# Find all compatable and non compatable edges of a machine

Given a machine in the formal language of N states and M pairs of output combinations in the form of 2D array arr[][]. Each row(say r) of arr[][] denotes the nodes from ‘A’ to ‘Z’ and each pair of a column(say (a, b)) denotes the change of state of node r to node a via state b. The task is to find the compatible and non compatible edges of the formal language.

Note: Edge(A, B) is said to be compatible as all the next state and output are either equal or unspecified in A, B corresponding to each column.

Example:

Input: N = 6, M = 4,
arr[][] = { { ‘-‘, ‘-‘, ‘C’, ‘1’, ‘E’, ‘1’, ‘B’, ‘1’ },
{ ‘E’, ‘0’, ‘-‘, ‘-‘, ‘-‘, ‘-‘, ‘-‘, ‘-‘ },
{ ‘F’, ‘0’, ‘F’, ‘1’, ‘-‘, ‘-‘, ‘-‘, ‘-‘ },
{ ‘-‘, ‘-‘, ‘-‘, ‘-‘, ‘B’, ‘1’, ‘-‘, ‘-‘ },
{ ‘-‘, ‘-‘, ‘F’, ‘0’, ‘A’, ‘0’, ‘D’, ‘1’ },
{ ‘C’, ‘0’, ‘-‘, ‘-‘, ‘B’, ‘0’, ‘C’, ‘1’ } }
Output:
Not Compatable Edges
(A, E) (A, F) (B, F) (C, E) (D, E) (D, F)
Compatable Edges
(A, B)(A, C)(A, D)(B, C)(B, D)(B, E)(C, D)(C, F)(E, F)

Input: N = 4, M = 4,
arr[][] = { { ‘-‘, ‘-‘, ‘C’, ‘1’, ‘E’, ‘1’, ‘B’, ‘1’ },
{ ‘-‘, ‘-‘, ‘-‘, ‘-‘, ‘B’, ‘1’, ‘-‘, ‘-‘ },
{ ‘-‘, ‘-‘, ‘F’, ‘0’, ‘A’, ‘0’, ‘D’, ‘1’ },
{ ‘C’, ‘0’, ‘-‘, ‘-‘, ‘B’, ‘0’, ‘C’, ‘1’ } }
Output:
Not Compatable Edges
(A, C) (A, D) (B, C) (B, D)
Compatable Edges
(A, B)(C, D)

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

Approach:

1. For all the possible combinations(say (a,b)) of the nodes, check if there is any possible path present in the formal language through any number of states as:
• If state via Node a is empty, then check for the next pair of nodes.
• If current traversed state(say Node b) via Node a is not empty and if the output state via Node a to Node b is not same then recursively check for a path from Node a to Node b.
• If the output state is the same, then it has a direct edge between Node a and Node b.
2. If the path is found between any pair of nodes,then the pair of nodes are a part of compatible node.
3. Store the above pair of compatible nodes in a matrix Mat[][].
4. Traverse the Mat[][] for all the possible pairs, and if that pair is present in Mat[][] then print it as a Compatible Nodes Else it is Not Compatible node.

Below is the implementation of the above approach:

