K-center Clustering Research Articles

Towards Metric DBSCAN: Exact, Approximate, and Streaming Algorithms

DBSCAN is a popular density-based clustering algorithm that has many different applications in practice. However, the running time of DBSCAN in high-dimensional space or general metric space (\em e.g., clustering a set of texts by using edit distance) can be as large as quadratic in the input size. Moreover, most of existing accelerating techniques for DBSCAN are only available for low-dimensional Euclidean space. In this paper, we study the DBSCAN problem under the assumption that the inliers (the core points and border points) have a low intrinsic dimension (which is a realistic assumption for many high-dimensional applications), where the outliers can locate anywhere in the space without any assumption. First, we propose a k-center clustering based algorithm that can reduce the time-consuming labeling and merging tasks of DBSCAN to be linear. Further, we propose a linear time approximate DBSCAN algorithm, where the key idea is building a novel small-size summary for the core points. Also, our algorithm can be efficiently implemented for streaming data and the required memory is independent of the input size. Finally, we conduct our experiments and compare our algorithms with several popular DBSCAN algorithms. The experimental results suggest that our proposed approach can significantly reduce the computational complexity in practice.

Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and Variants

Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and Variants

Efficient Constrained K-center Clustering with Background Knowledge

Center-based clustering has attracted significant research interest from both theory and practice. In many practical applications, input data often contain background knowledge that can be used to improve clustering results. In this work, we build on widely adopted k-center clustering and model its input background knowledge as must-link (ML) and cannot-link (CL) constraint sets. However, most clustering problems including k-center are inherently NP-hard, while the more complex constrained variants are known to suffer severer approximation and computation barriers that significantly limit their applicability. By employing a suite of techniques including reverse dominating sets, linear programming (LP) integral polyhedron, and LP duality, we arrive at the first efficient approximation algorithm for constrained k-center with the best possible ratio of 2. We also construct competitive baseline algorithms and empirically evaluate our approximation algorithm against them on a variety of real datasets. The results validate our theoretical findings and demonstrate the great advantages of our algorithm in terms of clustering cost, clustering quality, and running time.

Critical station identification and robustness analysis of urban rail transit networks based on comprehensive vote-rank algorithm

Urban rail transit networks have become the most important traffic modal and greatly improve the traffic environment and travel efficiency of metropolis, and the critical station identification and robustness assessment of urban rail transit networks have become more and more important. In this paper, we propose a comprehensive vote-rank (CVR) algorithm to identify the critical stations, and adopt mutual information and k-center clustering algorithm to analyze the similar characteristics of passenger flow on different dates of urban rail transit networks. Meanwhile, Shanghai metro network is chosen as the example and we find that the optimal classification of passenger flow on different dates is 2 categories, and the passenger flows have the significant differences between conventional day and unconventional day. The results show that the critical stations on unconventional day are farther from the city center than the critical stations on conventional day, and some critical stations locate in the suburbs on unconventional day. Moreover, we study the structural and functional robustness characteristics of urban rail transit networks subjected to continuous malicious attacks to the top critical nodes, and the results show that Shanghai metro network has the worse robustness subjected to malicious attacks. Compared with the existed methods, we can declare that the proposed CVR algorithm is the most effective method for critical station identification and robustness assessment among the discussed methods in this paper, and this study will provide theoretical and practical support for the safe operation of urban rail transit networks.

Massively parallel and streaming algorithms for balanced clustering

Clustering is a fundamental problem with a wide range of applications. In this paper, we study a balanced version of the well-known k-center clustering problem, where the objective is to select k centers from a given set of points, and assign to each center at most L of the input points, so as to minimize the maximum distance from a point to the center to which it is assigned. We assume soft capacities, where points are allowed to be selected as center more than once. Motivated by applications involving massive datasets, we study the problem in the massively parallel computation (MPC) and streaming models. In particular, we present a two-round MPC algorithm for the balanced k-center problem, achieving an approximation factor of 5α+2, where α is the approximation factor of the corresponding sequential algorithm. This substantially improves the currently best approximation factor of 32α, available for the problem. We show that the approximation factor of our algorithm can be further improved to 5+ε in all spaces with bounded doubling dimension, including the Euclidean space. We also consider the balanced k-center problem in the streaming model, and present a constant-factor streaming algorithm in any metric space using O(knε) memory, and a factor 5+ε streaming algorithm using O(kpolylogn) memory in doubling metrics.

Clustering with faulty centers

In this paper we introduce and formally study the problem of k-clustering with faulty centers. Specifically, we study the faulty versions of k-center, k-median, and k-means clustering, where centers have some probability of not existing, as opposed to prior work where clients had some probability of not existing. For all three problems we provide fixed parameter tractable algorithms, in the parameters k, d, and ε, that (1+ε)-approximate the minimum expected cost solutions for points in d dimensional Euclidean space. For Faulty k-center we additionally provide a 5-approximation for general metrics. Significantly, all of our algorithms have only a linear dependence on n.

