We prove the first polynomial separation between randomized and deterministic time-space tradeoffs of multi-output functions. In particular, we present a total function that on the input of $n$ elements in $[n]$, outputs $O(n)$ elements, such that: (1) There exists a randomized oblivious algorithm with space $O(\log n)$, time $O(n\log n)$ and one-way access to randomness, that computes the function with probability $1-O(1/n)$; (2) Any deterministic oblivious branching program with space $S$ and time $T$ that computes the function must satisfy $T^2S\geq\Omega(n^{2.5}/\log n)$. This implies that logspace randomized algorithms for multi-output functions cannot be black-box derandomized without an $\widetilde{\Omega}(n^{1/4})$ overhead in time. Since previously all the polynomial time-space tradeoffs of multi-output functions are proved via the Borodin-Cook method, which is a probabilistic method that inherently gives the same lower bound for randomized and deterministic branching programs, our lower bound proof is intrinsically different from previous works. We also examine other natural candidates for proving such separations, and show that any polynomial separation for these problems would resolve the long-standing open problem of proving $n^{1+\Omega(1)}$ time lower bound for decision problems with $\mathrm{polylog}(n)$ space.

翻译：我们首次证明了多输出函数在随机化与确定性时间-空间权衡之间的多项式分离。具体而言，我们提出了一个全函数，该函数输入$[n]$中的$n$个元素，输出$O(n)$个元素，且满足以下性质：(1) 存在一个随机化的遗忘算法，其空间复杂度为$O(\log n)$，时间复杂度为$O(n\log n)$，并以单向方式访问随机性，能以概率$1-O(1/n)$计算该函数；(2) 任何计算该函数的确定性遗忘分支程序，若空间为$S$、时间为$T$，则必须满足$T^2S\geq\Omega(n^{2.5}/\log n)$。这一结果表明，多输出函数的对数空间随机化算法无法通过黑盒去随机化实现，除非付出时间上的$\widetilde{\Omega}(n^{1/4})$代价。由于此前所有多输出函数的多项式时间-空间权衡均通过Borodin-Cook方法证明，该概率方法本质上对随机化和确定性分支程序给出相同的下界，因此我们的下界证明与先前工作有本质区别。我们还考察了其他用于证明此类分离的自然候选问题，并表明对这些问题的任何多项式分离都将解决一个长期悬而未决的开放问题——即决策问题在$\mathrm{polylog}(n)$空间下需要$n^{1+\Omega(1)}$时间下界的证明。