On Differentially Private Longest Increasing Subsequence Computation in Data Stream

Many important applications require a continuous computation of statistics over data streams. Activities monitoring, surveillance and fraud detections are some settings where it is crucial for the monitoring applications to protect user’s sensitive information in addition to efficiently compute the required statistics. In the last two decades, a broad range of techniques for time-series and stream data monitoring has been developed to provide provable privacy guarantees employing the formal notion of differential privacy. Although these solutions are well established, they are mostly limited to count based statistics (e.g. number of distinct elements, heavy hitters) and do not apply in settings where more complex statistics are needed. In this paper, we consider a more general problem of estimating the sortedness of a data stream by privately computing the length of the longest increasing subsequence (LIS). This important statistic can be used to detect surprising trends in time-series data (e.g. finance) and perform approximate string matching in computational biology domains. Our proposed approaches employ the differential privacy notion which provides strong and provable privacy guarantees. Our solutions estimate the length of the LIS using block decomposition and local approximation techniques. We provide a rigorous analysis to bound the approximation error of our algorithms in terms of privacy level and length of the stream. Furthermore, we extend our solutions to computing the length of the LIS over sliding windows and we show the beneficial effects of this formulation on the final utility. An extensive experimental evaluation of our proposed solutions on real-world data streams demonstrates the effectiveness of our approaches for computing accurate statistics and detecting surprising trends.