Find the shortest path from vertex 1 to every other vertex using the following algorithms: (a) Dijkstra (b) Ford
Dijkstra’s algorithm and the Bellman-Ford algorithm (commonly referred to as Ford’s algorithm). I’ll describe how each of these algorithms can be used to find the shortest path from vertex 1 to every other vertex in the graph:
(a) Dijkstra’s Algorithm:
Dijkstra’s algorithm finds the shortest path from a single source vertex to all other vertices in a weighted graph. It works by iteratively selecting the vertex with the smallest tentative distance (shortest path estimate) from the source and updating the distances to its neighbors. Here’s how Dijkstra’s algorithm works:
- Initialize the distance of vertex 1 to 0 and the distances of all other vertices to infinity.
- Create a priority queue (min-heap) to keep track of vertices and their tentative distances.
- Add vertex 1 to the priority queue with distance 0.
- While the priority queue is not empty:
- Extract the vertex with the smallest tentative distance.
- For each neighboring vertex (v) of the extracted vertex:
- Calculate the tentative distance from vertex 1 to v through the extracted vertex.
- If the calculated distance is smaller than the current distance to v, update v’s distance and enqueue v with the new distance.
- Once the algorithm completes, the distances from vertex 1 to all other vertices represent the shortest paths.
(b) Bellman-Ford Algorithm (Ford’s Algorithm):
The Bellman-Ford algorithm finds the shortest path from a single source vertex to all other vertices in a weighted graph, even in the presence of negative edge weights (as long as there are no negative cycles). Here’s how the Bellman-Ford algorithm works:
- Initialize the distance of vertex 1 to 0 and the distances of all other vertices to infinity.
- Repeat the following for (V – 1) times (where V is the number of vertices):
- For each edge (u, v) with weight w:
- Relax the edge: If the distance to vertex u plus w is smaller than the current distance to vertex v, update v’s distance.
- After (V – 1) iterations, all shortest paths with at most (V – 1) edges have been found.
- Optionally, perform an additional iteration to check for negative cycles (if a distance update occurs in this iteration, a negative cycle exists).
- The distances from vertex 1 to all other vertices represent the shortest paths.
Both Dijkstra’s algorithm and the Bellman-Ford algorithm can be used to find the shortest path from vertex 1 to every other vertex. Dijkstra’s algorithm is more efficient for graphs with non-negative edge weights, while the Bellman-Ford algorithm handles graphs with negative edge weights (with certain constraints) and can detect negative cycles.