800 research outputs found
Identifying Correlated Heavy-Hitters in a Two-Dimensional Data Stream
We consider online mining of correlated heavy-hitters from a data stream.
Given a stream of two-dimensional data, a correlated aggregate query first
extracts a substream by applying a predicate along a primary dimension, and
then computes an aggregate along a secondary dimension. Prior work on
identifying heavy-hitters in streams has almost exclusively focused on
identifying heavy-hitters on a single dimensional stream, and these yield
little insight into the properties of heavy-hitters along other dimensions. In
typical applications however, an analyst is interested not only in identifying
heavy-hitters, but also in understanding further properties such as: what other
items appear frequently along with a heavy-hitter, or what is the frequency
distribution of items that appear along with the heavy-hitters. We consider
queries of the following form: In a stream S of (x, y) tuples, on the substream
H of all x values that are heavy-hitters, maintain those y values that occur
frequently with the x values in H. We call this problem as Correlated
Heavy-Hitters (CHH). We formulate an approximate formulation of CHH
identification, and present an algorithm for tracking CHHs on a data stream.
The algorithm is easy to implement and uses workspace which is orders of
magnitude smaller than the stream itself. We present provable guarantees on the
maximum error, as well as detailed experimental results that demonstrate the
space-accuracy trade-off
Fast and Accurate Mining of Correlated Heavy Hitters
The problem of mining Correlated Heavy Hitters (CHH) from a two-dimensional
data stream has been introduced recently, and a deterministic algorithm based
on the use of the Misra--Gries algorithm has been proposed by Lahiri et al. to
solve it. In this paper we present a new counter-based algorithm for tracking
CHHs, formally prove its error bounds and correctness and show, through
extensive experimental results, that our algorithm outperforms the Misra--Gries
based algorithm with regard to accuracy and speed whilst requiring
asymptotically much less space
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 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 factor. Finally, we use our weighted
sampling algorithm to improve the message complexity of distributed
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
Recursive Sketching For Frequency Moments
In a ground-breaking paper, Indyk and Woodruff (STOC 05) showed how to
compute (for ) in space complexity O(\mbox{\em poly-log}(n,m)\cdot
n^{1-\frac2k}), which is optimal up to (large) poly-logarithmic factors in
and , where is the length of the stream and is the upper bound on
the number of distinct elements in a stream. The best known lower bound for
large moments is . A follow-up work of
Bhuvanagiri, Ganguly, Kesh and Saha (SODA 2006) reduced the poly-logarithmic
factors of Indyk and Woodruff to . Further reduction of poly-log factors has been an elusive
goal since 2006, when Indyk and Woodruff method seemed to hit a natural
"barrier." Using our simple recursive sketch, we provide a different yet simple
approach to obtain a algorithm for constant (our bound is, in fact, somewhat
stronger, where the term can be replaced by any constant number
of iterations instead of just two or three, thus approaching .
Our bound also works for non-constant (for details see the body of
the paper). Further, our algorithm requires only -wise independence, in
contrast to existing methods that use pseudo-random generators for computing
large frequency moments
- …