发表机构
ETH Zürich; INSAIT, Sofia University “St. Kliment Ohridski”（苏黎世联邦理工学院; 索菲亚大学“圣克利门特奥赫里德斯基” INSAIT）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对现有低拉伸生成树算法复杂但简单Dijkstra启发式效果好的问题，通过平滑分析证明图权重微扰可使任意根最短路径树成为$\tilde{O}(1)$-近似LSST，且扰动可通过低直径分解高效计算，提出LSST求解新方法。
AI 中文摘要
给定无向加权图$G$，$\beta$-近似低拉伸生成树（LSST）$T \subseteq G$是一种在期望意义下以$\beta$因子近似$G$的距离度量的树。目前，现有的可证明优良的LSST求解算法需要从任意源点精心构造近似最短路径树，算法结构复杂。与之相对，从业者发现一种简单得多的启发式方法效果出奇地好：选择任意根节点，运行Dijkstra算法，将得到的最短路径树用作LSST。\n 本文中，我们对Dijkstra算法等最短路径树算法进行了平滑分析，解释了这一现象。我们证明，对输入图的权重施加微小扰动，就足以使所得图中以任意节点为根的最短路径树成为$\tilde{O}(1)$-近似LSST。\n 我们进一步证明，这类扰动集合可通过少量低直径分解（LDDs）高效计算。因此，我们的证明具有构造性，提出了一种计算LSST的新方法。
英文摘要
Given an undirected weighted graph $G$, a $γ$-approximate low-stretch spanning tree (LSST) $T \subseteq G$ is a tree that approximates the distance metric of $G$ up to a $γ$-factor in expectation. Currently, existing algorithms to find a provably good LSST carefully construct an approximate shortest-path tree from an arbitrary source. The resulting algorithms are intricate. In contrast, practitioners observed that a much simpler heuristic performs surprisingly well: choose an arbitrary root, run Dijkstra's algorithm, and use the resulting shortest-path tree as an LSST. In this paper, we give a smoothed analysis of shortest-path tree algorithms, such as Dijkstra's algorithm, that explains this behavior. We show that adding a small perturbation to the weights of the input graph suffices to turn the shortest path tree rooted at an arbitrary node in the resulting graph into an $\tilde{O}(1)$-approximate LSST. We further show that the set of perturbations can be computed efficiently from few low-diameter decompositions (LDDs). Thus, our proof is also constructive in the sense of giving a novel approach to computing LSSTs.
