Scalable Facility Location for Massive Graphs on Pregel-like Systems

We propose a new scalable algorithm for facility location. Facility location is a classic problem, where the goal is to select a subset of facilities to open, from a set of candidate facilities F , in order to serve a set of clients C. The objective is to minimize the total cost of opening facilities plus the cost of serving each client from the facility it is assigned to. In this work, we are interested in the graph setting, where the cost of serving a client from a facility is represented by the shortest-path distance on the graph. This setting allows to model natural problems arising in the Web and in social media applications. It also allows to leverage the inherent sparsity of such graphs, as the input is much smaller than the full pairwise distances between all vertices. To obtain truly scalable performance, we design a parallel algorithm that operates on clusters of shared-nothing machines. In particular, we target modern Pregel-like architectures, and we implement our algorithm on Apache Giraph. Our solution makes use of a recent result to build sketches for massive graphs, and of a fast parallel algorithm to find maximal independent sets, as building blocks. In so doing, we show how these problems can be solved on a Pregel-like architecture, and we investigate the properties of these algorithms. Extensive experimental results show that our algorithm scales gracefully to graphs with billions of edges, while obtaining values of the objective function that are competitive with a state-of-the-art sequential algorithm

Large-Scale Distributed Algorithms for Facility Location with Outliers

This paper presents fast, distributed, O(1)-approximation algorithms for metric facility location problems with outliers in the Congested Clique model, Massively Parallel Computation (MPC) model, and in the k-machine model. The paper considers Robust Facility Location and Facility Location with Penalties, two versions of the facility location problem with outliers proposed by Charikar et al. (SODA 2001). The paper also considers two alternatives for specifying the input: the input metric can be provided explicitly (as an n x n matrix distributed among the machines) or implicitly as the shortest path metric of a given edge-weighted graph. The results in the paper are: - Implicit metric: For both problems, O(1)-approximation algorithms running in O(poly(log n)) rounds in the Congested Clique and the MPC model and O(1)-approximation algorithms running in O~(n/k) rounds in the k-machine model. - Explicit metric: For both problems, O(1)-approximation algorithms running in O(log log log n) rounds in the Congested Clique and the MPC model and O(1)-approximation algorithms running in O~(n/k) rounds in the k-machine model. Our main contribution is to show the existence of Mettu-Plaxton-style O(1)-approximation algorithms for both Facility Location with outlier problems. As shown in our previous work (Berns et al., ICALP 2012, Bandyapadhyay et al., ICDCN 2018) Mettu-Plaxton style algorithms are more easily amenable to being implemented efficiently in distributed and large-scale models of computation

Coresets for Clustering in Graphs of Bounded Treewidth

We initiate the study of coresets for clustering in graph metrics, i.e., the shortest-path metric of edge-weighted graphs. Such clustering problems are essential to data analysis and used for example in road networks and data visualization. A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set, and it offers significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings. Our main result is a near-linear time construction of a coreset for k-Median in a general graph $G$, with size $O_{\epsilon, k}(\mathrm{tw}(G))$ where $\mathrm{tw}(G)$ is the treewidth of $G$, and we complement the construction with a nearly-tight size lower bound. The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of $O(\mathrm{tw}(G))$ on the shattering dimension under any point weights. We validate our coreset on real-world road networks, and our scalable algorithm constructs tiny coresets with high accuracy, which translates to a massive speedup of existing approximation algorithms such as local search for graph k-Median

Average Distance Queries through Weighted Samples in Graphs and Metric Spaces: High Scalability with Tight Statistical Guarantees

The average distance from a node to all other nodes in a graph, or from a query point in a metric space to a set of points, is a fundamental quantity in data analysis. The inverse of the average distance, known as the (classic) closeness centrality of a node, is a popular importance measure in the study of social networks. We develop novel structural insights on the sparsifiability of the distance relation via weighted sampling. Based on that, we present highly practical algorithms with strong statistical guarantees for fundamental problems. We show that the average distance (and hence the centrality) for all nodes in a graph can be estimated using $O(\epsilon^{-2})$ single-source distance computations. For a set $V$ of $n$ points in a metric space, we show that after preprocessing which uses $O(n)$ distance computations we can compute a weighted sample $S\subset V$ of size $O(\epsilon^{-2})$ such that the average distance from any query point $v$ to $V$ can be estimated from the distances from $v$ to $S$. Finally, we show that for a set of points $V$ in a metric space, we can estimate the average pairwise distance using $O(n+\epsilon^{-2})$ distance computations. The estimate is based on a weighted sample of $O(\epsilon^{-2})$ pairs of points, which is computed using $O(n)$ distance computations. Our estimates are unbiased with normalized mean square error (NRMSE) of at most $\epsilon$. Increasing the sample size by a $O(\log n)$ factor ensures that the probability that the relative error exceeds $\epsilon$ is polynomially small.Comment: 21 pages, will appear in the Proceedings of RANDOM 201