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Bellman-Ford
Bellman-Ford Algorithm Explained Bellman-Ford is a graph algorithm used to find the shortest paths from a single source node to all other nodes in the gr...
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Bellman-Ford Algorithm Explained Bellman-Ford is a graph algorithm used to find the shortest paths from a single source node to all other nodes in the gr...
Bellman-Ford is a graph algorithm used to find the shortest paths from a single source node to all other nodes in the graph. This algorithm works by iteratively updating each node in the graph with the shortest path from the source node.
Key Concepts:
d[v] = min(d[v], d[u] + w(u, v))
Duality: Bellman-Ford finds a relaxation of the graph where the shortest paths are represented by weights in the edges.
Relaxation: This means that the algorithm finds the shortest path from the source node to every other node, iteratively updating the distances.
Negative Cycle: If there are negative weights (as in directed graphs), the relaxation process can lead to the formation of negative cycles, where the path from the source to itself is shorter than the path to other nodes.
Algorithm Steps:
Start with the source node and initialize its shortest path distance to 0.
Repeat the following steps for all other nodes in the graph:
For each neighbor u of the current node:
Calculate the new shortest path distance from the source to u through the current node.
If this new distance is shorter than the previously stored shortest path distance, update it.
Update the shortest path distance from the source to u to the minimum of the old and the new distance.
Continue steps 2 until all nodes have been processed.
If any node reaches a negative shortest path distance from the source, that means a negative cycle is present in the graph.
Example:
Suppose we have the following graph:
A---B---C---D
The algorithm would work as follows:
Starting with the source node A, its shortest path distance is initially set to 0.
For B, the shortest path is calculated to be 2 (from A to C, then C to B). So, d[B] = 2.
Similarly, for C, the shortest path is calculated to be 4 (from A to B, then B to C). So, d[C] = 4.
Since d[D] is not negative, the algorithm continues with step 3.
For D, the shortest path is calculated to be 3 (from A to E, then E to D).
As d[D] becomes negative (1), a negative cycle is detected.
Therefore, the shortest path from A to D would be through E, with a weight of 3