A heap data structure is a specialized tree-based data structure that satisfies the heap property. It is commonly used to implement priority queues and heapsort algorithms. One of the most famous applications of the heap data structure is in the implementation of **Dijkstra’s algorithm**.

## Dijkstra’s Algorithm

Dijkstra’s algorithm is an algorithm used to find the shortest path between nodes in a graph. It was developed by Dutch computer scientist Edsger Dijkstra in 1956.

The algorithm works by maintaining a priority queue, which is typically implemented using a heap data structure. The priority queue stores vertices along with their respective distances from the source vertex. Initially, all vertices are assigned a distance of infinity except for the source vertex, which is assigned a distance of 0.

The algorithm repeatedly extracts the vertex with the minimum distance from the priority queue and relaxes its neighboring vertices. Relaxation involves updating their distances if a shorter path through the current vertex is found.

### Heap Data Structure for Priority Queue

The heap data structure efficiently supports operations such as insertion and extraction of elements with minimum or maximum priority. In Dijkstra’s algorithm, we use a min-heap to store vertices based on their distances from the source vertex.

__Insertion:__When inserting a new vertex into the priority queue, we compare its distance with that of its parent node. If it violates the heap property (parent’s distance > child’s distance), we swap them until it satisfies the property.__Extraction:__To extract the vertex with minimum distance, we simply remove it from the top of the heap. Then, we restore the heap property by swapping it with its smallest child until it satisfies the property.

This efficient insertion and extraction process ensures that Dijkstra’s algorithm can find the shortest path in a graph efficiently.

## Example

Let’s consider an example to understand the application of heaps in Dijkstra’s algorithm:

Suppose we have a graph with vertices A, B, C, D, E, and F. The edges between these vertices have weights as follows:

- A -> B: 4
- A -> C: 2
- B -> E: 3
- B -> D: 2
- C -> D: 1
- C -> F: 4
- D -> E: 3

We start by initializing the distances from the source vertex (A) to all other vertices as infinity, except for A itself which is assigned a distance of 0. We insert all the vertices into the min-heap.

The algorithm proceeds as follows:

- Extract vertex A with distance 0 from the heap.
- Relax adjacent vertices B and C (distances become B=4, C=2).
- Heapify – rearrange the heap based on updated distances.

*Note: The diagram above is just an illustration and may not represent the actual state of data during execution.*