A unique sink orientation (USO) is an orientation of the $n$-dimensional hypercube graph such that every non-empty face contains a unique sink. Schurr showed that given any $n$-dimensional USO and any dimension $i$, the set of edges $E_i$ in that dimension can be decomposed into equivalence classes (so-called phases), such that flipping the orientation of a subset $S$ of $E_i$ yields another USO if and only if $S$ is a union of a set of these phases. In this paper we prove various results on the structure of phases. Using these results, we show that all phases can be computed in $O(3^n)$ time, significantly improving upon the previously known $O(4^n)$ trivial algorithm. Furthermore, we show that given a boolean circuit of size $poly(n)$ succinctly encoding an $n$-dimensional (acyclic) USO, it is PSPACE-complete to determine whether two given edges are in the same phase. The problem is thus equally difficult as determining whether the hypercube orientation encoded by a given circuit is an acyclic USO [G\"artner and Thomas, STACS'15].

翻译：唯一沉没定向（USO）是$n$维超立方体图的一种定向，使得每个非空面包含唯一的沉没点。Schurr 证明，给定任意$n$维USO和任意维度$i$，该维度中的边集$E_i$可分解为等价类（称为相位），使得翻转$E_i$的某个子集$S$的定向得到另一个USO当且仅当$S$是这些相位的并集。本文证明了关于相位结构的多个结果。利用这些结果，我们表明所有相位可在$O(3^n)$时间内计算，显著改进了此前已知的$O(4^n)$平凡算法。此外，我们证明，给定一个大小为$poly(n)$的布尔电路（简洁编码一个$n$维（无环）USO），判断两条给定边是否属于同一相位是PSPACE完全的。因此，该问题与判断给定电路编码的超立方体定向是否为无环USO [Gärtner and Thomas, STACS'15] 具有同等难度。