Approximating Large Frequency Moments with $O(n^{1-2\mathrm{/}k})$ Bits

In this paper we consider the problem of approximating frequency moments in the streaming model. Given a stream $D=\{p_1,p_2,\dots ,p_m\}$ of numbers from $\{1,\dots ,n\}$, a frequency of $i$ is defined as $f_i=\mathrm{\mid }\{j:p_j=i\}\mathrm{\mid }$. The $k$-th \emph{frequency moment} of $D$ is defined as $F_k = \sum_{i=1}^n f_i^k$. In this paper we give an upper bound on the space required to find a $k$-th frequency moment of $O(n^{1-2\mathrm{/}k})$ bits that matches, up to a constant factor, the lower bound of Woodruff and Zhang (STOC 12) for constant $ϵ$ and constant $k$. Our algorithm makes a single pass over the stream and works for any constant $k>3$.