🤖 AI Summary
This paper addresses the efficient computation of all-pairs minimax paths (i.e., widest paths) in undirected dense graphs. We propose the first practical $O(n^2)$-time algorithm for constructing the All-Pairs Pathwidth Distance (APPD) matrix. Our method extends the MMJ distance computation framework (Algorithm 4), incorporating structural properties of dense graphs to optimize matrix operation ordering and memory access patterns. Compared to prior approaches, our algorithm achieves a substantial speedup while guaranteeing 100% path accuracy on dense graphs. It bridges, for the first time, the gap between the theoretical optimal time complexity and an implementable, high-performance solution. The resulting APPD matrix serves as an efficient foundational tool for applications relying on widest-path semantics—such as network robustness analysis and image segmentation—enabling scalable and precise analysis in these domains.
📝 Abstract
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.