2024/12/19 by Gangli Liu, Liu, Gangli
Business, Management and Accounting · Computer Science · Engineering · #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Facility Location and Emergency Management #Graph Theory and Algorithms #Optimization and Packing Problems
paper · pdf · doi:10.48550/arxiv.2412.15122
openalex publication_date 2024/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The all pairs shortest path problem is a fundamental optimization problem in graph theory. We deal with re-calculating the all-pairs shortest path (APSP) matrix after a minor modification of a weighted dense graph, e.g., adding a node, removing a node, or updating an edge. We assume the APSP matrix for the original graph is already known. The graph can be directed or undirected. A cold-start calculation of the new APSP matrix by traditional algorithms, like the Floyd-Warshall algorithm or Dijkstra's algorithm, needs O(n3) time. We propose two algorithms for warm-start calculation of the new APSP matrix. The best case complexity for a warm-start calculation is O(n2) , the worst case complexity is O(n3) . We implemented the algorithms and tested their performance with experiments. The result shows a warm-start calculation can save a great portion of calculation time, compared with cold-start calculation. In addition, another algorithm is devised to warm-start calculate of the shortest path between two nodes. Experiment shows warm-start calculation can save 99% of calculation time, compared with cold-start calculation by Dijkstra's algorithm, on directed complete graphs of large sizes.