Graph

The Floyd-Warshall algorithm is a dynamic programming-based algorithm that finds the shortest paths between all pairs of vertices in a graph. Independently published by Robert Floyd and Stephen Warshall in 1962, it differs from Dijkstra’s and Bellman-Ford algorithms by computing shortest paths for all pairs simultaneously in a single execution. With O(V³) time complexity, it can handle edges with negative weights and detect the presence of negative cycles, making it a core component in various fields including network diameter calculation, transitive closure operations, and database query optimization. ...

The Bellman-Ford algorithm is an algorithm that finds the shortest paths from a single source vertex to all other vertices in a weighted graph. Independently discovered by Richard Bellman and Lester Ford Jr. in the 1950s, it has powerful characteristics that distinguish it from Dijkstra’s algorithm: it can handle edges with negative weights and detect whether negative cycles exist in the graph. Using a dynamic programming approach with O(VE) time complexity, it is used as a core component in various real-world applications including network routing protocols, arbitrage detection in financial markets, and minimum cost flow problems. ...

Dijkstra’s algorithm is the quintessential algorithm for finding the shortest paths from a starting vertex to all other vertices in a weighted graph, invented by Dutch computer scientist Edsger Wybe Dijkstra in 1956 and still serving as a core component in numerous fields including network routing, GPS navigation, and game AI. The algorithm uses a greedy algorithm approach to make optimal choices at each step, and achieves an efficient time complexity of O(E log V) through priority queue implementation, making it practical for large-scale graphs. ...