全为 1 的最大尺寸方形子矩阵

给定一个二进制矩阵，找出最大尺寸为全 1 的方形子矩阵。
例如，考虑下面的二进制矩阵。

算法：
令给定的二进制矩阵为 M[R][C]。该算法的思想是构造一个辅助大小矩阵 S[][]，其中每个条目 S[i][j] 表示包含 M[i][j] 的所有 1 的方形子矩阵的大小，其中 M[ i][j] 是子矩阵中最右边和最底部的条目。

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[][]

对于上例中给定的 M[R][C]，构造的 S[R][C] 将是：

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

上述矩阵中最大条目的值为3，条目的坐标为(4, 3)。使用最大值及其坐标，我们可以找出所需的子矩阵。

C++

// C++ code for Maximum size square
// sub-matrix with all 1s
#include 
#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 is code is contributed by rathbhupendra

C

// C code for Maximum size square
// sub-matrix with all 1s
#include
#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 = [[0 for k in range(C)] for l in range(R)]
    # here we have set the first row and column of S[][]
 
    # Construct 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

 $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

// 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);

输出：

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

时间复杂度： O(m*n)，其中 m 是行数，n 是给定矩阵中的列数。
辅助空间： O(m*n) 其中 m 是行数，n 是给定矩阵中的列数。
算法范式：动态规划

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