Quantiles are fundamental statistics in various data science tasks, but costly to compute, e.g., by loading the entire data in memory for ranking. With limited memory space, prevalent in end devices or databases with heavy loads, it needs to scan the data in multiple passes. The idea is to gradually shrink the range of the queried quantile till it is small enough to fit in memory for ranking the result. Existing methods use deterministic sketches to determine the exact range of quantile, known as deterministic filter, which could be inefficient in range shrinking. In this study, we propose to shrink the ranges more aggressively, using randomized summaries such as KLL sketch. That is, with a high probability the quantile lies in a smaller range, namely probabilistic filter, determined by the randomized sketch. Specifically, we estimate the expected passes for determining the exact quantiles with probabilistic filters, and select a proper probability that can minimize the expected passes. Analyses show that our exact quantile determination method can terminate in P passes with 1-δ confidence, storing O(N 1/P logP-1/2P (1/δ)) items, close to the lower bound Ømega(N1/P) for a fixed δ. The approach has been deployed as a function in an LSM-tree based time-series database Apache IoTDB. Remarkably, the randomized sketches can be pre-computed for the immutable SSTables in LSM-tree. Moreover, multiple quantile queries could share the data passes for probabilistic filters in range estimation. Extensive experiments on real and synthetic datasets demonstrate the superiority of our proposal compared to the existing methods with deterministic filters. On average, our method takes 0.48 fewer passes and 18% of the time compared with the state-of-the-art deterministic sketch (GK sketch).

Columnar storage is now an industry standard design in most open-source or commercial time series database products, making them HTAP systems. The time column of a time series serves as the key for identifying the other value column, namely single-column storage scheme. When multiple time series share a similar set of timestamps, very likely in a module of multiple sensors, it is natural to group them together, i.e., one time column identifies multiple value columns in a single-group storage scheme. While multiple value columns sharing the same time column reduce the space cost of repeating timestamps, it may introduce extra space cost for recording null values. The reason is that time series may not be exactly aligned on each timestamp, owing to missing values, distinct data collection frequencies, unsynchronized clocks and so on. The columngroups storage scheme is thus to divide columns into multiple groups, within which the value columns share the same time column. Unfortunately, the problem of finding the optimal column groups for the minimum space cost is highly challenging, NP-hard according to our analysis. Thereby, we propose a heuristic algorithm for automatically grouping time series for efficient columnar storage. The column groups storage has been deployed in Apache IoTDB, an open-source time series database. The extensive performance analysis, over real-world data from our industrial partners, demonstrates that the proposed column groups achieve near optimal storage, more concise than the storage of single-column or single-group schemes. Interestingly, both the flushing and querying time costs of column groups are comparable to those of single-column or singlegroup, i.e., without incurring extra time cost.

Median absolute deviation (MAD), the median of the absolute deviations from the median, has been found useful in various applications such as outlier detection. Together with median, MAD is more robust to abnormal data than mean and standard deviation (SD). Unfortunately, existing methods return only approximate MAD that may be far from the exact one, and thus mislead the downstream applications. Computing exact MAD is costly, however, especially in space, by storing the entire dataset in memory. In this paper, we propose COnstruction-REfinement Sketch (CORE-Sketch) for computing exact MAD. The idea is to construct some sketch within limited space, and gradually refine the sketch to find the MAD element, i.e., the element with distance to the median exactly equal to MAD. Mergeability and convergence of the method is analyzed, ensuring the correctness of the proposal and enabling parallel computation. Extensive experiments demonstrate that CORE-Sketch achieves significantly less space occupation compared to the aforesaid baseline of No-Sketch, and has time and space costs relatively comparable to the DD-Sketch method for approximate MAD.