In this paper, we present an advanced analysis of near optimal deterministic algorithms using a small space budget to solve the frequency estimation, heavy hitters, frequent items, and top-k approximation in the bounded deletion model. We define the family of SpaceSaving$\pm$ algorithms and explain why the original SpaceSaving$\pm$ algorithm only works when insertions and deletions are not interleaved. Next, we introduce the new DoubleSpaceSaving$\pm$ and the IntegratedSpaceSaving$\pm$ and prove their correctness. They show similar characteristics and both extend the popular space-efficient SpaceSaving algorithm. However, these two algorithms represent different trade-offs, in which DoubleSpaceSaving$\pm$ distributes the operations to two independent summaries while Integrated-SpaceSaving$\pm$ fully synchronizes deletions with insertions. Since data streams are often skewed, we present an improved analysis of these two algorithms and show that errors do not depend on the hot items and are only dependent on the cold and warm items. We also demonstrate how to achieve the relative error guarantee under mild assumptions. Moreover, we establish that the important mergeability property exists on these two algorithms which is desirable in distributed settings.

翻译：本文中，我们提出了对使用小空间预算在带删除模型下解决频率估计、高频项、频繁项及top-k近似问题的近最优确定性算法的高级分析。我们定义了SpaceSaving$\pm$算法族，并解释了原始SpaceSaving$\pm$算法仅在插入与删除不交错时有效的原因。随后，我们引入了新的DoubleSpaceSaving$\pm$和IntegratedSpaceSaving$\pm$算法，并证明了其正确性。这两种算法展现出相似特征，均扩展了广受欢迎的高空间效率SpaceSaving算法。然而，它们代表了不同的权衡：DoubleSpaceSaving$\pm$将操作分布到两个独立的概要中，而IntegratedSpaceSaving$\pm$则将删除与插入完全同步。由于数据流通常存在倾斜，我们提出了对这些算法的改进分析，表明误差不依赖于热项，仅依赖于冷项和暖项。我们还展示了在温和假设下如何实现相对误差保证。此外，我们证实了这两种算法具备重要的可合并性属性，这在分布式场景中具有优势。