Speeding Up Floyd–Warshall’s Algorithm to Compute All-Pairs Shortest Paths and the Transitive Closure of a Graph

• Published in: Algorithms Vol. 18; no. 9; p. 560
• Main Authors: Lancia, Giuseppe, Dalpasso, Marcello
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 04.09.2025
• Subjects: Algorithms; all-pairs shortest paths; Apexes; Boolean; Complexity; Costs; Data structures; FastSet data structure; Floyd–Warshall graph algorithm; Graph theory; Graphs; Performance enhancement; Shortest-path problems; SmartForce technique; transitive closure; Traveling salesman problem
• ISSN: 1999-4893; 1999-4893
• DOI: 10.3390/a18090560

Floyd–Warshall’s algorithm is a widely-known procedure for computing all-pairs shortest paths in a graph of n vertices in Θ(n3) time complexity. A simplified version of the same algorithm computes the transitive closure of the graph with the same time complexity. The algorithm operates on an n×n matrix, performing n inspections and no more than n updates of each matrix cell, until the final matrix is computed. In this paper, we apply a technique called SmartForce, originally devised as a performance enhancement for solving the traveling salesman problem, to avoid the inspection and checking of cells that do not need to be updated, thus reducing the overall computation time when the number, u, of cell updates is substantially smaller than n3. When the ratio u/n3 is not small enough, the performance of the proposed procedure might be worse than that of the Floyd–Warshall algorithm. To speed up the algorithm independently of the input instance type, we introduce an effective hybrid approach. Finally, a similar procedure, which exploits suitable fast data structures, can be used to achieve a speedup over the Floyd–Warshall simplified algorithm that computes the transitive closure of a graph.