Adversarially robust streaming algorithms are required to process a stream of elements and produce correct outputs, even when each stream element can be chosen as a function of 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 $\ell$ with elements in $\{1,\ldots,n\}$, and $\le 1/\text{poly}(\ell)$ error, we show that when $\ell = O(2^{\sqrt{\log n}})$, "random seed" adversarially robust algorithms, which only use randomness at initialization, require $\ell^{\Omega(1)}$ bits of space, while "random tape" adversarially robust algorithms, which may make random decisions at any time, may use $O(\text{polylog}(\ell))$ space. When $\ell$ is between $n^{\Omega(1)}$ and $O(\sqrt{n})$, "random tape" adversarially robust algorithms need $\ell^{\Omega(1)}$ space, while "random oracle" adversarially robust algorithms, which can read from a long random string for free, may use $O(\text{polylog}(\ell))$ 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：对抗性鲁棒流算法需要处理一个元素流并产生正确输出，即使每个流元素都可以根据算法先前的输出被选择。与经典流算法（仅需对最坏情况的固定流保证正确性）类似，能够访问随机性的对抗性鲁棒算法可以比确定性算法使用显著更少的空间。我们证明，对于流式环境中的缺失项寻找问题，其空间复杂度也显著取决于对抗性鲁棒算法被允许如何使用随机性。（相比之下，经典流算法的空间复杂度并不如此强烈地依赖于随机性的使用方式。）对于在长度为 $\ell$、元素取自 $\{1,\ldots,n\}$ 的流上进行缺失项寻找，且误差 $\le 1/\text{poly}(\ell)$ 的情况，我们证明：当 $\ell = O(2^{\sqrt{\log n}})$ 时，仅初始化时使用随机性的“随机种子”对抗性鲁棒算法需要 $\ell^{\Omega(1)}$ 比特的空间，而可以随时做出随机决策的“随机带”对抗性鲁棒算法可能仅需 $O(\text{polylog}(\ell))$ 空间。当 $\ell$ 介于 $n^{\Omega(1)}$ 和 $O(\sqrt{n})$ 之间时，“随机带”对抗性鲁棒算法需要 $\ell^{\Omega(1)}$ 空间，而可以免费读取长随机字符串的“随机预言机”对抗性鲁棒算法可能仅需 $O(\text{polylog}(\ell))$ 空间。“随机种子”情况下的空间下界，通过先前工作中给出的归约，源自本文给出的伪确定性流算法的下界。