We provide an efficient $ O(n^2) $ implementation for solving the all pairs minimax path problem or widest path problem in an undirected dense graph. It is a code implementation of the Algorithm 4 (MMJ distance by Calculation and Copy) in a previous paper. The distance matrix is also called the all points path distance (APPD). We conducted experiments to test the implementation and algorithm, compared it with several other algorithms for solving the APPD matrix. Result shows Algorithm 4 works good for solving the widest path or minimax path APPD matrix. It can drastically improve the efficiency for computing the APPD matrix. There are several theoretical outcomes which claim the APPD matrix can be solved accurately in $ O(n^2) $ . However, they are impractical because there is no code implementation of these algorithms. It seems Algorithm 4 is the first algorithm that has an actual code implementation for solving the APPD matrix of minimax path or widest path problem in $ O(n^2) $, in an undirected dense graph.

翻译：我们提供了一种高效的$ O(n^2) $实现，用于解决无向稠密图中的全对最小最大路径问题（亦称最宽路径问题）。该实现是先前一篇论文中算法4（通过计算与复制的MMJ距离）的代码实现。该距离矩阵也被称为全点路径距离（APPD）矩阵。我们通过实验测试了该实现与算法，并将其与求解APPD矩阵的其他几种算法进行了比较。结果表明，算法4在求解最宽路径或最小最大路径的APPD矩阵方面表现良好，能显著提升计算APPD矩阵的效率。尽管存在若干理论成果声称APPD矩阵可在$ O(n^2) $时间内精确求解，但由于缺乏这些算法的实际代码实现，它们并不具备实用性。算法4似乎是首个在无向稠密图中以$ O(n^2) $时间复杂度求解最小最大路径或最宽路径问题APPD矩阵、并拥有实际代码实现的算法。