 `// C++ implementation of the above approach ` `#include ` `using` `namespace` `std; ` `const` `int` `M = 8; ` ` `  `// Function to find the compatible and ` `// non-compatible for a given formal language ` `void` `findEdges(``char` `arr[][M], ``int` `n, ``int` `m) ` `{ ` ` `  `    ``// To store the compatible edges ` `    ``char` `mat = { ``'x'` `}; ` ` `  `    ``// Loop over every pair of nodes in the ` `    ``// given formal language ` `    ``for` `(``int` `i = 0; i < n; i++) { ` `        ``for` `(``int` `j = i + 1; j < n; j++) { ` ` `  `            ``// Traverse through the output ` `            ``// column and compare it between ` `            ``// each set of pairs of nodes ` `            ``for` `(``int` `k = 0; k < 2 * m; k += 2) { ` ` `  `                ``// If the the output is not ` `                ``// specified then leave the ` `                ``// edge unprocessed ` `                ``if` `(arr[i][k + 1] == ``'-'` `                    ``|| arr[j][k + 1] == ``'-'``) { ` `                    ``continue``; ` `                ``} ` ` `  `                ``// If the output of states ` `                ``// doesn't match then not ` `                ``// compatable. ` `                ``if` `(arr[i][k + 1] != arr[j][k + 1]) { ` ` `  `                    ``// Mark the not compatable ` `                    ``// edges in the maxtrix with ` `                    ``// character 'v' ` `                    ``mat[i][j] = ``'v'``; ` `                    ``mat[j][i] = ``'v'``; ` `                    ``break``; ` `                ``} ` `            ``} ` `        ``} ` `    ``} ` ` `  `    ``int` `nn = n; ` ` `  `    ``// Loop over all node to find other non ` `    ``// compatable edges ` `    ``while` `(nn--) { ` ` `  `        ``// Loop over every pair of nodes in ` `        ``// the given formal language ` `        ``for` `(``int` `i = 0; i < n; i++) { ` `            ``for` `(``int` `j = i + 1; j < n; j++) { ` ` `  `                ``int` `k; ` `                ``for` `(k = 0; k < m; k += 2) { ` ` `  `                    ``// If the the output is ` `                    ``// not specified then ` `                    ``// leave edge unprocessed ` `                    ``if` `(arr[i][k + 1] == ``'-'` `                        ``|| arr[j][k + 1] == ``'-'``) { ` `                        ``continue``; ` `                    ``} ` ` `  `                    ``// If output is not equal ` `                    ``// then break as non-compatable ` `                    ``if` `(arr[i][k + 1] != arr[j][k + 1]) { ` `                        ``break``; ` `                    ``} ` `                ``} ` ` `  `                ``if` `(k < m) { ` `                    ``continue``; ` `                ``} ` ` `  `                ``for` `(k = 0; k < m; k += 2) { ` ` `  `                    ``// If next states are unspecified ` `                    ``// then continue ` `                    ``if` `(arr[i][k] == ``'-'` `                        ``|| arr[j][k] == ``'-'``) { ` `                        ``continue``; ` `                    ``} ` ` `  `                    ``// If the states are not equal ` `                    ``if` `(arr[i][k] != arr[j][k]) { ` `                        ``int` `x = arr[i][k] - ``'A'``; ` `                        ``int` `y = arr[j][k] - ``'A'``; ` ` `  `                        ``// If the dependent edge ` `                        ``// is not compatable then ` `                        ``// this edge is also not ` `                        ``// compatable ` `                        ``if` `(mat[x][y] == ``'v'``) { ` `                            ``mat[i][j] = ``'v'``; ` `                            ``mat[j][i] = ``'v'``; ` `                            ``break``; ` `                        ``} ` `                    ``} ` `                ``} ` `            ``} ` `        ``} ` `    ``} ` ` `  `    ``// Output all Non-compatable Edges ` `    ``printf``(``"Not Compatable Edges \n"``); ` `    ``for` `(``int` `i = 0; i < n; i++) { ` `        ``for` `(``int` `j = i + 1; j < n; j++) { ` `            ``if` `(mat[i][j] == ``'v'``) { ` `                ``printf``(``"(%c, %c) "``, i + 65, j + 65); ` `            ``} ` `        ``} ` `    ``} ` `    ``printf``(``"\n"``); ` ` `  `    ``// Output all Compatable Edges ` `    ``printf``(``"Compatable Edges \n"``); ` `    ``for` `(``int` `i = 0; i < n; i++) { ` `        ``for` `(``int` `j = i + 1; j < n; j++) { ` `            ``if` `(mat[i][j] != ``'v'``) { ` `                ``printf``(``"(%c, %c)"``, i + 65, j + 65); ` `            ``} ` `        ``} ` `    ``} ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `    ``int` `n = 6, m = 4; ` ` `  `    ``char` `arr[] = { { ``'-'``, ``'-'``, ``'C'``, ``'1'``, ``'E'``, ``'1'``, ``'B'``, ``'1'` `}, ` `                      ``{ ``'E'``, ``'0'``, ``'-'``, ``'-'``, ``'-'``, ``'-'``, ``'-'``, ``'-'` `}, ` `                      ``{ ``'F'``, ``'0'``, ``'F'``, ``'1'``, ``'-'``, ``'-'``, ``'-'``, ``'-'` `}, ` `                      ``{ ``'-'``, ``'-'``, ``'-'``, ``'-'``, ``'B'``, ``'1'``, ``'-'``, ``'-'` `}, ` `                      ``{ ``'-'``, ``'-'``, ``'F'``, ``'0'``, ``'A'``, ``'0'``, ``'D'``, ``'1'` `}, ` `                      ``{ ``'C'``, ``'0'``, ``'-'``, ``'-'``, ``'B'``, ``'0'``, ``'C'``, ``'1'` `} }; ` ` `  `    ``findEdges(arr, n, m); ` `    ``return` `0; ` `} `

Output:

```Not Compatable Edges
(A, E) (A, F) (B, F) (C, E) (D, E) (D, F)
Compatable Edges
(A, B)(A, C)(A, D)(B, C)(B, D)(B, E)(C, D)(C, F)(E, F)
```

Time Complexity: O(M*N3), where N is the number of states and M is the Output for every states.

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