Adversarially robust streaming algorithms are required to process a stream of elements and produce correct outputs, even when each stream element can be chosen depending on earlier algorithm outputs. As with classic streaming algorithms, which must only be correct for the worst-case fixed stream, adversarially robust algorithms with access to randomness can use significantly less space than deterministic algorithms. We prove that for the Missing Item Finding problem in streaming, the space complexity also significantly depends on how adversarially robust algorithms are permitted to use randomness. (In contrast, the space complexity of classic streaming algorithms does not depend as strongly on the way randomness is used.) For Missing Item Finding on streams of length $r$ with elements in $\{1,...n\}$, and $\le 1/\text{poly}(n)$ error, we show that when $r = O(2^{\sqrt{\log n}})$, "random seed" adversarially robust algorithms, which only use randomness at initialization, require $r^{\Omega(1)}$ bits of space, while "random tape" adversarially robust algorithms, which may make random decisions at any time, may use $O(\text{polylog}(r))$ random bits. When $r = \Theta(\sqrt{n})$, "random tape" adversarially robust algorithms need $r^{\Omega(1)}$ space, while "random oracle" adversarially robust algorithms, which can read from a long random string for free, may use $O(\text{polylog}(r))$ space. The space lower bound for the "random seed" case follows, by a reduction given in prior work, from a lower bound for pseudo-deterministic streaming algorithms given in this paper.

翻译：对抗鲁棒流式算法要求处理元素流并产生正确输出，即使每个流元素可能根据先前算法输出进行选择。与经典流式算法（仅需在最坏固定流上正确）类似，具备随机访问能力的对抗鲁棒算法可比确定性算法使用显著更少空间。我们证明：在流式处理中的缺失项查找问题上，空间复杂度也显著依赖于对抗鲁棒算法被允许使用随机性的方式。（相比之下，经典流式算法的空间复杂度对随机性使用方式的依赖性较弱。）对于元素取自集合$\{1,...,n\}$、长度为$r$且误差$\le 1/\text{poly}(n)$的流上缺失项查找问题，我们表明：当$r = O(2^{\sqrt{\log n}})$时，仅在初始化阶段使用随机性的“随机种子”对抗鲁棒算法需要$r^{\Omega(1)}$比特空间，而可在任意时刻执行随机决策的“随机纸带”对抗鲁棒算法仅需$O(\text{polylog}(r))$随机比特。当$r = \Theta(\sqrt{n})$时，“随机纸带”对抗鲁棒算法需要$r^{\Omega(1)}$空间，而可免费读取长随机字符串的“随机预言机”对抗鲁棒算法仅需$O(\text{polylog}(r))$空间。“随机种子”情形下的空间下界，通过先前工作中的归约，源自本文给出的伪确定性流式算法的下界。