Universal Perfect Samplers for Incremental Streams

Abstract

If $G:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, the $G$-moment of a vector $\mathbf{x}\in {\mathbb{R}}_{+}^n$ is $G(\mathbf{x})={\sum }_{v\in [n]}G(\mathbf{x}(v))$ and the $G$-sampling problem is to select an index $v_{\ast }\in [n]$ according to its contribution to the $G$-moment, i.e., such that $\mathrm{Pr}(v_{\ast }=v)=G(\mathbf{x}(v))/G(\mathbf{x})$. Approximate $G$-samplers may introduce multiplicative and/or additive errors to this probability, and some have a non-trivial probability of failure. In this paper we focus on the exact $G$-sampling problem, where $G$ is selected from the class $\mathcal{G}$ of Laplace exponents of non-negative, one-dimensional Lévy processes, which includes several well studied classes such as $p$th moments $G(z)=z^p$, $p\in [0,1]$, logarithms $G(z)=\log (1+z)$, Cohen and Geri's soft concave sublinear functions, which are used to approximate concave sublinear functions, including cap statistics. We develop $G$-samplers for a vector $\mathbf{x}\in {\mathbb{R}}_{+}^n$ that is presented as an incremental stream of positive updates. In particular: * For any $G\in \mathcal{G}$, we give a very simple $G$-sampler that uses 2 words of memory and stores at all times a $v_{\ast }\in [n]$, such that $\mathrm{Pr}(v_{\ast }=v)$ is exactly $G(\mathbf{x}(v))/G(\mathbf{x})$. * We give a ``universal'' $\mathcal{G}$-sampler that uses $O(\log n)$ words of memory w.h.p., and given any $G\in \mathcal{G}$ at query time, produces an exact $G$-sample. With an overhead of a factor of $k$, both samplers can be used to $G$-sample a sequence of $k$ indices with or without replacement. Our sampling framework is simple and versatile, and can easily be generalized to sampling from more complex objects like graphs and hypergraphs.