Batch is an important pattern in data streams, which refers to a group of identical items that arrive closely. We find that some special batches that arrive periodically are of great value. In this paper, we formally define a new pattern, namely periodic batches. A group of periodic batches refers to several batches of the same item, where these batches arrive periodically. Studying periodic batches is important in many applications, such as caches, financial markets, online advertisements, networks, etc. We propose a one-pass sketching algorithm, namely the HyperCalm sketch, which takes two phases to detect periodic batches in real time. In phase 1, we propose a time-aware Bloom filter, namely HyperBloomFilter (HyperBF), to detect the start of batches. In phase 2, we propose an enhanced top-k algorithm, called Calm Space-Saving (CalmSS), to report top-k periodic batches. We theoretically derive the error bounds for HyperBF and CalmSS. Extensive experiments show HyperCalm outperforms the strawman solutions 4× in term of average relative error and 13.2× in term of speed. We also apply HyperCalm to a cache system and integrate HyperCalm into Apache Flink. All related codes are open-sourced.

Oracle-guided inductive synthesis (OGIS) is a widely-used framework to apply program synthesis techniques in practice. The question selection problem aims at reducing the number of iterations in OGIS by selecting a proper input for each OGIS iteration. Theoretically, a question selector can generally improve the performance of OGIS solvers on both interactive and non-interactive tasks if it is not only effective for reducing iterations but also efficient. However, all existing effective question selectors fail in satisfying the requirement of efficiency. To ensure effectiveness, they convert the question selection problem into an optimization one, which is difficult to solve within a short time. In this paper, we propose a novel question selector, named LearnSy . LearnSy is both efficient and effective and thus achieves general improvement for OGIS solvers for the first time. Since we notice that the optimization tasks in previous studies are difficult because of the complex behavior of operators, we estimate these behaviors in LearnSy as simple random events. Subsequently, we provide theoretical results for the precision of this estimation and design an efficient algorithm for its calculation. According to our evaluation, when dealing with interactive tasks, LearnSy can offer competitive performance compared to existing selectors while being more efficient and more general. Moreover, when working on non-interactive tasks, LearnSy can generally reduce the time cost of existing CEGIS solvers by up to 43.0%.