# Maximum size of square such that all submatrices of that size have sum less than K

Given an N x M matrix of integers and an integer K, the task is to find the size of the maximum square sub-matrix (S x S), such that all square sub-matrices of the given matrix of that size have a sum less than K.
Examples:

```Input: K = 30
mat[N][M] = {{1, 2, 3, 4, 6},
{5, 3, 8, 1, 2},
{4, 6, 7, 5, 5},
{2, 4, 8, 9, 4} };
Output: 2
Explanation:
All Sub-matrices of size 2 x 2
have sum less than 30

Input : K = 100
mat[N][M] = { { 1, 2, 3, 4 },
{ 5, 6, 7, 8 },
{ 9, 10, 11, 12 },
{ 13, 14, 15, 16 } };
Output: 3
Explanation:
All Sub-matrices of size 3 x 3
have sum less than 100

```

Naive Approach The basic solution is to choose the size S of the submatrix and find all the submatrices of that size and check that the sum of all sub-matrices is less than the given sum whereas, this can be improved by computing the sum of the matrix using this approach. Therefore, the task will be to choose the maximum size possible and the starting and ending position of the every possible sub-matrices. Due to which the overall time complexity will be O(N3).
Below is the implementation of the above approach:

 `// C++ implementation to find the ` `// maximum size square submatrix ` `// such that their sum is less than K ` ` `  `#include ` ` `  `using` `namespace` `std; ` ` `  `// Size of matrix ` `#define N 4 ` `#define M 5 ` ` `  `// Function to preprocess the matrix ` `// for computing the sum of every  ` `// possible matrix of the given size ` `void` `preProcess(``int` `mat[N][M],  ` `                ``int` `aux[N][M]) ` `{ ` `    ``// Loop to copy the first row ` `    ``// of the matrix into the aux matrix ` `    ``for` `(``int` `i = 0; i < M; i++) ` `        ``aux[0][i] = mat[0][i]; ` `     `  `    ``// Computing the sum column-wise ` `    ``for` `(``int` `i = 1; i < N; i++) ` `        ``for` `(``int` `j = 0; j < M; j++) ` `            ``aux[i][j] = mat[i][j] +  ` `                      ``aux[i - 1][j]; ` ` `  `    ``// Computing row wise sum ` `    ``for` `(``int` `i = 0; i < N; i++) ` `        ``for` `(``int` `j = 1; j < M; j++) ` `            ``aux[i][j] += aux[i][j - 1]; ` `} ` ` `  `// Function to find the sum of a ` `// submatrix with the given indices ` `int` `sumQuery(``int` `aux[N][M], ``int` `tli,  ` `          ``int` `tlj, ``int` `rbi, ``int` `rbj) ` `{ ` `    ``// Overall sum from the top to  ` `    ``// right corner of matrix ` `    ``int` `res = aux[rbi][rbj]; ` `     `  `    ``// Removing the sum from the top ` `    ``// corer of the matrix ` `    ``if` `(tli > 0) ` `        ``res = res - aux[tli - 1][rbj]; ` `     `  `    ``// Remove the overlapping sum ` `    ``if` `(tlj > 0) ` `        ``res = res - aux[rbi][tlj - 1]; ` `     `  `    ``// Add the sum of top corner  ` `    ``// which is substracted twice ` `    ``if` `(tli > 0 && tlj > 0) ` `        ``res = res +  ` `           ``aux[tli - 1][tlj - 1]; ` ` `  `    ``return` `res; ` `} ` ` `  `// Function to find the maximum ` `// square size possible with the  ` `// such that every submatrix have  ` `// sum less than the given sum ` `int` `maximumSquareSize(``int` `mat[N][M], ``int` `K) ` `{ ` `    ``int` `aux[N][M]; ` `    ``preProcess(mat, aux); ` `     `  `    ``// Loop to choose the size of matrix ` `    ``for` `(``int` `i = min(N, M); i >= 1; i--) { ` ` `  `        ``bool` `satisfies = ``true``; ` `         `  `        ``// Loop to find the sum of the ` `        ``// matrix of every possible submatrix ` `        ``for` `(``int` `x = 0; x < N; x++) { ` `            ``for` `(``int` `y = 0; y < M; y++) { ` `                ``if` `(x + i - 1 <= N - 1 &&  ` `                     ``y + i - 1 <= M - 1) { ` `                    ``if` `(sumQuery(aux, x, y,  ` `                   ``x + i - 1, y + i - 1) > K) ` `                        ``satisfies = ``false``; ` `                ``} ` `            ``} ` `        ``} ` `        ``if` `(satisfies == ``true``) ` `            ``return` `(i); ` `    ``} ` `    ``return` `0; ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `    ``int` `K = 30; ` `    ``int` `mat[N][M] = { { 1, 2, 3, 4, 6 }, ` `                    ``{ 5, 3, 8, 1, 2 }, ` `                    ``{ 4, 6, 7, 5, 5 }, ` `                    ``{ 2, 4, 8, 9, 4 } }; ` ` `  `    ``cout << maximumSquareSize(mat, K); ` `    ``return` `0; ` `} `

