28. Module – 03
The chapter explores the Bellman-Ford algorithm as a method for finding the shortest paths in graphs, especially those containing negative edge weights. It discusses the limitations and assumptions of Dijkstra's algorithm and contrasts them with the reassurances provided by Bellman-Ford when negative cycles are not present. The emphasis is placed on the determination of shortest paths through systematic updates rather than greedy choices.
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What we have learnt
- The Bellman-Ford algorithm can compute shortest paths even with negative edge weights, provided there are no negative cycles.
- A shortest path will never loop back to a vertex, thereby limiting the maximum number of edges in a path to n-1, where n is the number of vertices.
- The update operation in the Bellman-Ford algorithm ensures that no incorrect lower paths are adopted in the process of calculating shortest distances.
Key Concepts
- -- Dijkstra's Algorithm
- An algorithm for finding the shortest paths between nodes in a graph, which fails if negative edge weights are present.
- -- BellmanFord Algorithm
- An algorithm that calculates shortest paths from a single source vertex to all other vertices in a weighted graph and accommodates negative edge weights.
- -- Negative Cycle
- A cycle in a graph where the sum of the edge weights is negative, which makes the shortest path undefined as it can be decreased indefinitely.
- -- Looping in Paths
- The occurrence of revisiting a vertex in a path, which, under constraints, cannot happen in a shortest path.
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