# Minimum Cost of Simple Path between two nodes in a Directed and Weighted Graph

Given a directed graph, which may contain cycles, where every edge has weight, the task is to find the minimum cost of any simple path from a given source vertex ‘s’ to a given destination vertex ‘t’. Simple Path is the path from one vertex to another such that no vertex is visited more than once. If there is no simple path possible then return INF(infinite).

The graph is given as adjacency matrix representation where value of graph[i][j] indicates the weight of an edge from vertex i to vertex j and a value INF(infinite) indicates no edge from i to j.

Examples:

```Input : V = 5, E = 6
s = 0, t = 2
graph[][] =      0   1   2   3   4
0  INF -1  INF  1  INF
1  INF INF -2  INF INF
2  -3  INF INF INF INF
3  INF INF -1  INF INF
4  INF INF INF  2  INF

Output : -3
Explanation :
The minimum cost simple path between 0 and 2 is given by:
0 -----> 1 ------> 2 whose cost is (-1) + (-2) = (-3).

Input : V = 5, E = 6
s = 0, t = 4
graph[][] =      0   1   2   3   4
0  INF -7  INF -2  INF
1  INF INF -11 INF INF
2  INF INF INF INF INF
3  INF INF INF  3  -4
4  INF INF INF INF INF

Output : -6
Explanation :
The minimum cost simple path between 0 and 2 is given by:
0 -----> 3 ------> 4 whose cost is (-2) + (-4) = (-6).
```

Approach :

The main idea to solve the above problem is to traverse through all simple paths from s to t using a modified version of Depth First Search and find the minimum cost path amongst them. One important observation about DFS is that it traverses one path at a time, hence we can traverse separate paths independently using DFS by marking the nodes as unvisited before leaving them.

A simple solution is to start from s, go to all adjacent vertices, and follow recursion for further adjacent vertices until we reach the destination. This algorithm will work even when negative weight cycles or self edges are present in the graph.

Below is the implementation of the above-mentioned approach:

 `// C++ code for printing Minimum Cost ` `// Simple Path between two given nodes ` `// in a directed and weighted graph ` `#include ` `using` `namespace` `std; ` ` `  `// Define number of vertices in ` `// the graph and infinite value ` `#define V 5 ` `#define INF INT_MAX ` ` `  `// Function to do DFS through the nodes ` `int` `minimumCostSimplePath(``int` `u, ``int` `destination, ` `                    ``bool` `visited[], ``int` `graph[][V]) ` `{ ` ` `  `    ``// check if we find the destination ` `    ``// then further cost will be 0 ` `    ``if` `(u == destination) ` `        ``return` `0; ` ` `  `    ``// marking the current node as visited ` `    ``visited[u] = 1; ` ` `  `    ``int` `ans = INF; ` ` `  `    ``// traverse through all ` `    ``// the adjacent nodes ` `    ``for` `(``int` `i = 0; i < V; i++) { ` `        ``if` `(graph[u][i] != INF && !visited[i]) { ` ` `  `            ``// cost of the further path ` `            ``int` `curr = minimumCostSimplePath(i, ` `                        ``destination, visited, graph); ` ` `  `            ``// check if we have reached the destination ` `            ``if` `(curr < INF) { ` ` `  `                ``// Taking the minimum cost path ` `                ``ans = min(ans, graph[u][i] + curr); ` `            ``} ` `        ``} ` `    ``} ` ` `  `    ``// unmarking the current node ` `    ``// to make it available for other ` `    ``// simple paths ` `    ``visited[u] = 0; ` ` `  `    ``// returning the minimum cost ` `    ``return` `ans; ` `} ` ` `  `// driver code ` `int` `main() ` `{ ` ` `  `    ``// initialising the graph ` `    ``int` `graph[V][V]; ` `    ``for` `(``int` `i = 0; i < V; i++) { ` `        ``for` `(``int` `j = 0; j < V; j++) { ` `            ``graph[i][j] = INF; ` `        ``} ` `    ``} ` ` `  `    ``// marking all nodes as unvisited ` `    ``bool` `visited[V] = { 0 }; ` ` `  `    ``// initialising the edges; ` `    ``graph = -1; ` `    ``graph = 1; ` `    ``graph = -2; ` `    ``graph = -3; ` `    ``graph = -1; ` `    ``graph = 2; ` ` `  `    ``// source and destination ` `    ``int` `s = 0, t = 2; ` ` `  `    ``// marking the source as visited ` `    ``visited[s] = 1; ` ` `  `    ``cout << minimumCostSimplePath(s, t,  ` `                            ``visited, graph); ` ` `  `    ``return` `0; ` `} `

Output:

```-3
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