We develop a novel algorithm to construct a congestion-approximator with polylogarithmic quality on a capacitated, undirected graph in nearly-linear time. Our approach is the first *bottom-up* hierarchical construction, in contrast to previous *top-down* approaches including that of Racke, Shah, and Taubig (SODA 2014), the only other construction achieving polylogarithmic quality that is implementable in nearly-linear time (Peng, SODA 2016). Similar to Racke, Shah, and Taubig, our construction at each hierarchical level requires calls to an approximate max-flow/min-cut subroutine. However, the main advantage to our bottom-up approach is that these max-flow calls can be implemented directly *without recursion*. More precisely, the previously computed levels of the hierarchy can be converted into a *pseudo-congestion-approximator*, which then translates to a max-flow algorithm that is sufficient for the particular max-flow calls used in the construction of the next hierarchical level. As a result, we obtain the first non-recursive algorithms for congestion-approximator and approximate max-flow that run in nearly-linear time, a conceptual improvement to the aforementioned algorithms that recursively alternate between the two problems.

翻译：我们提出了一种新颖算法，可在近线性时间内为带容量无向图构建具有多对数质量的拥塞近似器。我们的方法是首个*自底向上*的层次化构建方法，与此前包括Racke、Shah和Taubig（SODA 2014）在内的*自顶向下*方法形成对比——后者是唯一其他可实现多对数质量且能在近线性时间内实现的构建方法（Peng，SODA 2016）。与Racke、Shah和Taubig的方法类似，我们在每个层次级别上的构建都需要调用近似最大流/最小割子程序。然而，我们自底向上方法的主要优势在于这些最大流调用可以直接实现*无需递归*。更准确地说，先前计算的层次级别可转化为*伪拥塞近似器*，进而转化为足以满足下一层次级别构建中特定最大流调用的最大流算法。因此，我们首次获得了在近线性时间内运行的非递归拥塞近似器算法和近似最大流算法，这对前述递归交替求解这两个问题的算法形成了概念性改进。