Approximating Large Frequency Moments with Pick-and-Drop Sampling

Given data stream D = {p 1,p 2,…,p m } of size m of numbers from {1,…, n}, the frequency of i is defined as f i = |{j: p j = i}|. The k-th frequency moment of D is defined as \(F_k = \sum_{i=1}^n f_i^k\). We consider the problem of approximating frequency moments in insertion-only streams for k ≥ 3. For any constant c we show an O(n 1 − 2/k log(n)log(c)(n)) upper bound on the space complexity of the problem. Here log(c)(n) is the iterative log function. Our main technical contribution is a non-uniform sampling method on matrices. We call our method a pick-and-drop sampling; it samples a heavy element (i.e., element i with frequency Ω(F k )) with probability Ω(1/n 1 − 2/k ) and gives approximation \(\tilde{f_i} \ge (1-\epsilon)f_i\). In addition, the estimations never exceed the real values, that is \( \tilde{f_j} \le f_j\) for all j. For constant e, we reduce the space complexity of finding a heavy element to O(n 1 − 2/k log(n)) bits. We apply our method of recursive sketches and resolve the problem with O(n 1 − 2/k log(n)log(c)(n)) bits. We reduce the ratio between the upper and lower bounds from O(log2(n)) to O(log(n)log(c)(n)). Thus, we provide a (roughly) quadratic improvement of the result of Andoni, Krauthgamer and Onak (FOCS 2011).