Abstract Clustering is a fundamental tool for analyzing large data sets. A rich body of work has been devoted to designing data-stream algorithms for the relevant optimization problems such as k-center, k-median, and k-means. Such algorithms need to be both time and and space efficient. In this paper, we address the problem of correlation clustering in the dynamic data stream model. The stream consists of updates to the edge weights of a graph on n nodes and the goal is to find a node-partition such that the end-points of negative-weight edges are typically in different clusters whereas the end-points of positive-weight edges are typically in the same cluster. We present polynomial-time, $$O(n\cdot {{\,\mathrm{polylog}\,}}n)$$ O ( n · polylog n ) -space approximation algorithms for natural problems that arise. We first develop data structures based on linear sketches that allow the “quality” of a given node-partition to be measured. We then combine these data structures with convex programming and sampling techniques to solve the relevant approximation problem. Unfortunately, the standard LP and SDP formulations are not obviously solvable in $$O(n\cdot {{\,\mathrm{polylog}\,}}n)$$ O ( n · polylog n ) -space. Our work presents space-efficient algorithms for the convex programming required, as well as approaches to reduce the adaptivity of the sampling.

In this paper, we address the problem of \emphcorrelation clustering in the dynamic data stream model. The stream consists of updates to the edge weights of a graph on n nodes and the goal is to find a node-partition such that the end-points of negative-weight edges are typically in different clusters whereas the end-points of positive-weight edges are typically in the same cluster. We present polynomial-time, O(n⋅\textpolylog n)-space approximation algorithms for natural problems that arise. We first develop data structures based on linear sketches that allow the “quality” of a given node-partition to be measured. We then combine these data structures with convex programming and sampling techniques to solve the relevant approximation problem. However the standard LP and SDP formulations are not obviously solvable in O(n⋅\textpolylog n)-space. Our work presents space-efficient algorithms for the convex programming required, as well as approaches to reduce the adaptivity of the sampling. Note that the improved space and running-time bounds achieved from streaming algorithms are also useful for offline settings such as MapReduce models.

In this paper we consider graph algorithms in models of computation where the space usage (random accessible storage, in addition to the read only input) is sublinear in the number of edges m and the access to input data is constrained. These questions arises in many natural settings, and in particular in the analysis of MapReduce or similar algorithms that model constrained parallelism with sublinear central processing. In SPAA 2011, Lattanzi etal. provided a O(1) approximation of maximum matching using O(p) rounds of iterative filtering via mapreduce and O(n1+1/p) space of central processing for a graph with n nodes and m edges.

We present the first fully polynomial approximation schemes for the maximum weighted (uncapacitated or capacitated) b–Matching problem for nonbipartite graphs that run in time (near) linear in the number of edges, that is, given any δ > 0 the algorithm produces a (1 – δ) approximation in O(mpoly(δ−1, logn)) time. We provide fractional solutions for the standard linear programming formulations for these problems and subsequently also provide fully polynomial (near) linear time approximation schemes for rounding the fractional solutions. Through these problems as a vehicle, we also present several ideas in the context of solving linear programs approximately using fast primal-dual algorithms. First, we show that approximation algorithms can be used to reduce the width of the formulation, and as a consequence we induce faster convergence. Second, even though the dual of these problems have exponentially many variables and an efficient exact computation of dual weights is infeasible, we can efficiently compute and use a sparse approximation of the dual weights using a combination of (i) adding perturbation to the constraints of the polytope and (ii) amplification followed by thresholding of the dual weights. These algorithms also have the advantage that they use O(npoly(δ−1, logn)) storage space and only make O(δ−4log (1/δ)logn) (or better) passes over a read only list of edges. These algorithms therefore can be run in the semi-streaming model and serve as exemplars where algorithms and ideas developed for the streaming model gives us algorithms for combinatorial optimization problems that were not known in absence of the streaming constraints.

When processing massive data sets, a core task is to construct synopses of the data. To be useful, a synopsis data structure should be easy to construct while also yielding good approximations of the relevant properties of the data set. A particularly useful class of synopses are sketches, i.e., those based on linear projections of the data. These are applicable in many models including various parallel, stream, and compressed sensing settings. A rich body of analytic and empirical work exists for sketching numerical data such as the frequencies of a set of entities. Our work investigates graph sketching where the graphs of interest encode the relationships between these entities. The main challenge is to capture this richer structure and build the necessary synopses with only linear measurements.

We initiate the study of graph sketching, i.e., algorithms that use a limited number of linear measurements of a graph to determine the properties of the graph. While a graph on n nodes is essentially O(n2)-dimensional, we show the existence of a distribution over random projections into d-dimensional “sketch” space (d ≪ n2) such that the relevant properties of the original graph can be inferred from the sketch with high probability. Specifically, we show that: 1. d = O(n · polylog n) suffices to evaluate properties including connectivity, k-connectivity, bipartiteness, and to return any constant approximation of the weight of the minimum spanning tree. 2. d = O(n1+γ) suffices to compute graph sparsifiers, the exact MST, and approximate the maximum weighted matchings if we permit O(1/γ)-round adaptive sketches, i.e., a sequence of projections where each projection may be chosen dependent on the outcome of earlier sketches.