# Find maximum of minimums from Layers of Matrix using numbers 1 to N^2

Given a square matrix of size N*N using numbers 1 to N^2, the task is to find the maximum of minimums of each layer of the matrix.

The layers of the matrix are the boundary elements of the sub-matrix starting at (i, i) and ending at (N – i + 1, N – i + 1), where 1<= i<= ceil(N/2).

Examples:

Input: Below is the given matrix:
1      2     3     4
5     6     7     8
9    10   11    12
13  14   15    16
Output: 6
Explanation: The layers are {1, 2, 3, 4, 8, 12, 16, 15, 14, 13, 9, 5} with minimum 1 and {6, 7, 10, 11} with minimum 6. The maximum of 1 and 6 is 6.

Input: Below is the given matrix:
1    2    3
4   30   5
1   2     3
Output: 30
Explanation: The layers are {1, 2, 3, 5, 3, 2, 1, 4, 1} with minimum 1 and {30} with minimum 30. The maximum of 1 and 30 is 30.

Approach: The idea is to observe carefully, for the ith layer, the elements of the top, left, right and bottom boundary are at indexes:

• top boundary is at indexes (i, j)
• left boundary is at indexes (j, i)
• right boundary is at indexes (j, n – i + 1)
• bottom boundary is at indexes (n – i + 1, j), where i <= j <= n – i + 1

Thus, find the minimums of each layer and store the maximum of these minimums.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach ` `#include ` `using` `namespace` `std; ` ` `  `// Function to return the minimum ` `// of the boundary elements ` `int` `getBoundaryMin(``int` `a[][4], ``int` `n, ` `                   ``int` `index) ` `{ ` `    ``int` `min1 = INT_MAX; ` ` `  `    ``for``(``int` `i = index; i < n - index; i++)  ` `    ``{ ` `         `  `        ``// Top boundary ` `        ``min1 = min(min1, ` `                   ``a[index][i]); ` ` `  `        ``// Left boundary ` `        ``min1 = min(min1, ` `                   ``a[i][index]); ` ` `  `        ``// Right boundary ` `        ``min1 = min(min1, ` `                    ``a[i][n - index - 1]); ` ` `  `        ``// Bottom boundary ` `        ``min1 = min(min1, ` `                   ``a[n - index - 1][i]); ` `    ``} ` `    ``return` `min1; ` `} ` ` `  `// Function to find the maximum of ` `// minimums of all layers ` `void` `MaximumOfMinimum(``int` `a[][4], ``int` `n) ` `{ ` `     `  `    ``// Calculate the layers ` `    ``int` `layers = n / 2 + n % 2; ` `    ``int` `max1 = INT_MIN; ` ` `  `    ``// Iterate for all the layers ` `    ``for``(``int` `i = 0; i < layers; i++) ` `    ``{ ` `         `  `        ``// Find maximum ` `        ``max1 = max(max1, ` `                   ``getBoundaryMin(a, n, i)); ` `    ``} ` `     `  `    ``// Print the answer ` `    ``cout << (max1); ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `     `  `    ``// Initialize the matrix ` `    ``int` `a[][4] = { { 1, 2, 3, 4 }, ` `                   ``{ 5, 6, 7, 8 }, ` `                   ``{ 9, 10, 11, 12 }, ` `                   ``{ 13, 14, 15, 16 } }; ` ` `  `    ``int` `n = ``sizeof``(a) / ``sizeof``(a[0]); ` ` `  `    ``// Function call ` `    ``MaximumOfMinimum(a, n); ` `} ` ` `  `// This code is contributed by chitranayal `

## Java

 `// Java Program for the above approach ` ` `  `class` `GFG { ` ` `  `    ``// Function to return the minimum ` `    ``// of the boundary elements ` `    ``static` `int` `    ``getBoundaryMin(``int` `a[][], ``int` `n, ` `                   ``int` `index) ` `    ``{ ` `        ``int` `min = Integer.MAX_VALUE; ` ` `  `        ``for` `(``int` `i = index; i < n - index; i++) { ` ` `  `            ``// Top boundary ` `            ``min = Math.min( ` `                ``min, ` `                ``a[index][i]); ` ` `  `            ``// Left boundary ` `            ``min = Math.min( ` `                ``min, ` `                ``a[i][index]); ` ` `  `            ``// Right boundary ` `            ``min = Math.min( ` `                ``min, ` `                ``a[i][n - index - ``1``]); ` ` `  `            ``// Bottom boundary ` `            ``min = Math.min( ` `                ``min, ` `                ``a[n - index - ``1``][i]); ` `        ``} ` ` `  `        ``return` `min; ` `    ``} ` ` `  `    ``// Function to find the maximum of ` `    ``// minimums of all layers ` `    ``static` `void` `MaximumOfMinimum( ` `        ``int` `a[][], ``int` `n) ` `    ``{ ` `        ``// Calculate the layers ` `        ``int` `layers = n / ``2` `+ n % ``2``; ` `        ``int` `max = Integer.MIN_VALUE; ` ` `  `        ``// Iterate for all the layers ` `        ``for` `(``int` `i = ``0``; i < layers; i++) { ` ` `  `            ``// Find maximum ` `            ``max ` `                ``= Math.max( ` `                    ``max, ` `                    ``getBoundaryMin(a, n, i)); ` `        ``} ` ` `  `        ``// Print the answer ` `        ``System.out.print(max); ` `    ``} ` ` `  `    ``// Driver Code ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``// Initialize the matrix ` `        ``int` `a[][] = { { ``1``, ``2``, ``3``, ``4` `}, ` `                      ``{ ``5``, ``6``, ``7``, ``8` `}, ` `                      ``{ ``9``, ``10``, ``11``, ``12` `}, ` `                      ``{ ``13``, ``14``, ``15``, ``16` `} }; ` ` `  `        ``int` `n = a.length; ` ` `  `        ``// Function call ` `        ``MaximumOfMinimum(a, n); ` `    ``} ` `} `

