From a high volume stream of weighted items, we want to maintain a generic sample of a certain limited size $k$ that we can later use to estimate the total weight of arbitrary subsets. This is the classic context of on-line reservoir sampling, thinki
ng of the generic sample as a reservoir. We present an efficient reservoir sampling scheme, $varoptk$, that dominates all previous schemes in terms of estimation quality. $varoptk$ provides {em variance optimal unbiased estimation of subset sums}. More precisely, if we have seen $n$ items of the stream, then for {em any} subset size $m$, our scheme based on $k$ samples minimizes the average variance over all subsets of size $m$. In fact, the optimality is against any off-line scheme with $k$ samples tailored for the concrete set of items seen. In addition to optimal average variance, our scheme provides tighter worst-case bounds on the variance of {em particular} subsets than previously possible. It is efficient, handling each new item of the stream in $O(log k)$ time. Finally, it is particularly well suited for combination of samples from different streams in a distributed setting.
