# Number of pair of positions in matrix which are not accessible

Given a positive integer N. Consider a matrix of N X N. No cell can be accessible from any other cell, except the given pair cell in the form of (x1, y1), (x2, y2) i.e there is a path (accessible) between (x2, y2) to (x1, y1). The task is to find the count of pairs (a1, b1), (a2, b2) such that cell (a2, b2) is not accessible from (a1, b1).

Examples:

```Input : N = 2
Allowed path 1: (1, 1) (1, 2)
Allowed path 2: (1, 2) (2, 2)
Output : 6
Cell (2, 1) is not accessible from any cell
and no cell is accessible from it.

(1, 1) - (2, 1)
(1, 2) - (2, 1)
(2, 2) - (2, 1)
(2, 1) - (1, 1)
(2, 1) - (1, 2)
(2, 1) - (2, 2)
```

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

Consider each cell as a node, numbered from 1 to N*N. Each cell (x, y) can be map to number using (x – 1)*N + y. Now, consider each given allowed path as an edge between nodes. This will form a disjoint set of the connected component. Now, using Depth First Traversal or Breadth First Traversal, we can easily find the number of nodes or size of a connected component, say x. Now, count of non-accessible paths are x*(N*N – x). This way we can find non-accessible paths for each connected path.

Below is implementation of this approach:

## C++

 `// C++ program to count number of pair of positions ` `// in matrix which are not accessible ` `#include ` `using` `namespace` `std; ` ` `  `// Counts number of vertices connected in a component ` `// containing x. Stores the count in k. ` `void` `dfs(vector<``int``> graph[], ``bool` `visited[], ` `                               ``int` `x, ``int` `*k) ` `{ ` `    ``for` `(``int` `i = 0; i < graph[x].size(); i++) ` `    ``{ ` `        ``if` `(!visited[graph[x][i]]) ` `        ``{ ` `            ``// Incrementing the number of node in ` `            ``// a connected component. ` `            ``(*k)++; ` ` `  `            ``visited[graph[x][i]] = ``true``; ` `            ``dfs(graph, visited, graph[x][i], k); ` `        ``} ` `    ``} ` `} ` ` `  `// Return the number of count of non-accessible cells. ` `int` `countNonAccessible(vector<``int``> graph[], ``int` `N) ` `{ ` `    ``bool` `visited[N*N + N]; ` `    ``memset``(visited, ``false``, ``sizeof``(visited)); ` ` `  `    ``int` `ans = 0; ` `    ``for` `(``int` `i = 1; i <= N*N; i++) ` `    ``{ ` `        ``if` `(!visited[i]) ` `        ``{ ` `            ``visited[i] = ``true``; ` ` `  `            ``// Initialize count of connected ` `            ``// vertices found by DFS starting ` `            ``// from i. ` `            ``int` `k = 1; ` `            ``dfs(graph, visited, i, &k); ` ` `  `            ``// Update result ` `            ``ans += k * (N*N - k); ` `        ``} ` `    ``} ` `    ``return` `ans; ` `} ` ` `  `// Inserting the edge between edge. ` `void` `insertpath(vector<``int``> graph[], ``int` `N, ``int` `x1, ` `                             ``int` `y1, ``int` `x2, ``int` `y2) ` `{ ` `    ``// Mapping the cell coordinate into node number. ` `    ``int` `a = (x1 - 1) * N + y1; ` `    ``int` `b = (x2 - 1) * N + y2; ` ` `  `    ``// Inserting the edge. ` `    ``graph[a].push_back(b); ` `    ``graph[b].push_back(a); ` `} ` ` `  `// Driven Program ` `int` `main() ` `{ ` `    ``int` `N = 2; ` ` `  `    ``vector<``int``> graph[N*N + 1]; ` ` `  `    ``insertpath(graph, N, 1, 1, 1, 2); ` `    ``insertpath(graph, N, 1, 2, 2, 2); ` ` `  `    ``cout << countNonAccessible(graph, N) << endl; ` `    ``return` `0; ` `} `

## Python3

 `# Python3 program to count number of pair of  ` `# positions in matrix which are not accessible ` ` `  `# Counts number of vertices connected in a  ` `# component containing x. Stores the count in k.  ` `def` `dfs(graph,visited, x, k): ` `    ``for` `i ``in` `range``(``len``(graph[x])): ` `        ``if` `(``not` `visited[graph[x][i]]): ` `             `  `            ``# Incrementing the number of node   ` `            ``# in a connected component.  ` `            ``k[``0``] ``+``=` `1` ` `  `            ``visited[graph[x][i]] ``=` `True` `            ``dfs(graph, visited, graph[x][i], k)  ` ` `  `# Return the number of count of  ` `# non-accessible cells.  ` `def` `countNonAccessible(graph, N): ` `    ``visited ``=` `[``False``] ``*` `(N ``*` `N ``+` `N) ` ` `  `    ``ans ``=` `0` `    ``for` `i ``in` `range``(``1``, N ``*` `N ``+` `1``): ` `        ``if` `(``not` `visited[i]): ` `            ``visited[i] ``=` `True` ` `  `            ``# Initialize count of connected  ` `            ``# vertices found by DFS starting  ` `            ``# from i.  ` `            ``k ``=` `[``1``] ` `            ``dfs(graph, visited, i, k)  ` ` `  `            ``# Update result  ` `            ``ans ``+``=` `k[``0``] ``*` `(N ``*` `N ``-` `k[``0``]) ` `    ``return` `ans ` ` `  `# Inserting the edge between edge.  ` `def` `insertpath(graph, N, x1, y1, x2, y2): ` `     `  `    ``# Mapping the cell coordinate  ` `    ``# into node number.  ` `    ``a ``=` `(x1 ``-` `1``) ``*` `N ``+` `y1  ` `    ``b ``=` `(x2 ``-` `1``) ``*` `N ``+` `y2  ` ` `  `    ``# Inserting the edge.  ` `    ``graph[a].append(b)  ` `    ``graph[b].append(a) ` ` `  `# Driver Code ` `if` `__name__ ``=``=` `'__main__'``: ` ` `  `    ``N ``=` `2` ` `  `    ``graph ``=` `[[] ``for` `i ``in` `range``(N``*``N ``+` `1``)]  ` ` `  `    ``insertpath(graph, N, ``1``, ``1``, ``1``, ``2``)  ` `    ``insertpath(graph, N, ``1``, ``2``, ``2``, ``2``)  ` ` `  `    ``print``(countNonAccessible(graph, N)) ` ` `  `# This code is contributed by PranchalK `

Output:

```6
```

Time Complexity : O(N * N).

## tags:

Graph Matrix DFS DFS Matrix Graph