Let $G=(V,E)$ be a simple, connected graph. One is often interested in a short path between two vertices $u,v$. We propose a spectral algorithm: construct the function $\phi:V \rightarrow \mathbb{R}_{\geq 0}$ $$ \phi = \arg\min_{f:V \rightarrow \mathbb{R} \atop f(u) = 0, f \not\equiv 0} \frac{\sum_{(w_1, w_2) \in E}{(f(w_1)-f(w_2))^2}}{\sum_{w \in V}{f(w)^2}}.$$ $\phi$ can also be understood as the smallest eigenvector of the Laplacian Matrix $L=D-A$ after the $u-$th row and column have been removed. We start in the point $v$ and construct a path from $v$ to $u$: at each step, we move to the neighbor for which $\phi$ is the smallest. This algorithm provably terminates and results in a short path from $v$ to $u$, often the shortest. The efficiency of this method is due to a discrete analogue of a phenomenon in Partial Differential Equations that is not well understood. We prove optimality for trees and discuss a number of open questions.

中文翻译：

---

最短路径问题的谱方法

令 $G=(V,E)$ 是一个简单的连通图。人们通常对两个顶点 $u,v$ 之间的短路径感兴趣。我们提出了一个谱算法：构造函数 $\phi:V \rightarrow \mathbb{R}_{\geq 0}$ $$ \phi = \arg\min_{f:V \rightarrow \mathbb{R} \atop f(u) = 0, f \not\equiv 0} \frac{\sum_{(w_1, w_2) \in E}{(f(w_1)-f(w_2))^2}}{\sum_{w \in V}{f(w)^2}}.$$ $\phi$也可以理解为拉普拉斯矩阵$L=DA$去掉$u-$th行和列后的最小特征向量. 我们从点 $v$ 开始，构造一条从 $v$ 到 $u$ 的路径：在每一步，我们移动到 $\phi$ 最小的邻居。该算法可证明终止并导致从 $v$ 到 $u$ 的短路径，通常是最短路径。这种方法的效率是由于偏微分方程中的一种现象的离散模拟还没有被很好地理解。我们证明了树的最优性并讨论了许多开放性问题。