Oblivious routing is a well-studied paradigm that uses static precomputed routing tables for selecting routing paths within a network. Existing oblivious routing schemes with polylogarithmic competitive ratio for general networks are tree-based, in the sense that routing is performed according to a convex combination of trees. However, this restriction to trees leads to a construction that has time quadratic in the size of the network and does not parallelize well. In this paper we study oblivious routing schemes based on electrical routing. In particular, we show that general networks with $n$ vertices and $m$ edges admit a routing scheme that has competitive ratio $O(\log^2 n)$ and consists of a convex combination of only $O(\sqrt{m})$ electrical routings. This immediately leads to an improved construction algorithm with time $\tilde{O}(m^{3/2})$ that can also be implemented in parallel with $\tilde{O}(\sqrt{m})$ depth.

翻译：遗忘路由是一种成熟的范式，利用静态预计算路由表在网络中选择路由路径。现有针对一般网络的对数级竞争比遗忘路由方案均基于树结构，即路由是根据树的凸组合执行的。然而，这种对树的限制导致构建所需时间与网络规模呈二次关系，且并行化程度不佳。本文研究基于电流路由的遗忘路由方案。具体而言，我们证明具有$n$个顶点和$m$条边的一般网络存在一种竞争比为$O(\log^2 n)$的路由方案，该方案仅由$O(\sqrt{m})$个电流路由的凸组合构成。这直接催生了一种改进的构建算法，时间复杂度为$\tilde{O}(m^{3/2})$，并且能以$\tilde{O}(\sqrt{m})$的深度并行实现。