AI 中文总结
研究带催化和常规内存的流模型，利用特定内存位精确计算频率矩及多项式，给出相关应用算法，通过与催化通信模型关联及新方法得出单遍算法限制和多遍算法下界，并设计出突破三遍障碍的两遍算法计算第二矩。
AI 中文摘要
我们引入一种同时使用催化和常规内存的流模型。在此模型中，展示了如何利用对数数量的常规内存位和多项式数量的催化内存位精确计算频率矩。更一般地，说明了如何在相同空间界限内精确计算项频率的任意多项式。作为应用，得到了能精确计算流中不同元素数量、图中三角形（或其他小子图）数量以及识别重尾元素的催化流算法。频率矩算法对流执行常数次数的遍历，多项式求值算法比多项式次数多一次遍历。通过将催化流模型与Pyne等人引入的催化通信模型相关联，表明催化内存对单遍流算法无用。对于多遍流算法的下界，Pyne等人的不可能性结果不够强。然而，使用不同技术表明在某些自然限制下，没有催化流算法能在少于三遍的情况下计算第二频率矩。这种受限的两遍算法定义指导我们设计出一种能规避这些限制并突破三遍障碍的精确计算第二矩的两遍算法。
英文摘要
We introduce a streaming model that uses both catalytic and regular memory. In this model, we show how to exactly compute the frequency moments using a logarithmic number of bits of regular memory and a polynomial number of bits of catalytic memory. More generally, we show how to compute arbitrary polynomials of the item frequencies exactly within the same space bounds. As an application, we obtain catalytic streaming algorithms that exactly compute the number of distinct elements in a stream, count the number of triangles (or any other small subgraph) in a graph whose edges arrive in a stream, and identify heavy hitters. Our algorithms for frequency moments perform a constant number of passes over the stream, and for polynomial evaluation, we require one more pass than the degree of the polynomial. By relating our catalytic streaming model to the catalytic communication model introduced in Pyne et al., we show that catalytic memory is not useful for any one-pass streaming algorithm. For lower bounds on multipass streaming algorithms, the impossibility results of Pyne et al. are not strong enough. However, using a different technique, we show that under certain natural restrictions, no catalytic streaming algorithm can compute the second frequency moment in fewer than three passes. This definition of the restricted class of two-pass algorithms then guides us in the design of a two-pass algorithm for computing the second moment exactly that circumvents these restrictions and breaks the three-pass barrier.
