Estimating small frequency moments of data stream: a characteristic function approach

A data stream is viewed as a sequence of $M$ updates of the form $(\text{index},i,v)$ to an $n$-dimensional integer frequency vector $f$, where the update changes $fi$ to $fi + v$, and $v$ is an integer and assumed to be in ${-m, ..., m}$. The $p$th frequency moment $Fp$ is defined as $\sum{i=1}ⁿ \abs{fi}^p$. We consider the problem of estimating $Fp$ to within a multiplicative approximation factor of $1\pm \epsilon$, for $p \in [0,2]$. Several estimators have been proposed for this problem, including Indyk's median estimator \cite{indy:focs00}, Li's geometric means estimator \cite{pinglib:2006}, an \Hss-based estimator \cite{gc:random07}. The first two estimators require space $\tilde{O}(\epsilon^{-2})$, where the $\tilde{O}$ notation hides polylogarithmic factors in $\epsilon^{-1}, m, n$ and $M$. Recently, Kane, Nelson and Woodruff in \cite{knw:soda10} present a space-optimal and novel estimator, called the log-cosine estimator. In this paper, we present an elementary analysis of the log-cosine estimator in a stand-alone setting. The analysis in \cite{knw:soda10} is more complicated.