How to Catch L_2-Heavy-Hitters on Sliding Windows

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 L2-heavy elements has remained completely open despite multiple papers and considerable success in finding L1-heavy elements. In this paper, we develop the first poly-logarithmic-memory algorithm for finding L2-heavy elements in sliding window model. Since L2 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 L2-heavy elements, but also heavy elements with respect to any Lp 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.