Maximum size square sub-matrix with all 1s

3.3

Given a binary matrix, find out the maximum size square sub-matrix with all 1s.

For example, consider the below binary matrix.
maximum-size-square-sub-matrix-with-all-1s

Algorithm:
Let the given binary matrix be M[R][C]. The idea of the algorithm is to construct an auxiliary size matrix S[][] in which each entry S[i][j] represents size of the square sub-matrix with all 1s including M[i][j] where M[i][j] is the rightmost and bottommost entry in sub-matrix.

1) Construct a sum matrix S[R][C] for the given M[R][C].
     a)    Copy first row and first columns as it is from M[][] to S[][]
     b)    For other entries, use following expressions to construct S[][]
         If M[i][j] is 1 then
            S[i][j] = min(S[i][j-1], S[i-1][j], S[i-1][j-1]) + 1
         Else /*If M[i][j] is 0*/
            S[i][j] = 0
2) Find the maximum entry in S[R][C]
3) Using the value and coordinates of maximum entry in S[i], print 
   sub-matrix of M[][]

For the given M[R][C] in above example, constructed S[R][C] would be:

   0  1  1  0  1
   1  1  0  1  0
   0  1  1  1  0
   1  1  2  2  0
   1  2  2  3  1
   0  0  0  0  0

The value of maximum entry in above matrix is 3 and coordinates of the entry are (4, 3). Using the maximum value and its coordinates, we can find out the required sub-matrix.

C/C++

// C/C++ code for Maximum size square sub-matrix with all 1s
#include<stdio.h>
#define bool int
#define R 6
#define C 5

void printMaxSubSquare(bool M[R][C])
{
  int i,j;
  int S[R][C];
  int max_of_s, max_i, max_j; 
 
  /* Set first column of S[][]*/
  for(i = 0; i < R; i++)
     S[i][0] = M[i][0];
 
  /* Set first row of S[][]*/     
  for(j = 0; j < C; j++)
     S[0][j] = M[0][j];
     
  /* Construct other entries of S[][]*/
  for(i = 1; i < R; i++)
  {
    for(j = 1; j < C; j++)
    {
      if(M[i][j] == 1) 
        S[i][j] = min(S[i][j-1], S[i-1][j], S[i-1][j-1]) + 1;
      else
        S[i][j] = 0;
    }    
  } 
  
  /* Find the maximum entry, and indexes of maximum entry 
     in S[][] */
  max_of_s = S[0][0]; max_i = 0; max_j = 0;
  for(i = 0; i < R; i++)
  {
    for(j = 0; j < C; j++)
    {
      if(max_of_s < S[i][j])
      {
         max_of_s = S[i][j];
         max_i = i; 
         max_j = j;
      }        
    }                 
  }     
  
  printf("Maximum size sub-matrix is: \n");
  for(i = max_i; i > max_i - max_of_s; i--)
  {
    for(j = max_j; j > max_j - max_of_s; j--)
    {
      printf("%d ", M[i][j]);
    }  
    printf("\n");
  }  
}     

/* UTILITY FUNCTIONS */
/* Function to get minimum of three values */
int min(int a, int b, int c)
{
  int m = a;
  if (m > b) 
    m = b;
  if (m > c) 
    m = c;
  return m;
}

/* Driver function to test above functions */
int main()
{
  bool M[R][C] =  {{0, 1, 1, 0, 1}, 
                   {1, 1, 0, 1, 0}, 
                   {0, 1, 1, 1, 0},
                   {1, 1, 1, 1, 0},
                   {1, 1, 1, 1, 1},
                   {0, 0, 0, 0, 0}};
               
  printMaxSubSquare(M);
  getchar();  
}  

Java

// JAVA Code for Maximum size square sub-matrix with all 1s
public class GFG
{
    // method for Maximum size square sub-matrix with all 1s
    static void printMaxSubSquare(int M[][])
    {
        int i,j;
        int R = M.length;         //no of rows in M[][]
        int C = M[0].length;     //no of columns in M[][]
        int S[][] = new int[R][C];     
        
        int max_of_s, max_i, max_j; 
      
        /* Set first column of S[][]*/
        for(i = 0; i < R; i++)
            S[i][0] = M[i][0];
      
        /* Set first row of S[][]*/    
        for(j = 0; j < C; j++)
            S[0][j] = M[0][j];
          
        /* Construct other entries of S[][]*/
        for(i = 1; i < R; i++)
        {
            for(j = 1; j < C; j++)
            {
                if(M[i][j] == 1) 
                    S[i][j] = Math.min(S[i][j-1],Math.min(S[i-1][j], S[i-1][j-1])) + 1;
                else
                    S[i][j] = 0;
            }    
        }     
       
        /* Find the maximum entry, and indexes of maximum entry 
             in S[][] */
        max_of_s = S[0][0]; max_i = 0; max_j = 0;
        for(i = 0; i < R; i++)
        {
            for(j = 0; j < C; j++)
            {
                if(max_of_s < S[i][j])
                {
                    max_of_s = S[i][j];
                    max_i = i; 
                    max_j = j;
                }        
            }                 
        }     
       
        System.out.println("Maximum size sub-matrix is: ");
        for(i = max_i; i > max_i - max_of_s; i--)
        {
            for(j = max_j; j > max_j - max_of_s; j--)
            {
                System.out.print(M[i][j] + " ");
            }  
            System.out.println();
        }  
    }    
    
    // Driver program 
    public static void main(String[] args) 
    {
        int M[][] =  {{0, 1, 1, 0, 1}, 
                      {1, 1, 0, 1, 0}, 
                      {0, 1, 1, 1, 0},
                      {1, 1, 1, 1, 0},
                      {1, 1, 1, 1, 1},
                      {0, 0, 0, 0, 0}};
             
        printMaxSubSquare(M);
    }

}


Output:

Maximum size sub-matrix is: 
1 1 1 
1 1 1 
1 1 1 

Time Complexity: O(m*n) where m is number of rows and n is number of columns in the given matrix.
Auxiliary Space: O(m*n) where m is number of rows and n is number of columns in the given matrix.
Algorithmic Paradigm: Dynamic Programming

Please write comments if you find any bug in above code/algorithm, or find other ways to solve the same problem

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