**Problem Statement:** Given a graph and a source vertex in the graph, find the shortest paths from the source to all vertices in the given graph.

### What is Dijkstra’s Algorithm?

Dijkstra’s algorithm is very similar to Prim’s algorithm for minimum spanning tree. Like Prim’s MST, we generate an* SPT (shortest path tree)* with a given source as the root. We maintain two sets, one set contains vertices included in the shortest path tree, other set includes vertices not yet included in the shortest path tree. At every step of the algorithm, we find a vertex that is in the other set (set of not yet included) and has a minimum distance from the source.

Below are the detailed steps used in Dijkstra’s algorithm to find the shortest path from a single source vertex to all other vertices in the given graph.

### Algorithm

- Create a set
sptSet(shortest path tree set) that keeps track of vertices included in the shortest path tree, i.e., whose minimum distance from the source is calculated and finalized. Initially, this set is empty.- Assign a distance value to all vertices in the input graph. Initialize all distance values as INFINITE. Assign the distance value as 0 for the source vertex so that it is picked first.
- While
sptSetdoesn’t include all vertices

- Pick a vertex u which is not there in
sptSetand has minimum distance value.- Include u to
sptSet.- Update the distance value of all adjacent vertices of u. To update the distance values, iterate through all adjacent vertices. For every adjacent vertex v, if the sum of the distance value of u (from source) and weight of edge u-v, is less than the distance value of v, then update the distance value of v.

### C++ Program for Dijkstra’s Single Source Shortest Path Algorithm

## C++

`// A C++ program for Dijkstra's single source shortest path` `// algorithm. The program is for adjacency matrix` `// representation of the graph` `#include <limits.h>` `#include <stdio.h>` `// Number of vertices in the graph` `#define V 9` `// A utility function to find the vertex with minimum` `// distance value, from the set of vertices not yet included` `// in shortest path tree` `int` `minDistance(` `int` `dist[], ` `bool` `sptSet[])` `{` ` ` `// Initialize min value` ` ` `int` `min = INT_MAX, min_index;` ` ` `for` `(` `int` `v = 0; v < V; v++)` ` ` `if` `(sptSet[v] == ` `false` `&& dist[v] <= min)` ` ` `min = dist[v], min_index = v;` ` ` `return` `min_index;` `}` `// A utility function to print the constructed distance` `// array` `void` `printSolution(` `int` `dist[], ` `int` `n)` `{` ` ` `printf` `(` `"Vertex Distance from Source\n"` `);` ` ` `for` `(` `int` `i = 0; i < V; i++)` ` ` `printf` `(` `"\t%d \t\t\t\t %d\n"` `, i, dist[i]);` `}` `// Function that implements Dijkstra's single source` `// shortest path algorithm for a graph represented using` `// adjacency matrix representation` `void` `dijkstra(` `int` `graph[V][V], ` `int` `src)` `{` ` ` `int` `dist[V]; ` `// The output array. dist[i] will hold the` ` ` `// shortest` ` ` `// distance from src to i` ` ` `bool` `sptSet[V]; ` `// sptSet[i] will be true if vertex i is` ` ` `// included in shortest` ` ` `// path tree or shortest distance from src to i is` ` ` `// finalized` ` ` `// Initialize all distances as INFINITE and stpSet[] as` ` ` `// false` ` ` `for` `(` `int` `i = 0; i < V; i++)` ` ` `dist[i] = INT_MAX, sptSet[i] = ` `false` `;` ` ` `// Distance of source vertex from itself is always 0` ` ` `dist[src] = 0;` ` ` `// Find shortest path for all vertices` ` ` `for` `(` `int` `count = 0; count < V - 1; count++) {` ` ` `// Pick the minimum distance vertex from the set of` ` ` `// vertices not yet processed. u is always equal to` ` ` `// src in the first iteration.` ` ` `int` `u = minDistance(dist, sptSet);` ` ` `// Mark the picked vertex as processed` ` ` `sptSet[u] = ` `true` `;` ` ` `// Update dist value of the adjacent vertices of the` ` ` `// picked vertex.` ` ` `for` `(` `int` `v = 0; v < V; v++)` ` ` `// Update dist[v] only if is not in sptSet,` ` ` `// there is an edge from u to v, and total` ` ` `// weight of path from src to v through u is` ` ` `// smaller than current value of dist[v]` ` ` `if` `(!sptSet[v] && graph[u][v]` ` ` `&& dist[u] != INT_MAX` ` ` `&& dist[u] + graph[u][v] < dist[v])` ` ` `dist[v] = dist[u] + graph[u][v];` ` ` `}` ` ` `// print the constructed distance array` ` ` `printSolution(dist, V);` `}` `// driver program to test above function` `int` `main()` `{` ` ` `/* Let us create the example graph discussed above */` ` ` `int` `graph[V][V] = { { 0, 4, 0, 0, 0, 0, 0, 8, 0 },` ` ` `{ 4, 0, 8, 0, 0, 0, 0, 11, 0 },` ` ` `{ 0, 8, 0, 7, 0, 4, 0, 0, 2 },` ` ` `{ 0, 0, 7, 0, 9, 14, 0, 0, 0 },` ` ` `{ 0, 0, 0, 9, 0, 10, 0, 0, 0 },` ` ` `{ 0, 0, 4, 14, 10, 0, 2, 0, 0 },` ` ` `{ 0, 0, 0, 0, 0, 2, 0, 1, 6 },` ` ` `{ 8, 11, 0, 0, 0, 0, 1, 0, 7 },` ` ` `{ 0, 0, 2, 0, 0, 0, 6, 7, 0 } };` ` ` `dijkstra(graph, 0);` ` ` `return` `0;` `}` |

**Output**

Vertex Distance from Source 0 0 1 4 2 12 3 19 4 21 5 11 6 9 7 8 8 14

### Complexity Analysis

**Time Complexity:** The time complexity of Dijkstra’s algorithm is O(V^2). This is because the algorithm uses two nested loops to traverse the graph and find the shortest path from the source node to all other nodes.

**Space Complexity: **The space complexity of Dijkstra’s algorithm is O(V), where V is the number of vertices in the graph. This is because the algorithm uses an array of size V to store the distances from the source node to all other nodes.

Please refer complete article on Dijkstra’s shortest path algorithm | Greedy Algo-7 for more details!