We study the distinct elements and $\ell_p$-heavy hitters problems in the sliding window model, where only the most recent $n$ elements in the data stream form the underlying set. We first introduce the composable histogram, a simple twist on the exponential (Datar et al., SODA 2002) and smooth histograms (Braverman and Ostrovsky, FOCS 2007) that may be of independent interest. We then show that the composable histogram along with a careful combination of existing techniques to track either the identity or frequency of a few specific items suffices to obtain algorithms for both distinct elements and $\ell_p$-heavy hitters that are nearly optimal in both $n$ and $\epsilon$. Applying our new composable histogram framework, we provide an algorithm that outputs a $(1+\epsilon)$-approximation to the number of distinct elements in the sliding window model and uses $\mathcal{O}\left(\frac{1}{\epsilon^2}\log n\log\frac{1}{\epsilon}\log\log n+\frac{1}{\epsilon}\log^2 n\right)$ bits of space. For $\ell_p$-heavy hitters, we provide an algorithm using space $\mathcal{O}\left(\frac{1}{\epsilon^p}\log^3 n\left(\log\log n+\log\frac{1}{\epsilon}\right)\right)$ for $0<p\le 2$, improving upon the best-known algorithm for $\ell_2$-heavy hitters (Braverman et al., COCOON 2014), which has space complexity $\mathcal{O}\left(\frac{1}{\epsilon^4}\log^3 n\right)$. We also show lower bounds of $\Omega\left(\frac{1}{\epsilon}\log^2 n+\frac{1}{\epsilon^2}\log n\right)$ for distinct elements and $\Omega\left(\frac{1}{\epsilon^p}\log^2 n\right)$ for $\ell_p$-heavy hitters.

翻译：我们研究滑动窗口模型中的不同元素与$\ell_p$-重击者问题，在该模型中仅数据流最近$n$个元素构成基础集合。首先引入可组合直方图——对指数直方图（Datar等，SODA 2002）与平滑直方图（Braverman和Ostrovsky，FOCS 2007）的简洁改进，该结构本身可能具有独立研究价值。随后证明，可组合直方图配合对若干特定项的身份或频率追踪技术的精心组合，足以获得对$n$和$\epsilon$均近最优的不同元素与$\ell_p$-重击者算法。通过应用新型可组合直方图框架，我们提出一种算法，可在滑动窗口模型下输出不同元素数量的$(1+\epsilon)$-近似值，空间复杂度为$\mathcal{O}\left(\frac{1}{\epsilon^2}\log n\log\frac{1}{\epsilon}\log\log n+\frac{1}{\epsilon}\log^2 n\right)$比特。对于$\ell_p$-重击者问题，我们给出空间复杂度$\mathcal{O}\left(\frac{1}{\epsilon^p}\log^3 n\left(\log\log n+\log\frac{1}{\epsilon}\right)\right)$的算法（适用于$0<p\le 2$），改进了现有最优$\ell_2$-重击者算法（Braverman等，COCOON 2014）的$\mathcal{O}\left(\frac{1}{\epsilon^4}\log^3 n\right)$空间复杂度。同时证明不同元素问题的下界为$\Omega\left(\frac{1}{\epsilon}\log^2 n+\frac{1}{\epsilon^2}\log n\right)$，$\ell_p$-重击者问题的下界为$\Omega\left(\frac{1}{\epsilon^p}\log^2 n\right)$。