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# Minimum number of integers required to fill the NxM grid

• Last Updated : 05 Feb, 2019

Given a grid of size (NxM) is to be filled with integers.

Filling of cells in the grid should be done in the following manner:

1. let A, B and C are three cell and, B and C shared a side with A.
2. Value of cell B and C must be distinct.
3. Let L be the number of distinct integers in a grid.
4. Each cell should contain value from 1 to L.

The task is to find the minimum value of L and any resulting grid.

Examples:

```Input: N = 1, M = 2
Output:
L = 2
grid = {1, 2}

Input: 2 3
Output:
L = 3
grid = {{1, 2, 3},
{1, 2, 3}}
Explanation: Integers in the neighbors
of cell (2, 2) are 1, 2 and 3.
All numbers are pairwise distinct.
```

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

Approach:
It is given that two cells shared a side with another cell must be distinct. For each such cell, there will be a possible maximum of 8 cells in a grid from whom its value must be different.
It will follow the 4 colour problem: Maximum colour required to fill the regions will be 4.

1. For N<4 or M<4
Number of integers required may vary from 1 to 4.
Checking 8 cells and then fill the current cell.
If number of distinct integers in 8 cells is less than L then fill the current cell with any remaining integer, otherwise fill the current cells with L+1 integer.
2. For N>=4 and M>=4
Number of integers required must be 4 according to 4 colour problem.
Use the 4×4 matrix to fill the NxM matrix.
```1 2 3 4
1 2 3 4
3 4 1 2
3 4 1 2```

Below is the implementation of the above approach:

Implementation:

 `# Python 3 implementation of``# above approach`` ` ` ` `# Function to display the matrix``def` `display_matrix(A):``    ``for` `i ``in` `A:``        ``print``(``*``i)`` ` ` ` `# Function for calculation``def` `cal_main(A, L, x, i, j):``    ``s ``=` `set``()`` ` `    ``# Checking 8 cells and``    ``# then fill the current cell.``    ``if` `(i ``-` `2``) >``=` `0``:``        ``s.add(A[i ``-` `2``][j])``    ``if` `(i ``+` `2``) < N:``        ``s.add(A[i ``+` `2``][j])``    ``if` `(j ``-` `2``) >``=` `0``:``        ``s.add(A[i][j ``-` `2``])``    ``if` `(j ``+` `2``) < M:``        ``s.add(A[i][j ``+` `2``])``    ``if` `(i ``-` `1``) >``=` `0` `and` `(j ``-` `1``) >``=` `0``:``        ``s.add(A[i ``-` `1``][j ``-` `1``])``    ``if` `(i ``-` `1``) >``=` `0` `and` `(j ``+` `1``) < M:``        ``s.add(A[i ``-` `1``][j ``+` `1``])``    ``if` `(i ``+` `1``) < N ``and` `(j ``-` `1``) >``=` `0``:``        ``s.add(A[i ``+` `1``][j ``-` `1``])``    ``if` `(i ``+` `1``) < N ``and` `(j ``+` `1``) < M:``        ``s.add(A[i ``+` `1``][j ``+` `1``])``     ` `    ``# Set to contain distinct value``    ``# of integers in 8 cells.``    ``s ``=` `s.difference({``0``})`` ` `    ``if` `len``(s) < L:`` ` `        ``# Set contain remaining integers``        ``w ``=` `x.difference(s)`` ` `        ``# fill the current cell``        ``# with maximum remaining integer``        ``A[i][j] ``=` `max``(w)``    ``else``:`` ` `        ``# fill the current cells with L + 1 integer.``        ``A[i][j] ``=` `L ``+` `1``        ``L ``+``=` `1`` ` `        ``# Increase the value of L``        ``x.add(L)``    ``return` `A, L, x`` ` ` ` `# Function to find the number``# of distinct integers``def` `solve(N, M):`` ` `    ``# initialise the list (NxM) with 0.``    ``A ``=` `[]``    ``for` `i ``in` `range``(N):``        ``K ``=` `[]``        ``for` `j ``in` `range``(M):``            ``K.append(``0``)``        ``A.append(K)``     ` `    ``# Set to contain distinct``    ``# value of integers from 1-L``    ``x ``=` `set``()``    ``L ``=` `0`` ` `    ``# Number of integer required``    ``# may vary from 1 to 4.``    ``if` `N < ``4` `or` `M < ``4``:``        ``if` `N > M:  ``# if N is greater``            ``for` `i ``in` `range``(N):``                ``for` `j ``in` `range``(M):``                    ``cal_main(A, L, x, i, j)`` ` `        ``else``:``            ``# if M is greater``            ``for` `j ``in` `range``(M):``                ``for` `i ``in` `range``(N):``                    ``cal_main(A, L, x, i, j)``    ``else``:`` ` `        ``# Number of integer required``        ``# must be 4``        ``L ``=` `4`` ` `        ``# 4×4 matrix to fill the NxM matrix.``        ``m4 ``=` `[[``1``, ``2``, ``3``, ``4``], ``            ``[``1``, ``2``, ``3``, ``4``], ``            ``[``3``, ``4``, ``1``, ``2``], ``            ``[``3``, ``4``, ``1``, ``2``]]`` ` `        ``for` `i ``in` `range``(``4``):``            ``for` `j ``in` `range``(``4``):``                ``A[i][j] ``=` `m4[i][j]``        ``for` `i ``in` `range``(``4``, N):``            ``for` `j ``in` `range``(``4``):``                ``A[i][j] ``=` `m4[i ``%` `4``][j]``        ``for` `j ``in` `range``(``4``, M):``            ``for` `i ``in` `range``(N):``                ``A[i][j] ``=` `A[i][j ``%` `4``]``    ``print``(L)``    ``display_matrix(A)`` ` ` ` `# Driver Code``if` `__name__ ``=``=` `"__main__"``:`` ` `    ``# sample input``    ``# Number of rows and columns``    ``N, M ``=` `10``, ``5``    ``solve(N, M)`
Output:
```4
1 2 3 4 1 2 3 4 1 2
1 2 3 4 1 2 3 4 1 2
3 4 1 2 3 4 1 2 3 4
3 4 1 2 3 4 1 2 3 4
1 2 3 4 1 2 3 4 1 2
1 2 3 4 1 2 3 4 1 2
3 4 1 2 3 4 1 2 3 4
3 4 1 2 3 4 1 2 3 4
1 2 3 4 1 2 3 4 1 2
1 2 3 4 1 2 3 4 1 2
```

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