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. Since the L2-heavy element problem doesn't satisfy certain conditions, existing methods for sliding windows algorithms, such as smooth histograms or exponential histograms are not directly applicable to it. In this paper, we develop the first polylogarithmic-memory algorithm for finding L2-heavy elements in the sliding window model. Our technique allows us not only to find L2-heavy elements, but also heavy elements with respect to any Lp with 0<p≤2 on sliding windows. By this we completely "close the gap" and resolve the question of finding Lp-heavy elements in the sliding window model with polylogarithmic memory, since it is well known that for p>2 this task is impossible. We demonstrate a broader applicability of our method on two additional examples: we show how to obtain a sliding window approximation of the similarity of two streams, and of the fraction of elements that appear exactly a specified number of times within the window (the α-rarity problem). In these two illustrative examples of our method, we replace the current expected memory bounds with worst case bounds.
|Number of pages||13|
|Journal||Theoretical Computer Science|
|State||Published - 2014|
Bibliographical noteFunding Information:
V.B. is supported in part by DARPA grant N660001-1-2-4014 . R.O. is supported in part by NSF grants CNS-0830803 ; CCF-0916574 ; IIS-1065276 ; CCF-1016540 ; CNS-1118126 ; CNS-1136174 ; US-Israel BSF grant 2008411 , OKAWA Foundation Research Award , IBM Faculty Research Award , Xerox Faculty Research Award , B. John Garrick Foundation Award , Teradata Research Award , and Lockheed-Martin Corporation Research Award . This material is also based upon work supported by the Defense Advanced Research Projects Agency through the US Office of Naval Research under Contract N00014-11-1-0392 . The views expressed are those of the author and do not reflect the official policy or position of the Department of Defense or the US Government.
© 2014 Elsevier B.V.
- Approximation algorithms
- Data streams
- Heavy hitters
- Sliding window model