Papers › Solving the all pairs shortest path problem after minor update of a large dense graph
Solving the all pairs shortest path problem after minor update of a large dense graph
Gangli Liu
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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.
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