Weighted Reservoir Sampling from Distributed Streams

We consider message-efficient continuous random sampling from a distributed stream, where the probability of inclusion of an item in the sample is proportional to a weight associated with the item. The unweighted version, where all weights are equal, is well studied, and admits tight upper and lower bounds on message complexity. For weighted sampling with replacement, there is a simple reduction to unweighted sampling with replacement. However, in many applications the stream has only a few heavy items which may dominate a random sample when chosen with replacement. Weighted sampling \textit{without replacement} (weighted SWOR) eludes this issue, since such heavy items can be sampled at most once. In this work, we present the first message-optimal algorithm for weighted SWOR from a distributed stream. Our algorithm also has optimal space and time complexity. As an application of our algorithm for weighted SWOR, we derive the first distributed streaming algorithms for tracking \textit{heavy hitters with residual error}. Here the goal is to identify stream items that contribute significantly to the residual stream, once the heaviest items are removed. Residual heavy hitters generalize the notion of $\ell_1$ heavy hitters and are important in streams that have a skewed distribution of weights. In addition to the upper bound, we also provide a lower bound on the message complexity that is nearly tight up to a $\log(1/\epsilon)$ factor. Finally, we use our weighted sampling algorithm to improve the message complexity of distributed $L_1$ tracking, also known as count tracking, which is a widely studied problem in distributed streaming. We also derive a tight message lower bound, which closes the message complexity of this fundamental problem.Comment: To appear in PODS 201

Stream Aggregation Through Order Sampling

This is paper introduces a new single-pass reservoir weighted-sampling stream aggregation algorithm, Priority-Based Aggregation (PBA). While order sampling is a powerful and e cient method for weighted sampling from a stream of uniquely keyed items, there is no current algorithm that realizes the benefits of order sampling in the context of stream aggregation over non-unique keys. A naive approach to order sample regardless of key then aggregate the results is hopelessly inefficient. In distinction, our proposed algorithm uses a single persistent random variable across the lifetime of each key in the cache, and maintains unbiased estimates of the key aggregates that can be queried at any point in the stream. The basic approach can be supplemented with a Sample and Hold pre-sampling stage with a sampling rate adaptation controlled by PBA. This approach represents a considerable reduction in computational complexity compared with the state of the art in adapting Sample and Hold to operate with a fixed cache size. Concerning statistical properties, we prove that PBA provides unbiased estimates of the true aggregates. We analyze the computational complexity of PBA and its variants, and provide a detailed evaluation of its accuracy on synthetic and trace data. Weighted relative error is reduced by 40% to 65% at sampling rates of 5% to 17%, relative to Adaptive Sample and Hold; there is also substantial improvement for rank queriesComment: 10 page

Geoadditive Regression Modeling of Stream Biological Condition

Indices of biotic integrity (IBI) have become an established tool to quantify the condition of small non-tidal streams and their watersheds. To investigate the effects of watershed characteristics on stream biological condition, we present a new technique for regressing IBIs on watershed-specific explanatory variables. Since IBIs are typically evaluated on anordinal scale, our method is based on the proportional odds model for ordinal outcomes. To avoid overfitting, we do not use classical maximum likelihood estimation but a component-wise functional gradient boosting approach. Because component-wise gradient boosting has an intrinsic mechanism for variable selection and model choice, determinants of biotic integrity can be identified. In addition, the method offers a relatively simple way to account for spatial correlation in ecological data. An analysis of the Maryland Biological Streams Survey shows that nonlinear effects of predictor variables on stream condition can be quantified while, in addition, accurate predictions of biological condition at unsurveyed locations are obtained

Graph Sample and Hold: A Framework for Big-Graph Analytics

Sampling is a standard approach in big-graph analytics; the goal is to efficiently estimate the graph properties by consulting a sample of the whole population. A perfect sample is assumed to mirror every property of the whole population. Unfortunately, such a perfect sample is hard to collect in complex populations such as graphs (e.g. web graphs, social networks etc), where an underlying network connects the units of the population. Therefore, a good sample will be representative in the sense that graph properties of interest can be estimated with a known degree of accuracy. While previous work focused particularly on sampling schemes used to estimate certain graph properties (e.g. triangle count), much less is known for the case when we need to estimate various graph properties with the same sampling scheme. In this paper, we propose a generic stream sampling framework for big-graph analytics, called Graph Sample and Hold (gSH). To begin, the proposed framework samples from massive graphs sequentially in a single pass, one edge at a time, while maintaining a small state. We then show how to produce unbiased estimators for various graph properties from the sample. Given that the graph analysis algorithms will run on a sample instead of the whole population, the runtime complexity of these algorithm is kept under control. Moreover, given that the estimators of graph properties are unbiased, the approximation error is kept under control. Finally, we show the performance of the proposed framework (gSH) on various types of graphs, such as social graphs, among others

Sublinear Estimation of Weighted Matchings in Dynamic Data Streams

This paper presents an algorithm for estimating the weight of a maximum weighted matching by augmenting any estimation routine for the size of an unweighted matching. The algorithm is implementable in any streaming model including dynamic graph streams. We also give the first constant estimation for the maximum matching size in a dynamic graph stream for planar graphs (or any graph with bounded arboricity) using $\tilde{O}(n^{4/5})$ space which also extends to weighted matching. Using previous results by Kapralov, Khanna, and Sudan (2014) we obtain a $\mathrm{polylog}(n)$ approximation for general graphs using $\mathrm{polylog}(n)$ space in random order streams, respectively. In addition, we give a space lower bound of $\Omega(n^{1-\varepsilon})$ for any randomized algorithm estimating the size of a maximum matching up to a $1+O(\varepsilon)$ factor for adversarial streams