 `// Java implementation to find the ` `// maximum size square submatrix ` `// such that their sum is less than K ` `class` `GFG{ ` `  `  `// Size of matrix ` `static` `final` `int` `N = ``4``; ` `static` `final` `int` `M = ``5``; ` `  `  `// Function to preprocess the matrix ` `// for computing the sum of every  ` `// possible matrix of the given size ` `static` `void` `preProcess(``int` `[][]mat,  ` `                ``int` `[][]aux) ` `{ ` `    ``// Loop to copy the first row ` `    ``// of the matrix into the aux matrix ` `    ``for` `(``int` `i = ``0``; i < M; i++) ` `        ``aux[``0``][i] = mat[``0``][i]; ` `      `  `    ``// Computing the sum column-wise ` `    ``for` `(``int` `i = ``1``; i < N; i++) ` `        ``for` `(``int` `j = ``0``; j < M; j++) ` `            ``aux[i][j] = mat[i][j] +  ` `                      ``aux[i - ``1``][j]; ` `  `  `    ``// Computing row wise sum ` `    ``for` `(``int` `i = ``0``; i < N; i++) ` `        ``for` `(``int` `j = ``1``; j < M; j++) ` `            ``aux[i][j] += aux[i][j - ``1``]; ` `} ` `  `  `// Function to find the sum of a ` `// submatrix with the given indices ` `static` `int` `sumQuery(``int` `[][]aux, ``int` `tli,  ` `          ``int` `tlj, ``int` `rbi, ``int` `rbj) ` `{ ` `    ``// Overall sum from the top to  ` `    ``// right corner of matrix ` `    ``int` `res = aux[rbi][rbj]; ` `      `  `    ``// Removing the sum from the top ` `    ``// corer of the matrix ` `    ``if` `(tli > ``0``) ` `        ``res = res - aux[tli - ``1``][rbj]; ` `      `  `    ``// Remove the overlapping sum ` `    ``if` `(tlj > ``0``) ` `        ``res = res - aux[rbi][tlj - ``1``]; ` `      `  `    ``// Add the sum of top corner  ` `    ``// which is substracted twice ` `    ``if` `(tli > ``0` `&& tlj > ``0``) ` `        ``res = res +  ` `           ``aux[tli - ``1``][tlj - ``1``]; ` `  `  `    ``return` `res; ` `} ` `  `  `// Function to find the maximum ` `// square size possible with the  ` `// such that every submatrix have  ` `// sum less than the given sum ` `static` `int` `maximumSquareSize(``int` `[][]mat, ``int` `K) ` `{ ` `    ``int` `[][]aux = ``new` `int``[N][M]; ` `    ``preProcess(mat, aux); ` `      `  `    ``// Loop to choose the size of matrix ` `    ``for` `(``int` `i = Math.min(N, M); i >= ``1``; i--) { ` `  `  `        ``boolean` `satisfies = ``true``; ` `          `  `        ``// Loop to find the sum of the ` `        ``// matrix of every possible submatrix ` `        ``for` `(``int` `x = ``0``; x < N; x++) { ` `            ``for` `(``int` `y = ``0``; y < M; y++) { ` `                ``if` `(x + i - ``1` `<= N - ``1` `&&  ` `                     ``y + i - ``1` `<= M - ``1``) { ` `                    ``if` `(sumQuery(aux, x, y,  ` `                   ``x + i - ``1``, y + i - ``1``) > K) ` `                        ``satisfies = ``false``; ` `                ``} ` `            ``} ` `        ``} ` `        ``if` `(satisfies == ``true``) ` `            ``return` `(i); ` `    ``} ` `    ``return` `0``; ` `} ` `  `  `// Driver Code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `K = ``30``; ` `    ``int` `mat[][] = { { ``1``, ``2``, ``3``, ``4``, ``6` `}, ` `                    ``{ ``5``, ``3``, ``8``, ``1``, ``2` `}, ` `                    ``{ ``4``, ``6``, ``7``, ``5``, ``5` `}, ` `                    ``{ ``2``, ``4``, ``8``, ``9``, ``4` `} }; ` `  `  `    ``System.out.print(maximumSquareSize(mat, K)); ` `} ` `} ` ` `  `// This code is contributed by PrinciRaj1992 `

