Accurately estimating the number of unique elements that appear in data streams in real time is a fundamental problem with applications including network traffic monitoring and real-time social media analytics. Traditional sketch-based algorithms such as FM Sketch and HyperLogLog offer memory-friendly solutions for cardinality estimation but fall short in scenarios where the stream elements are privacy-sensitive and require differential privacy. Although recent approaches have incorporated differential privacy into the above cardinality estimators, they are limited to single-query settings, restricting their applicability. Previous methods for private cardinality continual release settings-i.e., releasing the cardinality after each new element in the stream-demand large memory resources and are thus difficult to apply in practice. In this paper, we present a novel cardinality estimation framework, FC, which ensures differential privacy under continual releases while simultaneously achieving low memory usage, high accuracy, and efficient computation. Our approach innovatively leverages an efficient cardinality estimator and privacy-preserving mechanisms to overcome the limitations of existing methods. Comprehensive experiments demonstrate that our method reduces memory usage by up to 504 times compared to the best previous method while maintaining nearly the same accuracy. Additionally, under identical memory constraints, our method improves the estimation accuracy by orders of magnitude.

Computing set difference cardinalities is a critical task in database optimization, network management, and anomaly detection. Due to the limited computational and mem-ory resources, exactly calculating set difference cardinalities becomes impractical in real-world applications. To solve this issue, sketch methods such as Odd sketch, Tug-of-War sketch, and HyperLogLog sketch can be extended to provide approximate estimations of set difference cardinalities. They use a family of hash functions to compress all elements in a set into a compact data structure. Unfortunately, Odd sketch suffers from limited estimation range, while Tug-of-War sketch and HyperLogLog sketch unavoidably face the problems of large estimation errors and high computational costs. In this paper, we design a novel data structure of bit array GXBits to fast and accurately estimate set difference cardinalities in a large range. In GXBits, the prob-ability of each bit recording its corresponding elements follows a variant of geometric distributions and varies across different bits. We conduct extensive experiments on synthetic datasets and real-world datasets. Experimental results demonstrate that our method GXBits is more computationally and memory efficient, and significantly increases the estimation accuracy of existing methods by up to 221.3 times.

Estimating cardinality, i.e., the number of distinct elements, of a data stream is a fundamental problem in areas like databases, computer networks, and information retrieval. This study delves into a broader scenario where each element carries a positive weight. Unlike traditional cardinality estimation, limited research exists on weighted cardinality, with current methods requiring substantial memory and computational resources, challenging for devices with limited capabilities and real-time applications like anomaly detection. To address these issues, we propose QSketch, a memory-efficient sketch method for estimating weighted cardinality in streams. QSketch uses a quantization technique to condense continuous variables into a compact set of integer variables, with each variable requiring only 8 bits, making it 8 times smaller than previous methods. Furthermore, we leverage dynamic properties during QSketch generation to significantly enhance estimation accuracy and achieve a lower time complexity of O(1) for updating estimations upon encountering a new element. Experimental results on synthetic and real-world datasets show that QSketch is approximately 30% more accurate and two orders of magnitude faster than the state-of-the-art, using only 1/8 of the memory.

Given two sets of elements held by two different parties separately, computing the cardinality (i.e., the number of distinct elements) of their intersection set is a fundamental task in applications such as network monitoring and database systems. To handle large sets with limited space, computation, and communication costs, lightweight probabilistic methods (i.e., sketch methods) such as the Flajolet-Martin (FM) sketch and the HyperLogLog (HLL) sketch are extensively used. However, when a set's probabilistic data summary and the hash functions used to construct the sketch are disclosed to an untrusted third party, the set's privacy is compromised. Directly applyingLocal Differential Privacy (LDP) techniques to safeguard the sketch collection results in extremely large estimation errors of set intersection cardinalities. To address this issue, we propose a novel sketch method that makes it easier to incorporate noise into the constructed sketch to achieve differential privacy. More importantly, our sketch method is compatible with the LDP noise. In other words, the probabilistic model underlying our LDP-based data summary is quite basic, allowing us to eliminate the estimation error generated by the noise. We perform extensive experiments on various synthetic and real-world datasets and the experimental results demonstrate that our method is orders of magnitude more accurate and several times faster than state-of-the-art methods.

