This is a C++ Program to find the shortest path in linear time. This can be done by using Dijkstra’a Shortestpath algorithm.

Here is source code of the C++ Program to Find the Shortest Path from Source Vertex to All Other Vertices in Linear Time. The C++ program is successfully compiled and run on a Linux system. The program output is also shown below.

`#include <stdio.h>`

`#include <limits.h>`

`#include <iostream>`

using namespace std;

`// 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`

int printSolution(int dist[], int n)

`{`

cout << "Vertex Distance from Source\n";

for (int i = 0; i < V; i++)

printf("%d \t\t %d\n", i, dist[i]);

`}`

`// Funtion 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 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 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);

`}`

int main()

`{`

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, 0, 10, 0, 2, 0, 0 }, { 0, 0, 0, 14,

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:

$ g++ LinearTimeShortestPath.cpp $ a.out Vertex Distance from Source 0 0 1 4 2 12 3 19 4 21 5 11 6 9 7 8 8 14 ------------------ (program exited with code: 0) Press return to continue

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