 `# Python3 implementation to find the ` `# maximum size square submatrix ` `# such that their sum is less than K ` ` `  `# Size of matrix ` `N ``=` `4` `M ``=` `5` ` `  `# Function to preprocess the matrix ` `# for computing the sum of every  ` `# possible matrix of the given size ` `def` `preProcess(mat, aux): ` ` `  `    ``# Loop to copy the first row ` `    ``# of the matrix into the aux matrix ` `    ``for` `i ``in` `range` `(M): ` `        ``aux[``0``][i] ``=` `mat[``0``][i] ` `     `  `    ``# Computing the sum column-wise ` `    ``for` `i ``in` `range` `(``1``, N): ` `        ``for` `j ``in` `range` `(M): ` `            ``aux[i][j] ``=` `(mat[i][j] ``+` `                         ``aux[i ``-` `1``][j]) ` ` `  `    ``# Computing row wise sum ` `    ``for` `i ``in` `range` `(N): ` `        ``for` `j ``in` `range` `(``1``, M): ` `            ``aux[i][j] ``+``=` `aux[i][j ``-` `1``] ` ` `  `# Function to find the sum of a ` `# submatrix with the given indices ` `def` `sumQuery(aux, tli, tlj, rbi, rbj): ` ` `  `    ``# Overall sum from the top to  ` `    ``# right corner of matrix ` `    ``res ``=` `aux[rbi][rbj] ` `     `  `    ``# Removing the sum from the top ` `    ``# corer of the matrix ` `    ``if` `(tli > ``0``): ` `        ``res ``=` `res ``-` `aux[tli ``-` `1``][rbj] ` `     `  `    ``# Remove the overlapping sum ` `    ``if` `(tlj > ``0``): ` `        ``res ``=` `res ``-` `aux[rbi][tlj ``-` `1``] ` `     `  `    ``# Add the sum of top corner  ` `    ``# which is substracted twice ` `    ``if` `(tli > ``0` `and` `tlj > ``0``): ` `        ``res ``=` `(res ``+` `        ``aux[tli ``-` `1``][tlj ``-` `1``]) ` ` `  `    ``return` `res ` ` `  `# Function to find the maximum ` `# square size possible with the  ` `# such that every submatrix have  ` `# sum less than the given sum ` `def` `maximumSquareSize(mat, K): ` ` `  `    ``aux ``=` `[[``0` `for` `x ``in` `range` `(M)] ` `              ``for` `y ``in` `range` `(N)] ` `    ``preProcess(mat, aux) ` `     `  `    ``# Loop to choose the size of matrix ` `    ``for` `i ``in` `range` `(``min``(N, M), ``0``, ``-``1``): ` ` `  `        ``satisfies ``=` `True` `         `  `        ``# Loop to find the sum of the ` `        ``# matrix of every possible submatrix ` `        ``for` `x ``in` `range` `(N): ` `            ``for` `y ``in` `range` `(M) : ` `                ``if` `(x ``+` `i ``-` `1` `<``=` `N ``-` `1` `and` `                    ``y ``+` `i ``-` `1` `<``=` `M ``-` `1``): ` `                    ``if` `(sumQuery(aux, x, y,  ` `                                 ``x ``+` `i ``-` `1``,  ` `                                 ``y ``+` `i ``-` `1``) > K): ` `                        ``satisfies ``=` `False` `             `  `        ``if` `(satisfies ``=``=` `True``): ` `            ``return` `(i) ` `    ``return` `0` ` `  `# Driver Code ` `if` `__name__ ``=``=` `"__main__"``: ` ` `  `    ``K ``=` `30` `    ``mat ``=` `[[``1``, ``2``, ``3``, ``4``, ``6``], ` `           ``[``5``, ``3``, ``8``, ``1``, ``2``], ` `           ``[``4``, ``6``, ``7``, ``5``, ``5``], ` `           ``[``2``, ``4``, ``8``, ``9``, ``4``]] ` ` `  `    ``print``( maximumSquareSize(mat, K)) ` ` `  `# This code is contributed by Chitranayal `

