Many streaming algorithms provide only a high-probability relative approximation. These two relaxations, of allowing approximation and randomization, seem necessary -- for many streaming problems, both relaxations must be employed simultaneously, to avoid an exponentially larger (and often trivial) space complexity. A common drawback of these randomized approximate algorithms is that independent executions on the same input have different outputs, that depend on their random coins. Pseudo-deterministic algorithms combat this issue, and for every input, they output with high probability the same ``canonical'' solution. We consider perhaps the most basic problem in data streams, of counting the number of items in a stream of length at most $n$. Morris's counter [CACM, 1978] is a randomized approximation algorithm for this problem that uses $O(\log\log n)$ bits of space, for every fixed approximation factor (greater than $1$). Goldwasser, Grossman, Mohanty and Woodruff [ITCS 2020] asked whether pseudo-deterministic approximation algorithms can match this space complexity. Our main result answers their question negatively, and shows that such algorithms must use $\Omega(\sqrt{\log n / \log\log n})$ bits of space. Our approach is based on a problem that we call Shift Finding, and may be of independent interest. In this problem, one has query access to a shifted version of a known string $F\in\{0,1\}^{3n}$, which is guaranteed to start with $n$ zeros and end with $n$ ones, and the goal is to find the unknown shift using a small number of queries. We provide for this problem an algorithm that uses $O(\sqrt{n})$ queries. It remains open whether $poly(\log n)$ queries suffice; if true, then our techniques immediately imply a nearly-tight $\Omega(\log n/\log\log n)$ space bound for pseudo-deterministic approximate counting.

翻译：许多流式算法仅提供高概率的相对近似。这两种松弛（允许近似和随机化）似乎是必要的——对于许多流式问题，必须同时采用这两种松弛，以避免指数级更大的（且通常平凡的）空间复杂度。这些随机近似算法的一个常见缺陷是，在同一输入上的独立执行会产生不同的输出，这些输出依赖于它们的随机硬币。伪确定性算法解决了这一问题，对于每个输入，它们以高概率输出相同的“标准”解。我们考虑数据流中最基本的问题之一，即统计长度至多为$n$的流中项目的数量。Morris计数器[CACM, 1978]是解决该问题的随机近似算法，对于任意固定的近似因子（大于1），它使用$O(\log\log n)$比特的空间。Goldwasser、Grossman、Mohanty和Woodruff [ITCS 2020] 询问伪确定性近似算法是否能匹配这种空间复杂度。我们的主要结果否定了他们的提问，并表明此类算法必须使用$\Omega(\sqrt{\log n / \log\log n})$比特的空间。我们的方法基于一个我们称之为移位查找的问题，该问题可能具有独立的研究意义。在该问题中，可以查询已知字符串$F\in\{0,1\}^{3n}$的移位版本，该版本保证以$n$个零开头并以$n$个一结尾，目标是通过少量查询找到未知的移位。我们为此问题提供了一个使用$O(\sqrt{n})$次查询的算法。$poly(\log n)$次查询是否足够这一问题仍未解决；如果成立，那么我们的技术将立即得出伪确定性近似计数的接近紧致的$\Omega(\log n/\log\log n)$空间下界。