Hard

# Dijkstra’s – Shortest Path Algorithm (SPT) - Adjacency Matrix - Java Implementation

Earlier we have seen what Dijkstra’s algorithm is and how it works. In this article, we will see its implementation using the adjacency matrix.

brief: What is Dijkstra’s algorithm?

1. Dijkstra algorithm is a greedy algorithm.
2. It finds a shortest-path tree for a weighted undirected graph.
3. This means it finds the shortest paths between nodes in a graph, which may represent, for example, road networks
4. For a given source node in the graph, the algorithm finds the shortest path between the source node and every other node.
5. This algorithm also used for finding the shortest paths from a single node to a single destination node by stopping the algorithm once the shortest path to the destination node has been determined.
6. Dijkstra’s algorithm is very similar to Prim’s algorithm. In Prim’s algorithm, we create minimum spanning tree (MST) and in the Dijkstra algorithm, we create shortest-path tree (SPT) from the given source.

Example:

Implementation –

Ordering – Searching (Get the vertex with minimum distance among all vertices)

2. Maintain a set SPT[] to keep track to vertices included in SPT.
3. Assign a distance value to all the vertices, (say distance []) and initialize all the distances with +∞ (Infinity) except the source vertex. This will be used to keep track of distance of vertices from the source vertex. Distance of source vertex to source vertex will be 0.
4. Repeat the following steps until all vertices are processed.
1. Pick the vertex u which is not in SPT[] and has minimum distance. Here we will loop through the vertices and find the vertex with minimum distance.
2. Add vertex u to SPT[].
3. Loop over all the adjacent vertices of
4. For adjacent vertex v, if v is not in SPT[] and distance[v] > distance[u] + edge u-v weight then update distance[v] = distance[u] + edge u-v weight

Time Complexity: O(|V|2).

Total vertices: V, Total Edges : E

• O(V) – Iterate through all vertices to add them in SPT
• O(V) – Each time select a vertex with minimum distance.
• So over all complexity: O(|V|2).

Please see the animation below for better understanding.

Code:

Output:

```Dijkstra Algorithm: (Adjacency Matrix)
Source Vertex: 0 to vertex 0 distance: 0
Source Vertex: 0 to vertex 1 distance: 4
Source Vertex: 0 to vertex 2 distance: 3
Source Vertex: 0 to vertex 3 distance: 6
Source Vertex: 0 to vertex 4 distance: 8
Source Vertex: 0 to vertex 5 distance: 14
```