 `// C# implementation to find the ` `// maximum size square submatrix ` `// such that their sum is less than K ` `using` `System; ` ` `  `public` `class` `GFG{ ` `   `  `// Size of matrix ` `static` `readonly` `int` `N = 4; ` `static` `readonly` `int` `M = 5; ` `   `  `// Function to preprocess the matrix ` `// for computing the sum of every  ` `// possible matrix of the given size ` `static` `void` `preProcess(``int` `[,]mat,  ` `                ``int` `[,]aux) ` `{ ` `    ``// Loop to copy the first row ` `    ``// of the matrix into the aux matrix ` `    ``for` `(``int` `i = 0; i < M; i++) ` `        ``aux[0,i] = mat[0,i]; ` `       `  `    ``// Computing the sum column-wise ` `    ``for` `(``int` `i = 1; i < N; i++) ` `        ``for` `(``int` `j = 0; j < M; j++) ` `            ``aux[i,j] = mat[i,j] +  ` `                      ``aux[i - 1,j]; ` `   `  `    ``// Computing row wise sum ` `    ``for` `(``int` `i = 0; i < N; i++) ` `        ``for` `(``int` `j = 1; j < M; j++) ` `            ``aux[i,j] += aux[i,j - 1]; ` `} ` `   `  `// Function to find the sum of a ` `// submatrix with the given indices ` `static` `int` `sumQuery(``int` `[,]aux, ``int` `tli,  ` `          ``int` `tlj, ``int` `rbi, ``int` `rbj) ` `{ ` `    ``// Overall sum from the top to  ` `    ``// right corner of matrix ` `    ``int` `res = aux[rbi,rbj]; ` `       `  `    ``// Removing the sum from the top ` `    ``// corer of the matrix ` `    ``if` `(tli > 0) ` `        ``res = res - aux[tli - 1,rbj]; ` `       `  `    ``// Remove the overlapping sum ` `    ``if` `(tlj > 0) ` `        ``res = res - aux[rbi,tlj - 1]; ` `       `  `    ``// Add the sum of top corner  ` `    ``// which is substracted twice ` `    ``if` `(tli > 0 && tlj > 0) ` `        ``res = res +  ` `           ``aux[tli - 1,tlj - 1]; ` `   `  `    ``return` `res; ` `} ` `   `  `// Function to find the maximum ` `// square size possible with the  ` `// such that every submatrix have  ` `// sum less than the given sum ` `static` `int` `maximumSquareSize(``int` `[,]mat, ``int` `K) ` `{ ` `    ``int` `[,]aux = ``new` `int``[N,M]; ` `    ``preProcess(mat, aux); ` `       `  `    ``// Loop to choose the size of matrix ` `    ``for` `(``int` `i = Math.Min(N, M); i >= 1; i--) { ` `   `  `        ``bool` `satisfies = ``true``; ` `           `  `        ``// Loop to find the sum of the ` `        ``// matrix of every possible submatrix ` `        ``for` `(``int` `x = 0; x < N; x++) { ` `            ``for` `(``int` `y = 0; y < M; y++) { ` `                ``if` `(x + i - 1 <= N - 1 &&  ` `                     ``y + i - 1 <= M - 1) { ` `                    ``if` `(sumQuery(aux, x, y,  ` `                   ``x + i - 1, y + i - 1) > K) ` `                        ``satisfies = ``false``; ` `                ``} ` `            ``} ` `        ``} ` `        ``if` `(satisfies == ``true``) ` `            ``return` `(i); ` `    ``} ` `    ``return` `0; ` `} ` `   `  `// Driver Code ` `public` `static` `void` `Main(String[] args) ` `{ ` `    ``int` `K = 30; ` `    ``int` `[,]mat = { { 1, 2, 3, 4, 6 }, ` `                    ``{ 5, 3, 8, 1, 2 }, ` `                    ``{ 4, 6, 7, 5, 5 }, ` `                    ``{ 2, 4, 8, 9, 4 } }; ` `   `  `    ``Console.Write(maximumSquareSize(mat, K)); ` `} ` `} ` `  `  ` `  `// This code contributed by PrinciRaj1992 `