Kernel density estimation is a useful method for estimating the probability distribution of data. It is a challenge to achieve efficient kernel density estimation, especially for large-scale and high-dimension stream data. We proposerotation kernel, a novel kernel function for density estimation. The rotation kernel density can be fast estimated by a data structure namedRotation Kernel Density Sketch(RKDS). RKDS is a time- and memory-efficient method for kernel density estimation, even over data streams and distributed systems. RKDS is applicable for estimating density at specific points and also for representing data distribution. We provide theoretical analysis for rotation kernel and RKDS. Furthermore, we apply RKDS to outlier detection, concept drift detection, and personalized federated learning. Experiments show that our method improves time efficiency by up to$3\times 10^{3}$times compared with baselines. RKDS also provides comparable detecting precision and better delay on outlier detection and concept drift detection tasks.

Calculating the number of distinct values (i.e., NDV) in a column of a big table is costly yet fundamental to a variety of database applications such as data compression and profiling. To reduce the high time and space cost, a number of sketch methods (e.g., HyperLogLog) have been proposed, which estimate the NDV from a constructed compact data summary of distinct values. However, these methods fail or are costly to manage fully-dynamic scenarios where data is often inserted into and deleted from the table. To solve this issue, we propose a novel sketch method,Half-Xor. Our Half-Xor sketch consists of a compact bit matrix and a small counter array, and it needs to set a few bits and update a counter when handling a data insertion/deletion. Compared with the state-of-the-art mergeable method, our experimental results demonstrate that our method Half-Xor is up to 6.6 times more accurate under the same memory usage and reduces the memory usage by up to 16 times to achieve the same estimation accuracy.

Bipartite graphs widely exist in real-world scenarios and model binary relations like host-website, author-paper, and user-product. In bipartite graphs, a butterfly (i.e.,$2\times 2$bi-clique) is the smallest non-trivial cohesive structure and plays an important role in applications such as anomaly detection. Considerable efforts focus on counting butterflies in static bipartite graphs. However, they suffer from high time and space complexity when the bipartite graph of interest is given as a stream of edges. Although there are methods for approximately counting butterflies from bipartite graph streams, they suffer from either low accuracy or high time complexity. Therefore, it is still a challenge to accurately estimate butterfly counts from bipartite graph streams in a short time. To address this issue, we develop novel algorithms by exploiting the bipartite nature, which subtly integrates sampling and sketching techniques. We provide accurate estimators for butterfly counts and derive simple yet exact formulas for bounding their errors. We also conduct extensive experiments on a variety of real-world large bipartite graphs. Experimental results demonstrate that our algorithms are up to 20.0 times more accurate and up to 286.3 times faster than state-of-the-art methods under the same memory usage.

Kernel density estimation method is a powerful tool and is widely used in many important real-world applications such as anomaly detection and statistical learning. Unfortunately, current kernel methods suffer from high computational or space costs when dealing with large-scale, high-dimensional datasets, especially when the datasets of interest are given in a stream fashion. Although there are sketch methods designed for kernel density estimation over data streams, they still suffer from high computational costs. To address this problem, in this paper, we propose a novel Rotation Kernel. The Rotation Kernel is based on a Rotation Hash method and is much faster to compute. To achieve memory-efficient kernel density estimation over data streams, we design a method, RKD-Sketch, which compresses high dimensional data streams into a small array of integer counters. We conduct extensive experiments on both synthetic and real-world datasets, and experimental results demonstrate that our RKD-Sketch saves up to 216 times computational resources and up to 104 times space resources than state-of-the-arts. Furthermore, we apply our Rotation Kernel in active learning. Results show that our method achieves up to 256 times speedup and saves up to 13 times space to achieve the same accuracy as the baseline methods.

Given a set S, Membership Querying (MQ) answers whether a query element $q\in S$. It is a fundamental task in areas like database systems and computer networks. In this paper, we consider a more general problem, Multi-Set Multi-Membership Querying (MS-MMQ). Given n sets $S_0,łdots,S_n-1 $, MS-MMQ answers which sets contain element q. A direct way to address MS-MMQ is to build an MQ structure (e.g., Bloom Filter) for each set. However, the query and space complexities grow linearly with n and become prohibitive for a large n. To address this challenge, we propose a novel Circular Shift and Coalesce (CSC) framework to efficiently achieve approximate MS-MMQ. Instead of building an MQ data structure for each set, the CSC index encodes all n sets into a compact sketch and retrieves only a few bytes in the sketch for a query, which achieves high memory-efficiency and boosts the query speed by several times. CSC is compatible with mainstream data structures for Approximate MQ. We conduct experiments on real-world datasets and results demonstrate that our framework is up to 91.2 times faster and up to 48.9 times more accurate than state-of-the-art methods.