一种改进的交互式流媒体算法
An Improved Interactive Streaming Algorithm for the Distinct Elements Problem
摘要：
不同元素个数（频率矩$F_0$）的精确计算是数据流算法研究中的一个基本问题。我们用$n$表示流的长度，其中每个符号都是从大小为$m$的宇宙中提取的。虽然众所周知，矩$F_0,F_1,F_2$可以通过有效的流算法来近似，但是很容易看出，$F_0,F_2$的精确计算需要空间$Ω(m)$。因此，在以前的工作中，Cormode等人考虑了一个模型，其中数据流也由一个强大的助手处理，该助手提供了结果的交互式证明。他们给出了这样的协议，帮助者和验证者之间的通信轮数为NC中的所有函数的多对数。该轮数$\left(O(\log^2 m) \;\text{in the case of} \;F_0 \right)$可迅速使此类协议变得不切实际。Cormode等人也给出了一个协议，该协议使用$\log m +1$轮来精确计算$F_0$，其中空间复杂度是$O\left(\log m \log n+\log^2 m\right)$，但总通信量是$O\left(\sqrt{n}\log m\left(\log n+ \log m \right)\right)$。他们设法给$\log m$轮协议与$\operatorname{polylog}(m,n)$复杂性的许多其他有趣的问题，包括$F_2$，内积，范围和，但计算$F_0$正是与多对数空间和通信和$O(\log m)$轮仍然开放。在这项工作中，我们给出了一个流交互协议与$\log m$轮的$F_0$使用$O\left(\log m \left(\,\log n + \log m \log\log m\,\right)\right)$位的空间和通信是$O\left( \log m \left(\,\log n +\log^3 m (\log\log m)^2 \,\right)\right)$的精确计算。每个接收到的符号的验证器的更新时间是$O(\log^2 m)$。
英文摘要：
The exact computation of the number of distinct elements (frequency moment $F_0$) is a fundamental problem in the study of data streaming algorithms. We denote the length of the stream by $n$ where each symbol is drawn from a universe of size $m$. While it is well known that the moments $F_0,F_1,F_2$ can be approximated by efficient streaming algorithms, it is easy to see that exact computation of $F_0,F_2$ requires space $Ω(m)$. In previous work, Cormode et al. therefore considered a model where the data stream is also processed by a powerful helper, who provides an interactive proof of the result. They gave such protocols with a polylogarithmic number of rounds of communication between helper and verifier for all functions in NC. This number of rounds $\left(O(\log^2 m) \;\text{in the case of} \;F_0 \right)$ can quickly make such protocols impractical. Cormode et al. also gave a protocol with $\log m +1$ rounds for the exact computation of $F_0$ where the space complexity is $O\left(\log m \log n+\log^2 m\right)$ but the total communication $O\left(\sqrt{n}\log m\left(\log n+ \log m \right)\right)$. They managed to give $\log m$ round protocols with $\operatorname{polylog}(m,n)$ complexity for many other interesting problems including $F_2$, Inner product, and Range-sum, but computing $F_0$ exactly with polylogarithmic space and communication and $O(\log m)$ rounds remained open. In this work, we give a streaming interactive protocol with $\log m$ rounds for exact computation of $F_0$ using $O\left(\log m \left(\,\log n + \log m \log\log m\,\right)\right)$ bits of space and the communication is $O\left( \log m \left(\,\log n +\log^3 m (\log\log m)^2 \,\right)\right)$. The update time of the verifier per symbol received is $O(\log^2 m)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
