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Dijkstra’s Algorithm.cpp
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// DIJKSTRA’s ALGORITHM
// OVERVIEW OF THE ALGORITHM :
// Given a graph and a source vertex in the graph,
// find shortest paths from source to all vertices in the given graph.
// Dijkstra’s algorithm is very similar to Prim’s algorithm for minimum spanning tree.
// Like Prim’s MST, we generate a SPT (shortest path tree) with given source as root.
// We maintain two sets, one set contains vertices included in shortest path tree,
// other set includes vertices not yet included in shortest path tree.
// At every step of the algorithm, we find a vertex which 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
// 1) Create a set sptSet (shortest path tree set) that keeps track of vertices included in shortest path tree,
// i.e., whose minimum distance from source is calculated and finalized.
// Initially, this set is empty.
// 2) Assign a distance value to all vertices in the input graph.
// Initialize all distance values as INFINITE.
// Assign distance value as 0 for the source vertex so that it is picked first.
// 3) While sptSet doesn’t include all vertices
// ….a) Pick a vertex u which is not there in sptSet and has minimum distance value.
// ….b) Include u to sptSet.
// ….c) Update distance value of all adjacent vertices of u.
// To update the distance values, iterate through all adjacent vertices.
// For every adjacent vertex v, if sum of 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.
// IMPLEMENTATION OF DIJKSTRA'S ALGORITHM IN 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>
#include <stdlib.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[])
{
printf("Vertex \t\t Distance from Source\n");
for (int i = 0; i < V; i++)
printf("%d \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);
}
// 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);
}