Solving the all pairs shortest path problem after minor update of a large dense graph

TL;DR

This paper introduces two algorithms for efficiently updating the all-pairs shortest path matrix after minor changes in large dense graphs, significantly reducing computation time compared to traditional methods.

Contribution

The paper presents novel warm-start algorithms for updating the APSP matrix after graph modifications, with theoretical complexity analysis and experimental validation.

Findings

Warm-start algorithms reduce calculation time significantly.

Best case complexity is O(n^2), worst case is O(n^3).

Achieves up to 99% time savings in experiments.

Abstract

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(n^3) $ 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(n^2) $, the worst case complexity is $ O(n^3) $. 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.