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Maximum size square sub-matrix with all 1s

  • Difficulty Level : Medium
  • Last Updated : 30 Dec, 2021

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 the size of the square sub-matrix with all 1s including M[i][j] where M[i][j] is the rightmost and bottom-most 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 the 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 the maximum entry in the above matrix is 3 and the coordinates of the entry are (4, 3). Using the maximum value and its coordinates, we can find out the required sub-matrix. 

C++

// C++ code for Maximum size square 
// sub-matrix with all 1s 
#include <bits/stdc++.h>
#define bool int 
#define R 6 
#define C 5 
using namespace std;


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],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; 
            } 
        }             
    } 

    cout<<"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--) 
        { 
            cout << M[i][j] << " "; 
        } 
        cout << "\n"; 
    } 
} 


/* Driver code */
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); 
} 

// This code is contributed by rathbhupendra

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);
    }

}

Python3

# Python3 code for Maximum size
# square sub-matrix with all 1s

def printMaxSubSquare(M):
    R = len(M) # no. of rows in M[][]
    C = len(M[0]) # no. of columns in M[][]

    S = []
    for i in range(R):
      temp = []
      for j in range(C):
        if i==0 or j==0:
          temp += M[i][j],
        else:
          temp += 0,
      S += temp,
    # here we have set the first row and first column of S same as input matrix, other entries are set to 0

    # Update other entries
    for i in range(1, R):
        for j in range(1, C):
            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
    # indices of maximum entry in S[][]
    max_of_s = S[0][0]
    max_i = 0
    max_j = 0
    for i in range(R):
        for j in range(C):
            if (max_of_s < S[i][j]):
                max_of_s = S[i][j]
                max_i = i
                max_j = j

    print("Maximum size sub-matrix is: ")
    for i in range(max_i, max_i - max_of_s, -1):
        for j in range(max_j, max_j - max_of_s, -1):
            print (M[i][j], end = " ")
        print("")

# Driver Program
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)

# This code is contributed by Soumen Ghosh

C#

// C# Code for Maximum size square 
// sub-matrix with all 1s

using System;


public class GFG
{
    // method for Maximum size square sub-matrix with all 1s
    static void printMaxSubSquare(int [,]M)
    {
        int i,j;
        //no of rows in M[,]
        int R = M.GetLength(0);    
         //no of columns in M[,]
        int C = M.GetLength(1);    
        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;
                }     
            }                 
        }     
        
        Console.WriteLine("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--)
            {
                Console.Write(M[i,j] + " ");
            } 
            Console.WriteLine();
        } 
    } 
    
    // Driver program 
    public static void Main() 
    {
        int [,]M = new int[6,5]{{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);
    }

}


PHP

<?php
// PHP code for Maximum size square 
// sub-matrix with all 1s 

function printMaxSubSquare($M, $R, $C) 
{ 
    $S = array(array()) ;

    /* 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--) 
        { 
            echo $M[$i][$j], " " ; 
        } 
        echo "\n" ;
    } 
} 

# Driver code
$M = array(array(0, 1, 1, 0, 1), 
           array(1, 1, 0, 1, 0), 
           array(0, 1, 1, 1, 0), 
           array(1, 1, 1, 1, 0), 
           array(1, 1, 1, 1, 1), 
           array(0, 0, 0, 0, 0)); 
    
$R = 6 ;
$C = 5 ;         
printMaxSubSquare($M, $R, $C); 

// This code is contributed by Ryuga
?>

Javascript

<script>
// JavaScript code for Maximum size square 
// sub-matrix with all 1s 
let R = 6;
let C = 5;

function printMaxSubSquare(M) { 
    let i,j; 
    let S = [];

for ( var y = 0; y < R; y++ ) {
    S[ y ] = [];
    for ( var x = 0; x < C; x++ ) {
        S[ y ][ x ] = 0;
    }
}
    let 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; 
            } 
        }             
    } 

    document.write("Maximum size sub-matrix is: <br>"); 
    for(i = max_i; i > max_i - max_of_s; i--) 
    { 
        for(j = max_j; j > max_j - max_of_s; j--) 
        { 
            document.write( M[i][j] , " "); 
        } 
        document.write("<br>"); 
    } 
} 


/* Driver code */
let 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); 
</script>

Output: 

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

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

Space Optimized Solution: In order to compute an entry at any position in the matrix we only need the current row and the previous row.

C++

// C++ code for Maximum size square 
// sub-matrix with all 1s
// (space optimized solution)
#include <bits/stdc++.h>

using namespace std;

#define R 6 
#define C 5 

void printMaxSubSquare(bool M[R][C]) 
{
    int S[2][C], Max = 0;
  
    // set all elements of S to 0 first
    memset(S, 0, sizeof(S));

    // Construct the entries
    for (int i = 0; i < R;i++)
        for (int j = 0; j < C;j++){

            // Compute the entrie at the current position
            int Entrie = M[i][j];
            if(Entrie)
                if(j)
                    Entrie = 1 + min(S[1][j - 1], min(S[0][j - 1], S[1][j]));

            // Save the last entrie and add the new one
            S[0][j] = S[1][j];
            S[1][j] = Entrie;

            // Keep track of the max square length
            Max = max(Max, Entrie);
        }
    
    // Print the square
    cout << "Maximum size sub-matrix is: \n";
    for (int i = 0; i < Max; i++, cout << '\n')
        for (int j = 0; j < Max;j++)
            cout << "1 ";
} 

// Driver code
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);

    return 0;

    // This code is contributed
    // by Gatea David
}
Output
Maximum size sub-matrix is: 
1 1 1 
1 1 1 
1 1 1 

Time Complexity: O(m*n) where m is the number of rows and n is the number of columns in the given matrix. Auxiliary space: O(n) where n is the number of columns in the given matrix. 

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


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