Dijkstra Algorithm in Data Structure

The Dijkstra algorithm is a popular graph search algorithm that finds the shortest path between two vertices in a graph. It was developed by Dutch computer scientist Edsger Dijkstra in 1956 and has since become a fundamental concept in the field of data structures and algorithms.

## Understanding Graphs

Before we dive into the details of the Dijkstra algorithm, let’s first understand what graphs are. In computer science, a graph is a collection of nodes connected by edges.

These nodes can represent various entities, such as cities on a map or web pages on the internet. The edges between nodes represent relationships or connections between them.

Graphs can be classified into two main types: directed graphs and undirected graphs. In a directed graph, the edges have a direction, indicating that you can only traverse them in one specific way. In an undirected graph, the edges have no direction, allowing traversal in both directions.

## The Shortest Path Problem

Now that we know what graphs are, let’s discuss the shortest path problem. Given two nodes in a graph, the shortest path problem aims to find the shortest path between them. This problem has numerous real-world applications like finding optimal routes for navigation systems or determining the most efficient network routes.

### Dijkstra Algorithm Steps

The Dijkstra algorithm solves the shortest path problem by iteratively exploring all possible paths from the source node to other nodes in the graph. Here are the steps involved:

**Initialize:**Set an initial distance value for all nodes except the source node as infinity and set the distance of the source node as 0.**Select:**Choose the node with minimum distance from those not yet visited.**Visit:**Mark the selected node as visited and update the distances of its neighboring nodes.**Repeat:**Repeat steps 2 and 3 until all nodes have been visited or the destination node has been reached.

The algorithm uses a data structure called a priority queue to keep track of the nodes with their respective distances. This allows for efficient selection of the next node to visit.

### Example:

Let’s walk through an example to understand how the Dijkstra algorithm works. Consider a graph with the following nodes and edges:

- A – B (7)
- A – C (9)
- B – D (10)
- C – D (5)
- C – E (6)

If we want to find the shortest path from node A to node D, we start by initializing all distances as infinity except for A, which is set to 0. Then, we select the node with the minimum distance (A) and visit its neighboring nodes. We update their distances based on the edges connecting them.

In this case, after visiting A, we update B’s distance to 7 and C’s distance to 9. Next, we select C as it has the minimum distance among unvisited nodes and visit its neighbors: D and E. The distance from A to D becomes 14, while from A to E becomes 15.

We repeat this process until all nodes have been visited or until we reach our destination node (D). Finally, we obtain the shortest path from A to D by backtracking from D using each node’s parent pointer.

## Conclusion

The Dijkstra algorithm is a powerful tool for finding shortest paths in graphs. It efficiently solves the shortest path problem by iteratively exploring all possible paths from the source node. By understanding this algorithm, you’ll be equipped to solve a wide range of real-world problems like route planning and network optimization.