Fully Dynamic k-Center Clustering with Outliers

Fully Dynamic k-Center Clustering with Outliers

Scalable and Globally Optimal Generalized L₁ K-center Clustering via Constraint Generation in Mixed Integer Linear Programming

The k-center clustering algorithm, introduced over 35 years ago, is known to be robust to class imbalance prevalent in many clustering problems and has various applications such as data summarization, document clustering, and facility location determination. Unfortunately, existing k-center algorithms provide highly suboptimal solutions that can limit their practical application, reproducibility, and clustering quality. In this paper, we provide a novel scalable and globally optimal solution to a popular variant of the k-center problem known as generalized L_1 k-center clustering that uses L_1 distance and allows the selection of arbitrary vectors as cluster centers. We show that this clustering objective can be reduced to a mixed-integer linear program (MILP) that facilitates globally optimal clustering solutions. However, solving such a MILP may be intractable for large datasets; to remedy this, we present a scalable algorithm that leverages constraint generation to efficiently and provably converge to its global optimum. We further enhance outlier handling through a simple but elegant extension to our MILP objective. We first evaluate our algorithm on a variety of synthetic datasets to better understand its properties and then validate on 20 real benchmark datasets where we compare its performance to both traditional L_1 distance k-center and k-medians baselines. Our results demonstrate significant suboptimality of existing algorithms in comparison to our approach and further demonstrate that we can find optimal generalized L_1 k-center clustering solutions up to an unprecedented 1,000,000 data points.

Scalable and space-efficient Robust Matroid Center algorithms

Given a dataset V of points from some metric space, a popular robust formulation of the k-center clustering problem requires to select k points (centers) of V which minimize the maximum distance of any point of V from its closest center, excluding the z most distant points (outliers) from the computation of the maximum. In this paper, we focus on an important constrained variant of the robust k-center problem, namely, the Robust Matroid Center (RMC) problem, where the set of returned centers are constrained to be an independent set of a matroid of rank k built on V. Instantiating the problem with the partition matroid yields a formulation of the fair k-center problem, which has attracted the interest of the ML community in recent years. In this paper, we target accurate solutions of the RMC problem under general matroids, when confronted with large inputs. Specifically, we devise a coreset-based algorithm affording efficient sequential, distributed (MapReduce) and streaming implementations. For any fixed varepsilon >0, the algorithm returns solutions featuring a (3+varepsilon )-approximation ratio, which is a mere additive term varepsilon away from the 3-approximations achievable by the best known polynomial-time sequential algorithms. Moreover, the algorithm obliviously adapts to the intrinsic complexity of the dataset, captured by its doubling dimension D. For wide ranges of k,z,varepsilon , D, our MapReduce/streaming implementations require two rounds/one pass and substantially sublinear local/working memory. The theoretical results are complemented by an extensive set of experiments on real-world datasets, which provide clear evidence of the accuracy and efficiency of our algorithms and of their improved performance with respect to previous solutions.

Red–Blue k-Center Clustering with Distance Constraints

We consider a variant of the k-center clustering problem in IRd, where the centers can be divided into two subsets—one, the red centers of size p, and the other, the blue centers of size q, such that p+q=k, and each red center and each blue center must be a distance of at least some given α≥0 apart. The aim is to minimize the covering radius. We provide a bi-criteria approximation algorithm for the problem and a polynomial time algorithm for the constrained problem where all centers must lie on a given line ℓ. Additionally, we present a polynomial time algorithm for the case where only the orientation of the line is fixed in the plane (d=2), although the algorithm works even in IRd by constraining the line to lie in a plane and with a fixed orientation.

Parameterized Approximation Algorithms for K-center Clustering and Variants

k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.93, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm yields a 2^{O((klog k)/{epsilon})}dn-time (1+epsilon)-approximation for Euclidean k-center, where d is the dimension. In this work, we give a faster algorithm for small dimensions: roughly speaking an O^*(2^{O((1/epsilon)^{O(d)} k^{1-1/d} log k)})-time (1+epsilon)-approximation. In particular, the running time is roughly O^*(2^{O((1/epsilon)^{O(1)}sqrt{k}log k)}) in the plane. We complement our algorithmic result with a matching hardness lower bound. We also consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a 2^{O(klog k)}n^2 time 3-approximation for NUkC, and a 2^{O((klog k)/epsilon)}dn time (1+\epsilon)-approximation for Euclidean NUkC. The latter time bound matches the bound for k-center.

