273 research outputs found

    On Sampling Based Algorithms for k-Means

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    Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms

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    We present a technical survey on the state of the art approaches in data reduction and the coreset framework. These include geometric decompositions, gradient methods, random sampling, sketching and random projections. We further outline their importance for the design of streaming algorithms and give a brief overview on lower bounding techniques

    FPT Approximation for Constrained Metric k-Median/Means

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    The Metric kk-median problem over a metric space (X,d)(\mathcal{X}, d) is defined as follows: given a set LXL \subseteq \mathcal{X} of facility locations and a set CXC \subseteq \mathcal{X} of clients, open a set FLF \subseteq L of kk facilities such that the total service cost, defined as Φ(F,C)xCminfFd(x,f)\Phi(F, C) \equiv \sum_{x \in C} \min_{f \in F} d(x, f), is minimised. The metric kk-means problem is defined similarly using squared distances. In many applications there are additional constraints that any solution needs to satisfy. This gives rise to different constrained versions of the problem such as rr-gather, fault-tolerant, outlier kk-means/kk-median problem. Surprisingly, for many of these constrained problems, no constant-approximation algorithm is known. We give FPT algorithms with constant approximation guarantee for a range of constrained kk-median/means problems. For some of the constrained problems, ours is the first constant factor approximation algorithm whereas for others, we improve or match the approximation guarantee of previous works. We work within the unified framework of Ding and Xu that allows us to simultaneously obtain algorithms for a range of constrained problems. In particular, we obtain a (3+ε)(3+\varepsilon)-approximation and (9+ε)(9+\varepsilon)-approximation for the constrained versions of the kk-median and kk-means problem respectively in FPT time. In many practical settings of the kk-median/means problem, one is allowed to open a facility at any client location, i.e., CLC \subseteq L. For this special case, our algorithm gives a (2+ε)(2+\varepsilon)-approximation and (4+ε)(4+\varepsilon)-approximation for the constrained versions of kk-median and kk-means problem respectively in FPT time. Since our algorithm is based on simple sampling technique, it can also be converted to a constant-pass log-space streaming algorithm

    Observability of Dark Matter Substructure with Pulsar Timing Correlations

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    Dark matter substructure on small scales is currently weakly constrained, and its study may shed light on the nature of the dark matter. In this work we study the gravitational effects of dark matter substructure on measured pulsar phases in pulsar timing arrays (PTAs). Due to the stability of pulse phases observed over several years, dark matter substructure around the Earth-pulsar system can imprint discernible signatures in gravitational Doppler and Shapiro delays. We compute pulsar phase correlations induced by general dark matter substructure, and project constraints for a few models such as monochromatic primordial black holes (PBHs), and Cold Dark Matter (CDM)-like NFW subhalos. This work extends our previous analysis, which focused on static or single transiting events, to a stochastic analysis of multiple transiting events. We find that stochastic correlations, in a PTA similar to the Square Kilometer Array (SKA), are uniquely powerful to constrain subhalos as light as 1013 M\sim 10^{-13}~M_\odot, with concentrations as low as that predicted by standard CDM.Comment: 45 pages, 12 figure
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