in metric spaces. The running time of this algorithm is O(n) for any constant k and Ɛ, and it is o(n2) for all k = o(log n/ log log n). Since the description size of the input is Θ(n2), this is sublinear in the input size.
Our second result is the first pass-efficient data streaming algorithm for min-sum k-clustering in the distance oracle model, i.e., an algorithm that uses poly(log n, k) space and makes 2 passes over the input point set arriving as a data stream.
Our third result is a sublinear-time polylogarithmic-factor approximation algorithm for the min-sum k-clustering problem for arbitrary values of k.
To develop the coresets, we introduce the concept of a-preserving metric embeddings. Such an embedding satisfies properties that (a) the distance between any pair of points does not decrease, and (b) the cost of an optimal solution for the considered problem on input (P, d′) is within a constant factor of the optimal solution on input (P, d). In other words, the idea is find a metric embedding into a (structurally simpler) metric space that approximates the original metric up to a factor of a with respect to a certain problem. We believe that this concept is an interesting generalization of coresets.read more
Citations
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Journal Article
Property Testing and its connection to Learning and Approximation
TL;DR: In this paper, the authors consider the question of determining whether a function f has property P or is e-far from any function with property P. In some cases, it is also allowed to query f on instances of its choice.
Proceedings ArticleDOI
Approximate clustering without the approximation
TL;DR: If any c-approximation to the given clustering objective φ is e-close to the target, then this paper shows that this guarantee can be achieved for any constant c > 1, and for the min-sum objective the authors can do this for any Constant c > 2.
Journal ArticleDOI
Clustering under approximation stability
TL;DR: It is shown that for any constant c > 1, (c,ε)-approximation-stability of k-median or k-means objectives can be used to efficiently produce a clustering of error O(ε) with respect to the target clustering, as can stability of the min-sum objective if the target clusters are sufficiently large.
Book ChapterDOI
Sublinear-time algorithms
Artur Czumaj,Christian Sohler +1 more
TL;DR: In this article, the authors survey recent advances in the area of sublinear-time alg orithms, and present a survey of the most recent developments in this area.
Journal Article
Sublinear-time algorithms
Artur Czumaj,Christian Sohler +1 more
TL;DR: This paper surveys recent advances in the area of sublinear-time alg orithms and describes the current state of the art in this area.
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