Output:
```2

```

• Time complexity: O(N3)
• Auxiliary Space: O(N2)

Efficient Approach: The key observation is, if a square of side s is the maximum size satisfying the condition, then all sizes smaller than it will satisfy the condition. Using this we can reduce our search space at each step by half which is precisely the idea of Binary Search. Below is the illustration of the steps of the approach:

• Search Space: The search space for this problem will be from [1, min(N, M)]. That is the search space for binary search is defined as –

```low = 1
high = min(N, M)

```
•
• Next Search Space: In each iteration find the mid of the search space and then Finally, check that all subarrays of that size have the sum less than K. If all subarrays of that size have sum less than K. Then the next search space possible will be in the right of the middle. Otherwise, the next search space possible will be in the left of the middle. That is less than the middle.
• Case 1: Condition when the all the subarrays of size mid have sum less than K. Then –

```if checkSubmatrix(mat, mid, K):
low = mid + 1

```
•
• Case 2: Condition when the all the subarrays of size mid have sum greater than K. Then –

```if not checkSubmatrix(mat, mid, K):
high = mid - 1

```

Below is the implementation of the above approach:

 `// C++ implementation to find the ` `// maximum size square submatrix ` `// such that their sum is less than K ` ` `  `#include ` ` `  `using` `namespace` `std; ` ` `  `// Size of matrix ` `#define N 4 ` `#define M 5 ` ` `  `// Function to preprocess the matrix ` `// for computing the sum of every  ` `// possible matrix of the given size ` `void` `preProcess(``int` `mat[N][M],  ` `                     ``int` `aux[N][M]) ` `{ ` `    ``// Loop to copy the first row ` `    ``// of the matrix into the aux matrix ` `    ``for` `(``int` `i = 0; i < M; i++) ` `        ``aux[0][i] = mat[0][i]; ` ` `  `    ``// Computing the sum column-wise ` `    ``for` `(``int` `i = 1; i < N; i++) ` `        ``for` `(``int` `j = 0; j < M; j++) ` `            ``aux[i][j] = mat[i][j] +  ` `                       ``aux[i - 1][j]; ` ` `  `    ``// Computing row wise sum ` `    ``for` `(``int` `i = 0; i < N; i++) ` `        ``for` `(``int` `j = 1; j < M; j++) ` `            ``aux[i][j] += aux[i][j - 1]; ` `} ` ` `  `// Function to find the sum of a ` `// submatrix with the given indices ` `int` `sumQuery(``int` `aux[N][M], ``int` `tli,  ` `          ``int` `tlj, ``int` `rbi, ``int` `rbj) ` `{ ` `    ``// Overall sum from the top to  ` `    ``// right corner of matrix ` `    ``int` `res = aux[rbi][rbj]; ` ` `  `    ``// Removing the sum from the top ` `    ``// corer of the matrix ` `    ``if` `(tli > 0) ` `        ``res = res - aux[tli - 1][rbj]; ` ` `  `    ``// Remove the overlapping sum ` `    ``if` `(tlj > 0) ` `        ``res = res - aux[rbi][tlj - 1]; ` ` `  `    ``// Add the sum of top corner  ` `    ``// which is substracted twice ` `    ``if` `(tli > 0 && tlj > 0) ` `        ``res = res + aux[tli - 1][tlj - 1]; ` ` `  `    ``return` `res; ` `} ` ` `  `// Function to check whether square ` `// sub matrices of size mid satisfy  ` `// the condition or not ` `bool` `check(``int` `mid, ``int` `aux[N][M],  ` `                           ``int` `K) ` `{ ` ` `  `    ``bool` `satisfies = ``true``; ` `     `  `    ``// Iterating throught all possible ` `    ``// submatrices of given size ` `    ``for` `(``int` `x = 0; x < N; x++) { ` `        ``for` `(``int` `y = 0; y < M; y++) { ` `            ``if` `(x + mid - 1 <= N - 1 &&  ` `                  ``y + mid - 1 <= M - 1) { ` `                ``if` `(sumQuery(aux, x, y,  ` `          ``x + mid - 1, y + mid - 1) > K) ` `                    ``satisfies = ``false``; ` `            ``} ` `        ``} ` `    ``} ` `    ``return` `(satisfies == ``true``); ` `} ` `// Function to find the maximum ` `// square size possible with the  ` `// such that every submatrix have  ` `// sum less than the given sum ` `int` `maximumSquareSize(``int` `mat[N][M],  ` `                              ``int` `K) ` `{ ` `    ``int` `aux[N][M]; ` ` `  `    ``preProcess(mat, aux); ` `     `  `    ``// Search space ` `    ``int` `low = 1, high = min(N, M); ` `    ``int` `mid; ` `     `  `    ``// Binary search for size  ` `    ``while` `(high - low > 1) { ` `        ``mid = (low + high) / 2; ` `         `  `        ``// Check if the mid satisfies ` `        ``// the given condition ` `        ``if` `(check(mid, aux, K)) { ` `            ``low = mid; ` `        ``} ` `        ``else` `            ``high = mid; ` `    ``} ` `    ``if` `(check(high, aux, K)) ` `        ``return` `high; ` `    ``return` `low; ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `    ``int` `K = 30; ` `    ``int` `mat[N][M] = { { 1, 2, 3, 4, 6 }, ` `                    ``{ 5, 3, 8, 1, 2 }, ` `                    ``{ 4, 6, 7, 5, 5 }, ` `                    ``{ 2, 4, 8, 9, 4 } }; ` ` `  `    ``cout << maximumSquareSize(mat, K); ` `    ``return` `0; ` `} `

