On joining and caching stochastic streams

We consider the problem of joining data streams using limited cache memory, with the goal of producing as many result tuples as possible from the cache. Many cache replacement heuristics have been proposed in the past. Their performance often relies on implicit assumptions about the input streams, e.g., that the join attribute values follow a relatively stationary distribution. However, in general and in practice, streams often exhibit more complex behaviors, such as increasing trends and random walks, rendering these "hardwired" heuristics inadequate. In this paper, we propose a framework that is able to exploit known or observed statistical properties of input streams to make cache replacement decisions aimed at maximizing the expected number of result tuples. To illustrate the complexity of the solution space, we show that even an algorithm that considers, at every time step, all possible sequences of future replacement decisions may not be optimal. We then identify a condition between two candidate tuples under which an optimal algorithm would always choose one tuple over the other to replace. We develop a heuristic that behaves consistently with an optimal algorithm whenever this condition is satisfied. We show through experiments that our heuristic outperforms previous ones. As another evidence of the generality of our framework, we show that the classic caching/paging problem for static objects can be reduced to a stream join problem and analyzed under our framework, yielding results that agree with or extend classic ones.