Graph

The Floyd-Warshall algorithm is a dynamic programming algorithm for finding the shortest paths between every pair of vertices in a graph. Robert Floyd and Stephen Warshall published it independently in 1962. Unlike Dijkstra’s and Bellman-Ford algorithms, it computes all-pairs shortest paths in a single run. With O(V³) time complexity, it can handle negative edge weights and detect negative cycles, making it useful for tasks such as network diameter calculation, transitive closure, and database query optimization. ...

The Bellman-Ford algorithm 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 stands out from Dijkstra’s algorithm because it can handle negative edge weights and detect negative cycles in the graph. It uses a dynamic programming approach with O(VE) time complexity and serves as a core component in real-world applications such as network routing protocols, arbitrage detection in financial markets, and minimum cost flow problems. ...

Dijkstra’s algorithm is a fundamental method 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, it remains a core tool in fields such as network routing, GPS navigation, and game AI. The algorithm uses a greedy approach to make the best choice at each step, and with a priority queue it runs in O(E log V), making it practical for large-scale graphs. ...