Dijkstra's Algorithm: Finding Shortest Paths in Weighted Graphs

Learn Dijkstra's algorithm, a classic method for finding the shortest path between nodes in a weighted graph. This guide explains the algorithm's steps, its use of a distance table and visited node set, and provides a clear example to illustrate its functionality.

Dijkstra's Algorithm: Finding the Shortest Path in a Graph

Introduction to Dijkstra's Algorithm

Dijkstra's algorithm is a classic algorithm used to find the shortest path between nodes in a graph. A graph is a network of nodes (vertices) connected by edges. Each edge typically has a weight (or cost) associated with it, representing the distance, time, or cost of traversing that edge. Dijkstra's algorithm finds the shortest path from a single starting node (the source) to all other nodes in the graph.

How Dijkstra's Algorithm Works

Dijkstra's algorithm uses an iterative approach. It maintains a set S of visited nodes and a distance table that stores the shortest distance found so far from the source node to all other nodes. The algorithm works as follows:

  1. Initialization: Start with the source node in S and its distance set to 0. The distance to all other nodes is initialized to infinity.
  2. Iteration: Repeatedly select the node v with the shortest distance from the source node that is not yet in S. Add v to S. Update the distances to all nodes that are neighbors of v if a shorter path is found via v.
  3. Termination: Continue until all nodes are included in S.

The final distance table will contain the shortest distances from the source node to every other node in the graph.

Example: Applying Dijkstra's Algorithm

(An image of a sample graph showing weighted edges would be extremely beneficial here.)

Let's find the shortest path from node K to node L using Dijkstra's algorithm.

Step 1: Initialization

(Show the initial distance table with distances from K to other nodes.)

Step 2: Adding the Closest Vertex

(Show the updated distance table after adding the closest vertex to K (e.g. C) to the set S and updating the distances.)

Step 3: Adding the Next Closest Vertex

(Show the updated distance table after adding the next closest vertex to K (e.g., A) to S and updating distances.)

Step 4: Adding the Next Closest Vertex

(Show the updated distance table after adding the next closest vertex to K (e.g., B) to S and updating distances.)

Step 5: Adding the Final Vertex

(Show the final distance table after adding the last node, D, to S. The shortest distance from K to L and the shortest path should be clearly shown.)

Conclusion

Dijkstra's algorithm is a fundamental algorithm in graph theory, used to efficiently find the shortest paths between nodes. Its applications extend to various fields, including network routing, transportation planning, and resource allocation.