Bellman-Ford Algorithm

Dijkstra's shortest-path algorithm silently assumes every edge weight is non-negative. Hand it a graph with one negative edge and it can produce a wrong answer with full confidence, because the greedy choice that drives the algorithm no longer reflects the true cost. Real graphs (financial arbitrage detection, currency exchange, time-aware routing) routinely contain negative edges, and that single gap motivates an entire second wave of graph algorithms. **Graph Algorithms (Advanced)** covers that wave. Minimum-spanning-tree algorithms include Kruskal's (sort edges and union components) and Prim's (grow a tree using a priority queue), along with the cut property that explains why both produce the same minimum weight. Shortest paths gain Bellman-Ford for negative edges and negative-cycle detection, plus Floyd-Warshall for all-pairs shortest paths in `O(V^3)`. Strongly connected components are tackled by Kosaraju's (two DFS passes) and Tarjan's (single DFS with low-link values). The lesson also covers articulation points, bridges, Euler paths and circuits via Hierholzer's algorithm, and a first conceptual look at NP-complete Hamiltonian paths. In **Graph Algorithms (Core)**, you implemented Dijkstra, topological sort, and cycle detection. **Union-Find (Disjoint Set Union)** gave you the near-constant-time merge-and-query primitive that makes Kruskal's MST run in `O(E log E)`. From here, **Advanced Graph Algorithms (Network Flow)** turns to capacity-constrained optimization on directed graphs.