29,906 research outputs found
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
The quantum complexity of approximating the frequency moments
The 'th frequency moment of a sequence of integers is defined as , where is the number of times that occurs in the
sequence. Here we study the quantum complexity of approximately computing the
frequency moments in two settings. In the query complexity setting, we wish to
minimise the number of queries to the input used to approximate up to
relative error . We give quantum algorithms which outperform the best
possible classical algorithms up to quadratically. In the multiple-pass
streaming setting, we see the elements of the input one at a time, and seek to
minimise the amount of storage space, or passes over the data, used to
approximate . We describe quantum algorithms for , and
in this model which substantially outperform the best possible
classical algorithms in certain parameter regimes.Comment: 22 pages; v3: essentially published versio
On Practical Algorithms for Entropy Estimation and the Improved Sample Complexity of Compressed Counting
Estimating the p-th frequency moment of data stream is a very heavily studied
problem. The problem is actually trivial when p = 1, assuming the strict
Turnstile model. The sample complexity of our proposed algorithm is essentially
O(1) near p=1. This is a very large improvement over the previously believed
O(1/eps^2) bound. The proposed algorithm makes the long-standing problem of
entropy estimation an easy task, as verified by the experiments included in the
appendix
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