量子算法及无向图中离心率、半径和直径的下界
Quantum algorithms and lower bounds for eccentricity, radius, and diameter in undirected graphs
AI总结：
本研究提出量子算法计算无向加权图的直径和半径，时间复杂度为$\widetilde{O}(n\sqrt{m})$，并给出$2/3$近似直径的算法及$\Omega(\sqrt{nm})$查询下界。
摘要：
计算离心率、半径和直径的问题是图论中的基础问题。这些参数本质上基于图的距离度量来定义。在这项工作中，我们针对邻接表模型中的无向加权图，提出了用于直径和半径的量子算法。这些算法以$\widetilde{O}(n\sqrt{m})$的时间输出直径和半径及相应的路径。此外，对于直径问题，我们提出了一种量子算法，能在$\widetilde{O}(\sqrt{m}n^{3/4})$时间内将直径近似到$2/3$的比例。我们还通过从最小值寻找问题的归约，为上述所有问题建立了$\Omega(\sqrt{nm})$的量子查询下界。
英文摘要：
The problems of computing eccentricity, radius, and diameter are fundamental to graph theory. These parameters are intrinsically defined based on the distance metric of the graph. In this work, we propose quantum algorithms for the diameter and radius of undirected, weighted graphs in the adjacency list model. The algorithms output diameter and radius with the corresponding paths in $\widetilde{O}(n\sqrt{m})$ time. Additionally, for the diameter, we present a quantum algorithm that approximates the diameter within a $2/3$ ratio in $\widetilde{O}(\sqrt{m}n^{3/4})$ time. We also establish quantum query lower bounds of $Ω(\sqrt{nm})$ for all the aforementioned problems through a reduction from the minima finding problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
