You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
Given @math distributed data streams @math , we consider the problem of estimating the number of unique identifiers in streams defined by set expressions over @math . We identify a broad class of algorithms for solving this problem, and show that the estimators output by any algorithm in this class are perfectly unbiased and satisfy strong variance bounds. Our analysis unifies and generalizes a variety of earlier results in the literature. To demonstrate its generality, we describe several novel sampling algorithms in our class, and show that they achieve a novel tradeoff between accuracy, space usage, update speed, and applicability.

Cited Abstract(s):
[1]: The frequency moments of a sequence containingmielements of typei, 1?i?n, are the numbersFk=?ni=1mki. We consider the space complexity of randomized algorithms that approximate the numbersFk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, it turns out that the numbersF0,F1, andF2can be approximated in logarithmic space, whereas the approximation ofFkfork?6 requiresn?(1)space. Applications to data bases are mentioned as well.

[2]: Cardinality estimation has a wide range of applications and is of particular importance in database systems. Various algorithms have been proposed in the past, and the HyperLogLog algorithm is one of them. In this paper, we present a series of improvements to this algorithm that reduce its memory requirements and significantly increase its accuracy for an important range of cardinalities. We have implemented our proposed algorithm for a system at Google and evaluated it empirically, comparing it to the original HyperLogLog algorithm. Like HyperLogLog, our improved algorithm parallelizes perfectly and computes the cardinality estimate in a single pass.
[1] gave a streaming algorithm for the problem that outputs a @math -approximation with constant probability, using @math bits of space. This improves over the bit-complexity of HLL by roughly a @math factor (and avoids the assumption of truly random hash functions). Like HLL, it is not known how to extend the algorithm to handle @math queries for non-trivial properties @math , and the algorithm does not appear to have been implemented [2] .