## Python3

 `# Python3 program for the above approach ` `import` `sys ` ` `  `# Function to return the minimum ` `# of the boundary elements ` `def` `getBoundaryMin(a, n, index): ` `     `  `    ``min1 ``=` `sys.maxsize ` ` `  `    ``for` `i ``in` `range``(index, n ``-` `index): ` `         `  `        ``# Top boundary ` `        ``min1 ``=` `min``(min1, a[index][i]) ` ` `  `        ``# Left boundary ` `        ``min1 ``=` `min``(min1, a[i][index]) ` ` `  `        ``# Right boundary ` `        ``min1 ``=` `min``(min1, a[i][n ``-` `index ``-` `1``]) ` ` `  `        ``# Bottom boundary ` `        ``min1 ``=` `min``(min1, a[n ``-` `index ``-` `1``][i]) ` `     `  `    ``return` `min1 ` ` `  `# Function to find the maximum of ` `# minimums of all layers ` `def` `MaximumOfMinimum(a, n): ` `     `  `    ``# Calculate the layers ` `    ``layers ``=` `n ``/``/` `2` `+` `n ``%` `2` `    ``max1 ``=` `-``sys.maxsize ``-` `1` ` `  `    ``# Iterate for all the layers ` `    ``for` `i ``in` `range``(``0``, layers): ` `         `  `        ``# Find maximum ` `        ``max1 ``=` `max``(max1, getBoundaryMin(a, n, i)) ` `     `  `    ``# Print the answer ` `    ``print``(max1) ` ` `  `# Driver Code ` `     `  `# Initialize the matrix ` `a ``=` `[ [ ``1``, ``2``, ``3``, ``4` `], ` `      ``[ ``5``, ``6``, ``7``, ``8` `], ` `      ``[ ``9``, ``10``, ``11``, ``12` `] , ` `      ``[ ``13``, ``14``, ``15``, ``16` `] ]  ` ` `  `n ``=` `len``(a) ` ` `  `# Function call ` `MaximumOfMinimum(a, n) ` ` `  `# This code is contributed by sanjoy_62 `

## C#

 `// C# program for the above approach ` `using` `System; ` ` `  `class` `GFG{ ` ` `  `// Function to return the minimum ` `// of the boundary elements ` `static` `int` `getBoundaryMin(``int``[,]a, ``int` `n, ` `                          ``int` `index) ` `{ ` `    ``int` `min = ``int``.MaxValue; ` ` `  `    ``for``(``int` `i = index; i < n - index; i++)  ` `    ``{ ` `         `  `        ``// Top boundary ` `        ``min = Math.Min(min, a[index, i]); ` ` `  `        ``// Left boundary ` `        ``min = Math.Min(min, a[i, index]); ` ` `  `        ``// Right boundary ` `        ``min = Math.Min(min, a[i, n - index - 1]); ` ` `  `        ``// Bottom boundary ` `        ``min = Math.Min(min, a[n - index - 1, i]); ` `    ``} ` `    ``return` `min; ` `} ` ` `  `// Function to find the maximum of ` `// minimums of all layers ` `static` `void` `MaximumOfMinimum(``int``[,]a, ``int` `n) ` `{ ` `     `  `    ``// Calculate the layers ` `    ``int` `layers = n / 2 + n % 2; ` `    ``int` `max = ``int``.MinValue; ` ` `  `    ``// Iterate for all the layers ` `    ``for``(``int` `i = 0; i < layers; i++)  ` `    ``{ ` `         `  `        ``// Find maximum ` `        ``max = Math.Max(max, ` `                       ``getBoundaryMin(a, n, i)); ` `    ``} ` ` `  `    ``// Print the answer ` `    ``Console.Write(max); ` `} ` ` `  `// Driver Code ` `public` `static` `void` `Main(String[] args) ` `{ ` `     `  `    ``// Initialize the matrix ` `    ``int``[,]a = { { 1, 2, 3, 4 }, ` `                ``{ 5, 6, 7, 8 }, ` `                ``{ 9, 10, 11, 12 }, ` `                ``{ 13, 14, 15, 16 } }; ` ` `  `    ``int` `n = a.GetLength(0); ` ` `  `    ``// Function call ` `    ``MaximumOfMinimum(a, n); ` `} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

Output:

```6
```

Time Complexity: O(N2)
Auxiliary Space: O(1)

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