Consider a bounded-degree graph $G$ that belongs to a minor-closed family (such as planar graphs). Such a graph has a hyperfinite decomposition, wherein, for a sufficiently small $\varepsilon > 0$, one can remove $\varepsilon dn$ edges to obtain connected components of size independent of $n$. (As usual, $n$ is the number of vertices and $d$ is the degree bound.) In a seminal result, Hassidim-Kelner-Nguyen-Onak (FOCS 2009) introduced the partition oracle, a procedure that provides local access to a hyperfinite decomposition. The partition oracle computes the component containing an input vertex $v$ with query complexity (to $G$) independent of $n$. Remarkably, this is done without any preprocessing on $G$. The coordination is done purely through a shared random seed. Despite a line of work on optimizing the query complexity of partition oracles, there were no attempts to bound the size of the random seed. All existing partition oracles use a random seed of size $Ω(n)$, which technically implies a linear setup time. Any blackbox derandomization would likely need $Ω(\log^2n)$ uniform random bits. A natural question is whether the random seed can also have length independent of $n$. We prove the $poly(d\varepsilon^{-1})$-query partition oracles of Kumar-Seshadhri-Stolman can be implemented with a random seed of $poly(d\varepsilon^{-1}) \cdot \log n$ length. To get a deeper understanding on the randomness complexity, we consider a more general model where the vertex labels come from the universe $[N]$, where $N \geq n$. In this setting, we prove that any partition oracle even for cycles requires $ω_N(1)$ random bits.

翻译：考虑一个属于小闭族（如平面图）的有界度图 $G$。此类图具有超有限分解，即对于足够小的 $\varepsilon > 0$，可移除 $\varepsilon dn$ 条边以获得大小与 $n$ 无关的连通分量（通常 $n$ 为顶点数，$d$ 为度上界）。在一项开创性成果中，Hassidim-Kelner-Nguyen-Onak (FOCS 2009) 引入了分区预言机，该程序提供对超有限分解的局部访问。分区预言机计算包含输入顶点 $v$ 的分量，其对 $G$ 的查询复杂度与 $n$ 无关。值得注意地，这一过程无需对 $G$ 进行预处理，协调完全通过共享随机种子实现。尽管已有系列工作优化分区预言机的查询复杂度，但尚无研究尝试约束随机种子的大小。现有所有分区预言机均使用大小为 $Ω(n)$ 的随机种子，这技术上意味着线性设置时间。任何黑盒去随机化方法很可能需要 $Ω(\log^2n)$ 均匀随机比特。一个自然问题是随机种子长度能否也与 $n$ 无关。我们证明 Kumar-Seshadhri-Stolman 提出的 $poly(d\varepsilon^{-1})$-查询分区预言机可用长度为 $poly(d\varepsilon^{-1}) \cdot \log n$ 的随机种子实现。为深入理解随机性复杂度，我们考虑更一般的模型：顶点标签来自全集 $[N]$（其中 $N \geq n$）。在此设定下，我们证明即便是针对环图，任何分区预言机也需要 $ω_N(1)$ 个随机比特。