 `// Java implementation to find the ` `// maximum size square submatrix ` `// such that their sum is less than K ` `class` `GFG{ ` ` `  `// Size of matrix ` `static` `final` `int` `N = ``4``; ` `static` `final` `int` `M = ``5``; ` ` `  `// Function to preprocess the matrix ` `// for computing the sum of every  ` `// possible matrix of the given size ` `static` `void` `preProcess(``int` `[][]mat,  ` `                       ``int` `[][]aux) ` `{ ` `     `  `    ``// Loop to copy the first row of  ` `    ``// the matrix into the aux matrix ` `    ``for``(``int` `i = ``0``; i < M; i++) ` `       ``aux[``0``][i] = mat[``0``][i]; ` ` `  `    ``// Computing the sum column-wise ` `    ``for``(``int` `i = ``1``; i < N; i++) ` `       ``for``(``int` `j = ``0``; j < M; j++) ` `          ``aux[i][j] = mat[i][j] +  ` `                      ``aux[i - ``1``][j]; ` ` `  `    ``// Computing row wise sum ` `    ``for``(``int` `i = ``0``; i < N; i++) ` `       ``for``(``int` `j = ``1``; j < M; j++) ` `          ``aux[i][j] += aux[i][j - ``1``]; ` `} ` ` `  `// Function to find the sum of a ` `// submatrix with the given indices ` `static` `int` `sumQuery(``int` `[][]aux, ``int` `tli,  ` `                    ``int` `tlj, ``int` `rbi, ``int` `rbj) ` `{ ` `     `  `    ``// Overall sum from the top to  ` `    ``// right corner of matrix ` `    ``int` `res = aux[rbi][rbj]; ` ` `  `    ``// Removing the sum from the top ` `    ``// corer of the matrix ` `    ``if` `(tli > ``0``) ` `        ``res = res - aux[tli - ``1``][rbj]; ` ` `  `    ``// Remove the overlapping sum ` `    ``if` `(tlj > ``0``) ` `        ``res = res - aux[rbi][tlj - ``1``]; ` ` `  `    ``// Add the sum of top corner  ` `    ``// which is substracted twice ` `    ``if` `(tli > ``0` `&& tlj > ``0``) ` `        ``res = res + aux[tli - ``1``][tlj - ``1``]; ` ` `  `    ``return` `res; ` `} ` ` `  `// Function to check whether square ` `// sub matrices of size mid satisfy  ` `// the condition or not ` `static` `boolean` `check(``int` `mid, ``int` `[][]aux,  ` `                     ``int` `K) ` `{ ` ` `  `    ``boolean` `satisfies = ``true``; ` `     `  `    ``// Iterating throught all possible ` `    ``// submatrices of given size ` `    ``for``(``int` `x = ``0``; x < N; x++) ` `    ``{ ` `       ``for``(``int` `y = ``0``; y < M; y++) ` `       ``{ ` `          ``if` `(x + mid - ``1` `<= N - ``1` `&&  ` `              ``y + mid - ``1` `<= M - ``1``) ` `          ``{ ` `              ``if` `(sumQuery(aux, x, y,  ` `                           ``x + mid - ``1``, ` `                           ``y + mid - ``1``) > K) ` `                  ``satisfies = ``false``; ` `          ``} ` `       ``} ` `    ``} ` `    ``return` `(satisfies == ``true``); ` `} ` ` `  `// Function to find the maximum ` `// square size possible with the  ` `// such that every submatrix have  ` `// sum less than the given sum ` `static` `int` `maximumSquareSize(``int` `[][]mat,  ` `                             ``int` `K) ` `{ ` `    ``int` `[][]aux = ``new` `int``[N][M]; ` ` `  `    ``preProcess(mat, aux); ` `     `  `    ``// Search space ` `    ``int` `low = ``1``, high = Math.min(N, M); ` `    ``int` `mid; ` `     `  `    ``// Binary search for size  ` `    ``while` `(high - low > ``1``) ` `    ``{ ` `        ``mid = (low + high) / ``2``; ` `         `  `        ``// Check if the mid satisfies ` `        ``// the given condition ` `        ``if` `(check(mid, aux, K))  ` `        ``{ ` `            ``low = mid; ` `        ``} ` `        ``else` `            ``high = mid; ` `    ``} ` `    ``if` `(check(high, aux, K)) ` `        ``return` `high; ` `    ``return` `low; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `K = ``30``; ` `    ``int` `[][]mat = { { ``1``, ``2``, ``3``, ``4``, ``6` `}, ` `                    ``{ ``5``, ``3``, ``8``, ``1``, ``2` `}, ` `                    ``{ ``4``, ``6``, ``7``, ``5``, ``5` `}, ` `                    ``{ ``2``, ``4``, ``8``, ``9``, ``4` `} }; ` ` `  `    ``System.out.print(maximumSquareSize(mat, K)); ` `} ` `} ` ` `  `// This code is contributed by Rajput-Ji `

