Understanding the Problem
The Bellman-Ford Algorithm is used to find the shortest paths from a single source vertex to all other vertices in a weighted graph. Unlike Dijkstra's algorithm, Bellman-Ford supports negative weight edges and is capable of detecting negative weight cycles — cycles where the total sum of edge weights is negative.
This algorithm is especially useful in scenarios like financial networks or graphs with penalties, where such negative weights naturally occur.
Step-by-Step Solution with Example
Step 1: Initialize the Distance Array
Assume a graph with 5 vertices (numbered 0 to 4) and the following edges:
edges = [
[0, 1, 6],
[0, 2, 7],
[1, 2, 8],
[1, 3, 5],
[1, 4, -4],
[2, 3, -3],
[2, 4, 9],
[3, 1, -2],
[4, 3, 7]
]
source = 0
We initialize a dist[]
array such that dist[source] = 0
and all others as Infinity
.
Step 2: Relax All Edges V-1 Times
We iterate over all edges V - 1 times (where V is the number of vertices). During each pass, we attempt to "relax" each edge:
- If
dist[u] + weight < dist[v]
, update dist[v] = dist[u] + weight
.
After each pass, the algorithm slowly propagates the shortest path information throughout the graph.
Step 3: Detect Negative Weight Cycle
After performing V-1
iterations, we perform one extra pass over all edges. If any distance can still be reduced, it means a negative weight cycle exists in the graph.
In that case, the algorithm should return [-1]
to indicate the presence of such a cycle.
Step 4: Final Result
If no negative cycles are detected, return the dist[]
array, which contains the shortest distances from the source to every other vertex.
[0, 2, 7, 4, -2] // For the given example
Edge Cases
- Graph with no edges: All nodes remain at
Infinity
distance except the source.
- Negative edge but no cycle: Bellman-Ford handles it correctly.
- Disconnected graph: Vertices that are not reachable from the source remain at
Infinity
.
- Negative weight cycle: If a cycle exists that can endlessly decrease the path cost, the algorithm detects it and should return
[-1]
.
Finally
The Bellman-Ford algorithm is an essential tool when dealing with graphs that include negative edge weights. While it's not as fast as Dijkstra’s algorithm for graphs with only positive weights, its ability to detect negative cycles makes it extremely valuable.
By following a clear step-by-step relaxation process and checking for further improvements in the last iteration, we ensure both correctness and cycle detection. Understanding this approach builds a strong foundation for handling more complex graph problems.
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