Accurate MapReduce Algorithms for k-Median and k-Means in General Metric Spaces

Center-based clustering is a fundamental primitive for data analysis and becomes very challenging for large datasets. In this paper, we focus on the popular k-median and k-means variants which, given a set P of points from a metric space and a parameter k<|P|, require to identify a set S of k centers minimizing, respectively, the sum of the distances and of the squared distances of all points in P from their closest centers. Our specific focus is on general metric spaces, for which it is reasonable to require that the centers belong to the input set (i.e., S subseteq P). We present coreset-based 3-round distributed approximation algorithms for the above problems using the MapReduce computational model. The algorithms are rather simple and obliviously adapt to the intrinsic complexity of the dataset, captured by the doubling dimension D of the metric space. Remarkably, the algorithms attain approximation ratios that can be made arbitrarily close to those achievable by the best known polynomial-time sequential approximations, and they are very space efficient for small D, requiring local memory sizes substantially sublinear in the input size. To the best of our knowledge, no previous distributed approaches were able to attain similar quality-performance guarantees in general metric spaces

Distributed Clustering in General Metrics via Coresets

Center-based clustering is a fundamental primitive for data analysis and is very challenging for large datasets. We developed coreset based space/round-efficient MapReduce algorithms to solve the k-center, k-median, and k-means variants in general metrics. Remarkably, the algorithms obliviously adapt to the doubling dimension of the metric space, and attain approximation ratios that can be made arbitrarily close to those achievable by the best known polynomial-time sequential approximations

Distributed k-Means with Outliers in General Metrics

Center-based clustering is a pivotal primitive for unsupervised learning and data analysis. A popular variant is the k-means problem, which, given a set P of points from a metric space and a parameter k < |P|, requires finding a subset S ⊂ P of k points, dubbed centers, which minimizes the sum of all squared distances of points in P from their closest center. A more general formulation, introduced to deal with noisy datasets, features a further parameter z and allows up to z points of P (outliers) to be disregarded when computing the aforementioned sum. We present a distributed coreset-based 3-round approximation algorithm for k-means with z outliers for general metric spaces, using MapReduce as a computational model. Our distributed algorithm requires sublinear local memory per reducer, and yields a solution whose approximation ratio is an additive term O(γ) away from the one achievable by the best known polynomial-time sequential (possibly bicriteria) approximation algorithm, where γ can be made arbitrarily small. An important feature of our algorithm is that it obliviously adapts to the intrinsic complexity of the dataset, captured by its doubling dimension D. To the best of our knowledge, no previous distributed approaches were able to attain similar quality-performance tradeoffs for general metrics

Scalable Distributed Approximation of Internal Measures for Clustering Evaluation

The most widely used internal measure for clustering evaluation is the silhouette coefficient, whose naive computation requires a quadratic number of distance calculations, which is clearly unfeasible for massive datasets. Surprisingly, there are no known general methods to efficiently approximate the silhouette coefficient of a clustering with rigorously provable high accuracy. In this paper, we present the first scalable algorithm to compute such a rigorous approximation for the evaluation of clusterings based on any metric distances. Our algorithm hinges on a Probability Proportional to Size (PPS) sampling scheme, and, for any fixed $\varepsilon, \delta \in (0,1)$, it approximates the silhouette coefficient within a mere additive error $O(\varepsilon)$ with probability $1-\delta$, using a very small number of distance calculations. We also prove that the algorithm can be adapted to obtain rigorous approximations of other internal measures of clustering quality, such as cohesion and separation. Importantly, we provide a distributed implementation of the algorithm using the MapReduce model, which runs in constant rounds and requires only sublinear local space at each worker, which makes our estimation approach applicable to big data scenarios. We perform an extensive experimental evaluation of our silhouette approximation algorithm, comparing its performance to a number of baseline heuristics on real and synthetic datasets. The experiments provide evidence that, unlike other heuristics, our estimation strategy not only provides tight theoretical guarantees but is also able to return highly accurate estimations while running in a fraction of the time required by the exact computation, and that its distributed implementation is highly scalable, thus enabling the computation of internal measures for very large datasets for which the exact computation is prohibitive.Comment: 16 pages, 4 tables, 1 figur

