A persistent data structure , also known as a multiversion data structure in the database literature, is a data structure that preserves all its previous versions as it is updated over time. Every update (inserting, deleting, or changing a data record) to the data structure creates a new version, while all the versions are kept in the data structure so that any previous version can still be queried. Persistent data structures aim at recording all versions accurately, which results in a space requirement that is at least linear to the number of updates. In many of today’s big data applications, in particular, for high-speed streaming data, the volume and velocity of the data are so high that we cannot afford to store everything. Therefore, streaming algorithms have received a lot of attention in the research community, which uses only sublinear space by sacrificing slightly on accuracy. All streaming algorithms work by maintaining a small data structure in memory, which is usually called a sketch , summary , or synopsis . The summary is updated upon the arrival of every element in the stream, thus it is ephemeral , meaning that it can only answer queries about the current status of the stream. In this article, we aim at designing persistent summaries, thereby giving streaming algorithms the ability to answer queries about the stream at any prior time.

Given a time series S = ((x1, y1), (x2, y2), …) and a prescribed error bound ε, the piecewise linear approximation (PLA) problem with max-error guarantees is to construct a piecewise linear function f such that |f(xi)-yi| ≤ ε for all i. In addition, we would like to have an online algorithm that takes the time series as the records arrive in a streaming fashion, and outputs the pieces of f on-the-fly. This problem has applications wherever time series data is being continuously collected, but the data collection device has limited local buffer space and communication bandwidth, so that the data has to be compressed and sent back during the collection process. Prior work addressed two versions of the problem, where either f consists of disjoint segments, or f is required to be a continuous piecewise linear function. In both cases, existing algorithms can produce a function f that has the minimum number of pieces while meeting the prescribed error bound ε. However, we observe that neither minimizes the true representation size of f, i.e., the number of parameters required to represent f. In this paper, we design an online algorithm that generates the optimal PLA in terms of representation size while meeting the prescribed max-error guarantee. Our experiments on many real-world data sets show that our algorithm can reduce the representation size of f by around 15% on average compared with the current best methods, while still requiring O(1) processing time per data record and small space.

A persistent data structure, also known as a multiversion data structure in the database literature, is a data structure that preserves all its previous versions as it is updated over time. Every update (inserting, deleting, or changing a data record) to the data structure creates a new version, while all the versions are kept in the data structure so that any previous version can still be queried.

A fundamental problem in data management and analysis is to generate descriptions of the distribution of data. It is most common to give such descriptions in terms of the cumulative distribution, which is characterized by the quantiles of the data. The design and engineering of efficient methods to find these quantiles has attracted much study, especially in the case where the data is described incrementally, and we must compute the quantiles in an online, streaming fashion. Yet while such algorithms have proved to be tremendously useful in practice, there has been limited formal comparison of the competing methods, and no comprehensive study of their performance. In this paper, we remedy this deficit by providing a taxonomy of different methods, and describe efficient implementations. In doing so, we propose and analyze variations that have not been explicitly studied before, yet which turn out to perform the best. To illustrate this, we provide detailed experimental comparisons demonstrating the tradeoffs between space, time, and accuracy for quantile computation.