Provide C++ codes to find the length of the shortest path. A 2D array of 1s and

Question

Provide C++ codes to find the length of the shortest path. A 2D array of 1s and 0s of size N*N shown below. You must start from the top left (index (0,0)) and find the length of the shortest path to the bottom right (index (n,n)). You may only travel across elements that have a value "1". The Top left and bottom right indices are guaranteed to be "1". You may only move horizontally or vertically one index position with each step.

11001

01011

01100

11110

11011

The shortest path has a length of 8 (the bolded path) for the maze above. Your program will open an input file containing the maze.Your program should find N and output the length of the shortest path to the screen. If no path exists, your program should output "0". (Hint: use some fast algorithm like Dijkstra's or something similar).

Explanation / Answer

Answer :-

Dijkstra’s shortest path algorithm :

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 minimum distance from 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 sptSetand 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.

Program :-

#include <stdio.h>

#include <limits.h>

#define V 9
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;
}

int printSolution(int dist[], int n)
{
printf("Vertex Distance from Source ");
for (int i = 0; i < V; i++)
   printf("%d %d ", i, dist[i]);
}

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

  
   for (int i = 0; i < V; i++)
       dist[i] = INT_MAX, sptSet[i] = false;

   dist[src] = 0;

   for (int count = 0; count < V-1; count++)
   {
  
   int u = minDistance(dist, sptSet);

  
   sptSet[u] = true;

   for (int v = 0; v < V; 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()
{

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;
}

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.