 `// C# implementation to find the ` `// maximum size square submatrix ` `// such that their sum is less than K ` `using` `System; ` ` `  `class` `GFG{ ` ` `  `// Size of matrix ` `static` `readonly` `int` `N = 4; ` `static` `readonly` `int` `M = 5; ` ` `  `// Function to preprocess the matrix ` `// for computing the sum of every  ` `// possible matrix of the given size ` `static` `void` `preProcess(``int` `[,]mat,  ` `                       ``int` `[,]aux) ` `{ ` `     `  `    ``// Loop to copy the first row of  ` `    ``// the matrix into the aux matrix ` `    ``for``(``int` `i = 0; i < M; i++) ` `       ``aux[0, i] = mat[0, i]; ` ` `  `    ``// Computing the sum column-wise ` `    ``for``(``int` `i = 1; i < N; i++) ` `       ``for``(``int` `j = 0; j < M; j++) ` `          ``aux[i, j] = mat[i, j] +  ` `                      ``aux[i - 1, j]; ` ` `  `    ``// Computing row wise sum ` `    ``for``(``int` `i = 0; i < N; i++) ` `       ``for``(``int` `j = 1; j < M; j++) ` `          ``aux[i, j] += aux[i, j - 1]; ` `} ` ` `  `// Function to find the sum of a ` `// submatrix with the given indices ` `static` `int` `sumQuery(``int` `[,]aux, ``int` `tli,  ` `                    ``int` `tlj, ``int` `rbi, ``int` `rbj) ` `{ ` `     `  `    ``// Overall sum from the top to  ` `    ``// right corner of matrix ` `    ``int` `res = aux[rbi, rbj]; ` ` `  `    ``// Removing the sum from the top ` `    ``// corer of the matrix ` `    ``if` `(tli > 0) ` `        ``res = res - aux[tli - 1, rbj]; ` ` `  `    ``// Remove the overlapping sum ` `    ``if` `(tlj > 0) ` `        ``res = res - aux[rbi, tlj - 1]; ` ` `  `    ``// Add the sum of top corner  ` `    ``// which is substracted twice ` `    ``if` `(tli > 0 && tlj > 0) ` `        ``res = res + aux[tli - 1, tlj - 1]; ` ` `  `    ``return` `res; ` `} ` ` `  `// Function to check whether square ` `// sub matrices of size mid satisfy  ` `// the condition or not ` `static` `bool` `check(``int` `mid, ``int` `[,]aux,  ` `                  ``int` `K) ` `{ ` ` `  `    ``bool` `satisfies = ``true``; ` `     `  `    ``// Iterating throught all possible ` `    ``// submatrices of given size ` `    ``for``(``int` `x = 0; x < N; x++) ` `    ``{ ` `       ``for``(``int` `y = 0; y < M; y++) ` `       ``{ ` `          ``if` `(x + mid - 1 <= N - 1 &&  ` `              ``y + mid - 1 <= M - 1) ` `          ``{ ` `              ``if` `(sumQuery(aux, x, y,  ` `                           ``x + mid - 1, ` `                           ``y + mid - 1) > K) ` `                  ``satisfies = ``false``; ` `          ``} ` `       ``} ` `    ``} ` `    ``return` `(satisfies == ``true``); ` `} ` ` `  `// Function to find the maximum ` `// square size possible with the  ` `// such that every submatrix have  ` `// sum less than the given sum ` `static` `int` `maximumSquareSize(``int` `[,]mat,  ` `                             ``int` `K) ` `{ ` `    ``int` `[,]aux = ``new` `int``[N, M]; ` ` `  `    ``preProcess(mat, aux); ` `     `  `    ``// Search space ` `    ``int` `low = 1, high = Math.Min(N, M); ` `    ``int` `mid; ` `     `  `    ``// Binary search for size  ` `    ``while` `(high - low > 1) ` `    ``{ ` `        ``mid = (low + high) / 2; ` `         `  `        ``// Check if the mid satisfies ` `        ``// the given condition ` `        ``if` `(check(mid, aux, K))  ` `        ``{ ` `            ``low = mid; ` `        ``} ` `        ``else` `            ``high = mid; ` `    ``} ` `    ``if` `(check(high, aux, K)) ` `        ``return` `high; ` `    ``return` `low; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `Main(String[] args) ` `{ ` `    ``int` `K = 30; ` `    ``int` `[,]mat = { { 1, 2, 3, 4, 6 }, ` `                   ``{ 5, 3, 8, 1, 2 }, ` `                   ``{ 4, 6, 7, 5, 5 }, ` `                   ``{ 2, 4, 8, 9, 4 } }; ` ` `  `    ``Console.Write(maximumSquareSize(mat, K)); ` `} ` `} ` ` `  `// This code is contributed by Rajput-Ji `

Output:
```2

```

Performance Analysis:

• Time Complexity: O(N2 * log(N) )
• Auxiliary Space: O(N2)

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