K-Center Clustering with Outliers in Sliding Windows

Metric k-center clustering is a fundamental unsupervised learning primitive. Although widely used, this primitive is heavily affected by noise in the data, so a more sensible variant seeks for the best solution that disregards a given number z of points of the dataset, which are called outliers. We provide efficient algorithms for this important variant in the streaming model under the sliding window setting, where, at each time step, the dataset to be clustered is the window W of the most recent data items. For general metric spaces, our algorithms achieve O1 approximation and, remarkably, require a working memory linear in k+z and only logarithmic in |W|. For spaces of bounded doubling dimension, the approximation can be made arbitrarily close to 3. For these latter spaces, we show, as a by-product, how to estimate the effective diameter of the window W, which is a measure of the spread of the window points, disregarding a given fraction of noisy distances. We also provide experimental evidence of the practical viability of the improved clustering and diameter estimation algorithms.

On Perturbation Resilience of Non-uniform k-Center

The Non-Uniform k-center (NUkC) problem has recently been formulated by Chakrabarty et al. [ICALP, 2016; ACM Trans Algorithms 16(4):46:1–46:19, 2020] as a generalization of the classical k-center clustering problem. In NUkC, given a set of n points P in a metric space and non-negative numbers r_1, r_2, ldots , r_k, the goal is to find the minimum dilation alpha and to choose k balls centered at the points of P with radius alpha cdot r_i for 1le ile k, such that all points of P are contained in the union of the chosen balls. They showed that the problem is mathsf {NP}-hard to approximate within any factor even in tree metrics. On the other hand, they designed a “bi-criteria” constant approximation algorithm that uses a constant times k balls. Surprisingly, no true approximation is known even in the special case when the r_i’s belong to a fixed set of size 3. In this paper, we study the NUkC problem under perturbation resilience, which was introduced by Bilu and Linial (Comb Probab Comput 21(5):643–660, 2012). We show that the problem under 2-perturbation resilience is polynomial time solvable when the r_i’s belong to a constant-sized set. However, we show that perturbation resilience does not help in the general case. In particular, our findings imply that even with perturbation resilience one cannot hope to find any “good” approximation for the problem.

How to Use Model-Based Cluster Analysis Efficiently in Person-Oriented Research.

Model-based cluster analysis (MBCA) was created to automatize the often subjective model-selection procedure of traditional explorative clustering methods. It is a type of finite mixture modelling, assuming that the data come from a mixture of different subpopulations following given distributions, typically multivariate normal. In that case cluster analysis is the exploration of the underlying mixture structure. In MBCA finding the possible number of clusters and the best clustering model is a statistical model-selection problem, where the models with differing number and type of component distributions are compared. For fitting a certain model MBCA uses a likelihood based Bayesian Information Criterion (BIC) to evaluate its appropriateness and the model with the highest BIC value is accepted as the final solution. The aim of the present study is to investigate the adequacy of automatic model selection in MBCA using BIC, and suggested alternative methods, like the Integrated Completed Likelihood Criterion (ICL), or Baudry’s method. An additional aim is to refine these procedures by using so called quality coefficients (QCs), borrowed from methodological advances within the field of exploratory cluster analysis, to help in the choice of an appropriate cluster structure (CLS), and also to compare the efficiency of MBCA in identifying a theoretical CLS with those of various other clustering methods. The analyses are restricted to studying the performance of various procedures of the type described above for two classification situations, typical in person-oriented studies: (1) an example data set characterized by a perfect theoretical CLS with seven types (seven completely homogeneous clusters) was used to generate three data sets with varying degrees of measurement error added to the original values, and (2) three additional data sets based on another perfect theoretical CLS with four types. It was found that the automatic decision rarely led to an optimal solution. However, dropping solutions with irregular BIC curves, and using different QCs as an aid in choosing between different solutions generated by MBCA and by fusing close clusters, optimal solutions were achieved for the two classification situations studied. With this refined procedure the revealed cluster solutions of MBCA often proved to be at least as good as those of different hierarchical and k-center clustering methods. MBCA was definitely superior in identifying four-type CLS models. In identifying seven-type CLS models MBCA performed at a similar level as the best of other clustering methods (such as k-means) only when the reliability level of the input variables was high or moderate, otherwise it was slightly less efficient.

Fair colorful k-center clustering

An instance of colorfulk-center consists of points in a metric space that are colored red or blue, along with an integer k and a coverage requirement for each color. The goal is to find the smallest radius rho such that there exist balls of radius rho around k of the points that meet the coverage requirements. The motivation behind this problem is twofold. First, from fairness considerations: each color/group should receive a similar service guarantee, and second, from the algorithmic challenges it poses: this problem combines the difficulties of clustering along with the subset-sum problem. In particular, we show that this combination results in strong integrality gap lower bounds for several natural linear programming relaxations. Our main result is an efficient approximation algorithm that overcomes these difficulties to achieve an approximation guarantee of 3, nearly matching the tight approximation guarantee of 2 for the classical k-center problem which this problem generalizes. algorithms either opened more than k centers or only worked in the special case when the input points are in the plane.

