Why Relaxing Edges N-1 times, gives us Single Source Shortest Path?
In the worst-case scenario, a shortest path between two vertices can have at most N-1 edges, where N is the number of vertices. This is because a simple path in a graph with N vertices can have at most N-1 edges, as it’s impossible to form a closed loop without revisiting a vertex.
By relaxing edges N-1 times, the Bellman-Ford algorithm ensures that the distance estimates for all vertices have been updated to their optimal values, assuming the graph doesn’t contain any negative-weight cycles reachable from the source vertex. If a graph contains a negative-weight cycle reachable from the source vertex, the algorithm can detect it after N-1 iterations, since the negative cycle disrupts the shortest path lengths.
In summary, relaxing edges N-1 times in the Bellman-Ford algorithm guarantees that the algorithm has explored all possible paths of length up to N-1, which is the maximum possible length of a shortest path in a graph with N vertices. This allows the algorithm to correctly calculate the shortest paths from the source vertex to all other vertices, given that there are no negative-weight cycles.
Bellman–Ford Algorithm
Imagine you have a map with different cities connected by roads, each road having a certain distance. The Bellman–Ford algorithm is like a guide that helps you find the shortest path from one city to all other cities, even if some roads have negative lengths. It’s like a GPS for computers, useful for figuring out the quickest way to get from one point to another in a network. In this article, we’ll take a closer look at how this algorithm works and why it’s so handy in solving everyday problems.
Table of Content
- Bellman-Ford Algorithm
- The idea behind Bellman Ford Algorithm
- Principle of Relaxation of Edges for Bellman-Ford
- Why Relaxing Edges N-1 times, gives us Single Source Shortest Path?
- Why Does the Reduction of Distance in the N’th Relaxation Indicates the Existence of a Negative Cycle?
- Working of Bellman-Ford Algorithm to Detect the Negative cycle in the graph
- Algorithm to Find Negative Cycle in a Directed Weighted Graph Using Bellman-Ford
- Handling Disconnected Graphs in the Algorithm
- Complexity Analysis of Bellman-Ford Algorithm
- Bellman Ford’s Algorithm Applications
- Drawback of Bellman Ford’s Algorithm
Contact Us