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Let $a_1, a_2,\dotsc, a_m$ be a sequence of integers where each $a_j \in \{1,2,\dotsc,n\}$. For $i \in \{1,2,\dotsc,n\}$, let $m_i = |\{j : a_j = i\}|$. The $k$th frequency moment is defined to be

$\displaystyle F_k = \sum_{i=1}^n m_i^k.$

In their well-known paper, The space complexity of approximating the frequency moments, Alon et al. give a streaming algorithm that approximates $F_k$ using roughly $O(n^{1-\frac{1}{k}}(\log n + \log m))$ space. They also use communication complexity techniques to obtain a lower bound of $\Omega(n^{1-\frac{5}{k}})$ for $k > 5$. For $k = 0,1,2$, they provide more or less matching upper and lower bounds.

Have there been improvements to these bounds since then, and has there been progress for $k = 3,4,5$?

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There's been a fair bit of progress. On the specific problem of $F_k$, there is a matching upper and lower bound of $n^{1-2/k}$ for $k > 2$. The upper bounds come from this paper by Indyk and Woodruff (which appeared in STOC 2005) and the lower bounds are via the information complexity framework, due to Bar-Yossef et al and Chakrabarti et al.

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For k<=2

1) k=0, the bound is $O(1/\epsilon^2+log(n))$ from http://people.seas.harvard.edu/~minilek/papers/f0.pdf.

2) k=1, The paper by Alon et all gives a reference to paper by Morris which takes $\tilde{O}(log(log(n))$ space.

3) k=2, I think the AMS sketch from their paper is optimal

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Something related.

I think there has also been some related work on approximating $F_\alpha$ where $\alpha$ is not necessarily a integer. http://www.stat.cornell.edu/~li/SODA09_CC.pdf is something which I know of. I am not completely familiar with this but I think their main point of interest is dependence on $\epsilon$ and not $n$.

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    $\begingroup$ Note this is for $\alpha \in (1, 2)$ when the dependence on $n$ is known to be polylog so $\epsilon$ becomes the bottleneck $\endgroup$ – Sasho Nikolov Dec 14 '11 at 2:12

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