Extreme k-Center Clustering

Metric clustering is a fundamental primitive in machine learning with several applications for mining massive datasets. An important example of metric clustering is the k-center problem. While this problem has been extensively studied in distributed settings, all previous algorithms use Ω(k) space per machine and Ω(n k) total work. In this paper, we develop the first highly scalable approximation algorithm for k-center clustering, with O~(n^ε) space per machine and O~(n^(1+ε)) total work, for arbitrary small constant ε. It produces an O(log log log n)-approximate solution with k(1+o(1)) centers in O(log log n) rounds of computation.

Faster balanced clusterings in high dimension

The problem of constrained clustering has attracted significant attention in the past decades. In this paper, we study the balanced k-center, k-median, and k-means clustering problems where the size of each cluster is constrained by the given lower and upper bounds. The problems are motivated by the applications in processing large-scale data in high dimension. Existing methods often need to compute complicated matchings (or min cost flows) to satisfy the balance constraint, and thus suffer from high complexities especially in high dimension. We develop an effective framework for the three balanced clustering problems to address this issue, and our method is based on a novel spatial partition idea in geometry. For the balanced k-center clustering, we provide a 4-approximation algorithm that improves the existing approximation factors; for the balanced k-median and k-means clusterings, our algorithms yield constant and (1+ϵ)-approximation factors with any ϵ>0. More importantly, our algorithms achieve linear or nearly linear running times when k is a constant, and significantly improve the existing ones. Our results can be easily extended to metric balanced clusterings and the running times are sub-linear in terms of the complexity of n-point metric.

Identification of Opinion Spammers using Reviewer Reputation and Clustering Analysis

Online reviews have increasingly become a very important resource before making a purchasing decisions. Unfortunately, malicious sellers try to game the system by hiring a person or team (which is called spammers) to fabricate fake reviews to improve their reputation.Existing methods mainly take the problem as a general binary classification or focus on some heuristic rules. However, supervised learning methods relies heavily on a large number of labeled examples of deceptive and truthful opinions by domain experts, and most of features mentioned in the heuristic strategy ignore the characteristic of the group organization among spammers. In this paper, an effective method of identifying opinion spammers is proposed. Firstly, suspected spammers are detected by means of unsupervised learning based on reviewer’s reputation. We believe that the reviewer’s reputation has a direct relation with the quality of reviews. Generally, review written by user with lower reputation, shows lower quality and higher possibility to be fake. Therefore, the model assigns reputation score to each reviewer wherein the content based factors and activeness of reviewers are employed efficiently. On basis of all suspected spammers, k-center clustering algorithm is performed to further spot the spammers based on the observation of burst of review release time. Experimental results on Amazon’s dataset are encouraging and indicate that our approach poses high accuracy and recall, and good performance is achieved.

Fully Dynamic k-Center Clustering with Improved Memory Efficiency

Static and dynamic clustering algorithms are a fundamental tool in any machine learning library. Most of the efforts in developing dynamic machine learning and data mining algorithms have been focusing on the sliding window model (where at any given point in time only the most recent data items are retained) or more simplistic models. However, in many real-world applications one might need to deal with arbitrary deletions and insertions. We develop a $(2+\epsilon)$ -approximation algorithm for the k-center clustering problem with ``small'' amortized cost under the fully dynamic adversarial model. In such a model, points can be added or removed arbitrarily, provided that the adversary does not have access to the random choices of our algorithm. The amortized cost of our algorithm is poly-logarithmic when the ratio between the maximum and minimum distance between any two points in input is bounded by a polynomial, while k and ϵ are constant. Furthermore, we significantly improve the memory requirement of our fully dynamic algorithm, although at the cost of a worse approximation ratio of $4 +\epsilon$ . Our theoretical results are complemented with an extensive experimental evaluation on dynamic data from Twitter, Flickr, as well as trajectory data, demonstrating the effectiveness of our approach

High Dimensional Clustering with r-nets

Clustering, a fundamental task in data science and machine learning, groups a set of objects in such a way that objects in the same cluster are closer to each other than to those in other clusters. In this paper, we consider a well-known structure, so-called r-nets, which rigorously captures the properties of clustering. We devise algorithms that improve the runtime of approximating r-nets in high-dimensional spaces with1 and `2 metrics from, where . These algorithms are also used to improve a framework that provides approximate solutions to other high dimensional distance problems. Using this framework, several important related problems can also be solved efficiently, e.g.,pproximate kth-nearest neighbor distance-approximate Min-Max clustering,-approximate k-center clustering. In addition, we build an algorithm that-approximates greedy permutations in time O˜((dn+n2−α)·logΦ) where Φ is the spread of the input. This algorithm is used to -approximate k-center with the same time complexity.