DECENTRALIZED NETWORK BANDWIDTH PREDICTION AND NODE SEARCH

As modern computing becomes increasingly data-intensive and distributed, it is becoming crucial to effectively manage and exploit end-to-end network bandwidth information from hosts on wide-area networks. Inspired by the finding that Internet bandwidth can be represented approximately in a tree metric space, we focus on three specific research problems. First, we have designed a decentralized algorithm for network bandwidth prediction. The algorithm embeds the bandwidth information as distance in an edge-weighted tree, without performing full n-to-n measurements. No central and fixed infrastructure is required. Each joining node performs a limited number of sampling measurements. Second, we designed a decentralized algorithm to search for a centroid node that has high-bandwidth connections with a given set of nodes. The algorithm can find a centroid accurately and efficiently using the bandwidth data produced by the prediction algorithm. Last, we have designed another type of decentralized search algorithm to find a cluster of nodes that have high-bandwidth interconnections. While the clustering problem is NP-complete in a general graph, our algorithm runs in polynomial time with the bandwidth data predicted in a tree metric space. We provide proofs that our algorithms for bandwidth prediction and node search have perfect accuracy and high scalability when a network is modeled as a tree metric space. Also, experimental results with real-world data sets validate the high accuracy and scalability of our approaches

Parallel Algorithms for Geometric Graph Problems

We give algorithms for geometric graph problems in the modern parallel models inspired by MapReduce. For example, for the Minimum Spanning Tree (MST) problem over a set of points in the two-dimensional space, our algorithm computes a $(1+\epsilon)$-approximate MST. Our algorithms work in a constant number of rounds of communication, while using total space and communication proportional to the size of the data (linear space and near linear time algorithms). In contrast, for general graphs, achieving the same result for MST (or even connectivity) remains a challenging open problem, despite drawing significant attention in recent years. We develop a general algorithmic framework that, besides MST, also applies to Earth-Mover Distance (EMD) and the transportation cost problem. Our algorithmic framework has implications beyond the MapReduce model. For example it yields a new algorithm for computing EMD cost in the plane in near-linear time, $n^{1+o_\epsilon(1)}$. We note that while recently Sharathkumar and Agarwal developed a near-linear time algorithm for $(1+\epsilon)$-approximating EMD, our algorithm is fundamentally different, and, for example, also solves the transportation (cost) problem, raised as an open question in their work. Furthermore, our algorithm immediately gives a $(1+\epsilon)$-approximation algorithm with $n^{\delta}$ space in the streaming-with-sorting model with $1/\delta^{O(1)}$ passes. As such, it is tempting to conjecture that the parallel models may also constitute a concrete playground in the quest for efficient algorithms for EMD (and other similar problems) in the vanilla streaming model, a well-known open problem

Towards a Framework for DHT Distributed Computing

Distributed Hash Tables (DHTs) are protocols and frameworks used by peer-to-peer (P2P) systems. They are used as the organizational backbone for many P2P file-sharing systems due to their scalability, fault-tolerance, and load-balancing properties. These same properties are highly desirable in a distributed computing environment, especially one that wants to use heterogeneous components. We show that DHTs can be used not only as the framework to build a P2P file-sharing service, but as a P2P distributed computing platform. We propose creating a P2P distributed computing framework using distributed hash tables, based on our prototype system ChordReduce. This framework would make it simple and efficient for developers to create their own distributed computing applications. Unlike Hadoop and similar MapReduce frameworks, our framework can be used both in both the context of a datacenter or as part of a P2P computing platform. This opens up new possibilities for building platforms to distributed computing problems. One advantage our system will have is an autonomous load-balancing mechanism. Nodes will be able to independently acquire work from other nodes in the network, rather than sitting idle. More powerful nodes in the network will be able use the mechanism to acquire more work, exploiting the heterogeneity of the network. By utilizing the load-balancing algorithm, a datacenter could easily leverage additional P2P resources at runtime on an as needed basis. Our framework will allow MapReduce-like or distributed machine learning platforms to be easily deployed in a greater variety of contexts