如何在滑动窗上捕捉L_2级的大击球手
How to Catch L_2-Heavy-Hitters on Sliding Windows
摘要：
在流数据中找到重元素（heavy-hitters）是一个核心的、众所周知的任务。尽管这个问题的重要性，当考虑流的滑动窗口模型（其中元素最终到期）时，寻找L_2重元素的问题仍然是完全开放的，尽管多篇论文和相当大的成功，在寻找L_1重元素。本文提出了滑动窗口模型中寻找L_2重元素的第一个多对数记忆算法。由于L_2重元素在许多基本流问题（如频率矩）中起着核心作用，我们相信我们的方法对许多滑动窗口算法和应用是非常有用的。例如，我们的方法不仅可以在滑动窗口上找到L_2重元素，而且可以找到关于任意L_p（0<p<2）的重元素。因此，我们的论文完全解决了对于p的所有值具有多对数记忆的滑动窗口寻找L_p重元素的问题，因为众所周知，对于p>2，这一任务是不可能的。我们的方法也可以有其他应用。我们展示了更广泛的适用性，我们的新颖而简单的方法上两个额外的例子：我们展示了如何获得滑动窗口近似的其他属性，如两个流的相似性，或元素的分数，正好出现在窗口内指定的次数（罕见的问题）。在我们的方法的这两个说明性的例子中，我们用最坏情况下的界限替换当前预期的内存界限。
英文摘要：
Finding heavy-elements (heavy-hitters) in streaming data is one of the central, and well-understood tasks. Despite the importance of this problem, when considering the sliding windows model of streaming (where elements eventually expire) the problem of finding L_2-heavy elements has remained completely open despite multiple papers and considerable success in finding L_1-heavy elements. In this paper, we develop the first poly-logarithmic-memory algorithm for finding L_2-heavy elements in sliding window model. Since L_2 heavy elements play a central role for many fundamental streaming problems (such as frequency moments), we believe our method would be extremely useful for many sliding-windows algorithms and applications. For example, our technique allows us not only to find L_2-heavy elements, but also heavy elements with respect to any L_p for 0<p<2 on sliding windows. Thus, our paper completely resolves the question of finding L_p-heavy elements for sliding windows with poly-logarithmic memory for all values of p since it is well known that for p>2 this task is impossible. Our method may have other applications as well. We demonstrate a broader applicability of our novel yet simple method on two additional examples: we show how to obtain a sliding window approximation of other properties such as the similarity of two streams, or the fraction of elements that appear exactly a specified number of times within the window (the rarity problem). In these two illustrative examples of our method, we replace the current expected memory bounds with worst case bounds.
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