# Minimum cost to reverse edges such that there is path between every pair of nodes

Given a connected, directional graph. Each node is connected to exactly two other nodes. There is weight associated with each edge denoting the cost to reverse its direction. The task is to find the minimum cost to reverse some edges of the graph such that it is possible to go from each node to every other node.

Examples:

```Input:
5
1 2 7
5 1 8
5 4 5
3 4 1
3 2 10
Output: 15

Input:
6
1 5 4
5 3 8
2 4 15
1 6 16
2 3 23
4 6 42
Output: 39
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:

• In order to reach from each node to every other node, the graph must form a ring i.e Direct all edges on it in one of 2 directions either clockwise or anti-clockwise. Let us denote the cost of redirecting all the clockwise edges to anticlockwise direction as cost1 and vice versa as cost2. The answer is clearly the minimum of these two costs.
• Maintain two boolean arrays start and end. The start and end arrays denote whether there is an edge starting from or ending at a given node. Whenever we encounter an edge going from node a to node b, we first check the condition if there is an edge already starting from node a or ending at node b. If there is an edge that satisfying the condition, the edge is in the opposite direction to the edge already present. In this case, we update cost2 and store the edge is the opposite direction. Otherwise, we update the cost1. This way we are able to maintain the costs of both orientations. Finally, print the minimum cost.

Below is the implementation of above approach:

 `// C++ code to find ` `// the minimum cost to ` `// reverse the edges ` `#include ` `using` `namespace` `std; ` ` `  `// Function to calculate ` `// min cost for reversing ` `// the edges ` `int` `minCost(vector >& graph, ``int` `n) ` `{ ` ` `  `    ``int` `cost1 = 0, cost2 = 0; ` `    ``// bool array to mark ` `    ``// start and end node ` `    ``// of a graph ` `    ``bool` `start[n + 1] = { ``false` `}; ` `    ``bool` `end[n + 1] = { ``false` `}; ` ` `  `    ``for` `(``int` `i = 0; i < n; i++) { ` ` `  `        ``int` `a = graph[i]; ` `        ``int` `b = graph[i]; ` `        ``int` `c = graph[i]; ` ` `  `        ``// This edge must ` `        ``// start from b and end at a ` `        ``if` `(start[a] || end[b]) { ` `            ``cost2 += c; ` `            ``start[b] = ``true``; ` `            ``end[a] = ``true``; ` `        ``} ` ` `  `        ``// This edge must ` `        ``// start from a and end at b ` `        ``else` `{ ` `            ``cost1 += c; ` `            ``start[a] = ``true``; ` `            ``end[b] = ``true``; ` `        ``} ` `    ``} ` ` `  `    ``// Return minimum of ` `    ``// both posibilities ` `    ``return` `min(cost1, cost2); ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `n = 5; ` `    ``// Adjacency list representation ` `    ``// of a graph ` `    ``vector > graph = { ` `        ``{ 1, 2, 7 }, ` `        ``{ 5, 1, 8 }, ` `        ``{ 5, 4, 5 }, ` `        ``{ 3, 4, 1 }, ` `        ``{ 3, 2, 10 } ` `    ``}; ` ` `  `    ``int` `ans = minCost(graph, n); ` `    ``cout << ans << ``'\n'``; ` ` `  `    ``return` `0; ` `} `

 `// Java code to find the minimum cost to ` `// reverse the edges ` `class` `GFG  ` `{ ` ` `  `// Function to calculate min cost for  ` `// reversing the edges ` `static` `int` `minCost(``int``[][] graph, ``int` `n) ` `{ ` ` `  `    ``int` `cost1 = ``0``, cost2 = ``0``; ` `     `  `    ``// bool array to mark start and  ` `    ``// end node of a graph ` `    ``boolean` `[]start = ``new` `boolean``[n + ``1``]; ` `    ``boolean` `[]end = ``new` `boolean``[n + ``1``]; ` ` `  `    ``for` `(``int` `i = ``0``; i < n; i++)  ` `    ``{ ` `        ``int` `a = graph[i][``0``]; ` `        ``int` `b = graph[i][``1``]; ` `        ``int` `c = graph[i][``2``]; ` ` `  `        ``// This edge must start from b ` `        ``// and end at a ` `        ``if` `(start[a] || end[b])  ` `        ``{ ` `            ``cost2 += c; ` `            ``start[b] = ``true``; ` `            ``end[a] = ``true``; ` `        ``} ` ` `  `        ``// This edge must start from a ` `        ``// and end at b ` `        ``else`  `        ``{ ` `            ``cost1 += c; ` `            ``start[a] = ``true``; ` `            ``end[b] = ``true``; ` `        ``} ` `    ``} ` ` `  `    ``// Return minimum of both posibilities ` `    ``return` `Math.min(cost1, cost2); ` `} ` ` `  `// Driver code ` `public` `static` `void` `main(String[] args)  ` `{ ` `    ``int` `n = ``5``; ` `     `  `    ``// Adjacency list representation ` `    ``// of a graph ` `    ``int` `[][]graph = {{ ``1``, ``2``, ``7` `}, ` `                     ``{ ``5``, ``1``, ``8` `}, ` `                     ``{ ``5``, ``4``, ``5` `}, ` `                     ``{ ``3``, ``4``, ``1` `}, ` `                     ``{ ``3``, ``2``, ``10` `}}; ` ` `  `    ``int` `ans = minCost(graph, n); ` `    ``System.out.println(ans); ` `} ` `} ` ` `  `// This code is contributed by Rajput-Ji `

 `# Python code to find the minimum cost to ` `# reverse the edges ` ` `  `# Function to calculate min cost for  ` `# reversing the edges ` `def` `minCost(graph, n): ` ` `  `    ``cost1, cost2 ``=` `0``, ``0``; ` `     `  `    ``# bool array to mark start and  ` `    ``# end node of a graph ` `    ``start ``=` `[``False``]``*``(n ``+` `1``); ` `    ``end ``=` `[``False``]``*``(n ``+` `1``); ` ` `  `    ``for` `i ``in` `range``(n):  ` `        ``a ``=` `graph[i][``0``]; ` `        ``b ``=` `graph[i][``1``]; ` `        ``c ``=` `graph[i][``2``]; ` ` `  `        ``# This edge must start from b ` `        ``# and end at a ` `        ``if` `(start[a] ``or` `end[b]): ` `            ``cost2 ``+``=` `c; ` `            ``start[b] ``=` `True``; ` `            ``end[a] ``=` `True``; ` ` `  `        ``# This edge must start from a ` `        ``# and end at b ` `        ``else``: ` `            ``cost1 ``+``=` `c; ` `            ``start[a] ``=` `True``; ` `            ``end[b] ``=` `True``; ` ` `  `    ``# Return minimum of both posibilities ` `    ``return` `min``(cost1, cost2); ` ` `  `# Driver code ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``n ``=` `5``; ` `     `  `    ``# Adjacency list representation ` `    ``# of a graph ` `    ``graph ``=` `[[ ``1``, ``2``, ``7` `], ` `                    ``[ ``5``, ``1``, ``8` `], ` `                    ``[ ``5``, ``4``, ``5` `], ` `                    ``[ ``3``, ``4``, ``1` `], ` `                    ``[ ``3``, ``2``, ``10` `]]; ` ` `  `    ``ans ``=` `minCost(graph, n); ` `    ``print``(ans); ` `     `  `# This code is contributed by 29AjayKumar `

 `// C# code to find the minimum cost to ` `// reverse the edges ` `using` `System; ` `     `  `class` `GFG  ` `{ ` ` `  `// Function to calculate min cost for  ` `// reversing the edges ` `static` `int` `minCost(``int``[,] graph, ``int` `n) ` `{ ` `    ``int` `cost1 = 0, cost2 = 0; ` `     `  `    ``// bool array to mark start and  ` `    ``// end node of a graph ` `    ``Boolean []start = ``new` `Boolean[n + 1]; ` `    ``Boolean []end = ``new` `Boolean[n + 1]; ` ` `  `    ``for` `(``int` `i = 0; i < n; i++)  ` `    ``{ ` `        ``int` `a = graph[i, 0]; ` `        ``int` `b = graph[i, 1]; ` `        ``int` `c = graph[i, 2]; ` ` `  `        ``// This edge must start from b ` `        ``// and end at a ` `        ``if` `(start[a] || end[b])  ` `        ``{ ` `            ``cost2 += c; ` `            ``start[b] = ``true``; ` `            ``end[a] = ``true``; ` `        ``} ` ` `  `        ``// This edge must start from a ` `        ``// and end at b ` `        ``else` `        ``{ ` `            ``cost1 += c; ` `            ``start[a] = ``true``; ` `            ``end[b] = ``true``; ` `        ``} ` `    ``} ` ` `  `    ``// Return minimum of both posibilities ` `    ``return` `Math.Min(cost1, cost2); ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main(String[] args)  ` `{ ` `    ``int` `n = 5; ` `     `  `    ``// Adjacency list representation ` `    ``// of a graph ` `    ``int` `[,]graph = {{ 1, 2, 7 }, ` `                    ``{ 5, 1, 8 }, ` `                    ``{ 5, 4, 5 }, ` `                    ``{ 3, 4, 1 }, ` `                    ``{ 3, 2, 10 }}; ` ` `  `    ``int` `ans = minCost(graph, n); ` `    ``Console.WriteLine(ans); ` `} ` `} ` ` `  `// This code is contributed by PrinciRaj1992 `

Output:
```15
```

Time Complexity: O(N) where N is number of edges

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