We study the classical problem of moment estimation of an underlying vector whose $n$ coordinates are implicitly defined through a series of updates in a data stream. We show that if the updates to the vector arrive in the random-order insertion-only model, then there exist space efficient algorithms with improved dependencies on the approximation parameter $\varepsilon$. In particular, for any real $p > 2$, we first obtain an algorithm for $F_p$ moment estimation using $\tilde{\mathcal{O}}\left(\frac{1}{\varepsilon^{4/p}}\cdot n^{1-2/p}\right)$ bits of memory. Our techniques also give algorithms for $F_p$ moment estimation with $p>2$ on arbitrary order insertion-only and turnstile streams, using $\tilde{\mathcal{O}}\left(\frac{1}{\varepsilon^{4/p}}\cdot n^{1-2/p}\right)$ bits of space and two passes, which is the first optimal multi-pass $F_p$ estimation algorithm up to $\log n$ factors. Finally, we give an improved lower bound of $\Omega\left(\frac{1}{\varepsilon^2}\cdot n^{1-2/p}\right)$ for one-pass insertion-only streams. Our results separate the complexity of this problem both between random and non-random orders, as well as one-pass and multi-pass streams.
翻译:我们研究一个基础矢量的瞬间估计传统问题, 其美元坐标通过数据流中的一系列更新来隐含地定义。 我们显示, 如果矢量的更新以随机顺序插入的模型到达, 那么就会有空间高效算法, 其依赖性在近似参数$\ varepsilon$> 2美元上得到改善。 特别是, 对于任何真实的 $p > 2, 我们首先使用 $\ p$ 来获得一个 $_ p$ 秒估计的算法, 使用 $tilde_ palthcal{ Ormalde{ { left{ { left{ { left{ 1\\\\\\\\\\\\\\\\\\\\\\\\\\ r\ r\\\\\ r\\ rcr> 美元( rblent_ plickral_ plickral_ $__\ n_\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\