查找二进制矩阵中是否存在角为1的矩形

有一个给定的二进制矩阵，我们需要找到给定矩阵中是否存在四个角都等于的矩形或正方形

例子：

Input :
mat[][] = { 1 0 0 1 0
            0 0 1 0 1
            0 0 0 1 0
            1 0 1 0 1}
Output : Yes
as there exists-
1 0 1
0 1 0
1 0 1

蛮力方法-

每当我们在任何索引处找到 1 时，我们就开始扫描矩阵，然后我们尝试所有索引组合，以便我们可以形成矩形。
算法——

for i = 1 to rows
  for j = 1 to columns
   if matrix[i][j] == 1
     for k=i+1 to rows
       for l=j+1 to columns
         if (matrix[i][l]==1 &&
             matrix[k][j]==1 &&
             m[k][l]==1)
                return true
return false

C++

// A brute force approach based CPP program to
// find if there is a rectangle with 1 as corners.
#include 
using namespace std;
 
// Returns true if there is a rectangle with
// 1 as corners.
bool isRectangle(const vector >& m)
{
    // finding row and column size
    int rows = m.size();
    if (rows == 0)
        return false;
    int columns = m[0].size();
 
    // scanning the matrix
    for (int y1 = 0; y1 < rows; y1++)
      for (int x1 = 0; x1 < columns; x1++)
 
          // if any index found 1 then try
          // for all rectangles
          if (m[y1][x1] == 1)
            for (int y2 = y1 + 1; y2 < rows; y2++)
              for (int x2 = x1 + 1; x2 < columns; x2++)
                if (m[y1][x2] == 1 && m[y2][x1] == 1 &&
                                       m[y2][x2] == 1)
                  return true;
    return false;
}
 
// Driver code
int main()
{
    vector > mat = { { 1, 0, 0, 1, 0 },
                                 { 0, 0, 1, 0, 1 },
                                 { 0, 0, 0, 1, 0 },
                                 { 1, 0, 1, 0, 1 } };
    if (isRectangle(mat))
        cout << "Yes";
    else
        cout << "No";
}

Java

// A brute force approach based CPP program to
// find if there is a rectangle with 1 as corners.
public class FindRectangle {
    // Returns true if there is a rectangle with
    // 1 as corners.
    static boolean isRectangle(int m[][])
    {
        // finding row and column size
        int rows = m.length;
        if (rows == 0)
            return false;
        int columns = m[0].length;
 
        // scanning the matrix
        for (int y1 = 0; y1 < rows; y1++)
          for (int x1 = 0; x1 < columns; x1++)
            // if any index found 1 then try
            // for all rectangles
            if (m[y1][x1] == 1)
              for (int y2 = y1 + 1; y2 < rows; y2++)
                for (int x2 = x1 + 1; x2 < columns; x2++)
                  if (m[y1][x2] == 1 && m[y2][x1] == 1 &&
                                           m[y2][x2] == 1)
                    return true;
        return false;
    }
 
    public static void main(String args[])
    {
        int mat[][] = { { 1, 0, 0, 1, 0 },
                        { 0, 0, 1, 0, 1 },
                        { 0, 0, 0, 1, 0 },
                        { 1, 0, 1, 0, 1 } };
        if (isRectangle(mat))
            System.out.print("Yes");
        else
            System.out.print("No");
    }
}
// This code is contributed by Gaurav Tiwari

Python3

# A brute force approach based Python3 program to
# find if there is a rectangle with 1 as corners.
 
# Returns true if there is a rectangle
# with 1 as corners.
def isRectangle(m):
 
    # finding row and column size
    rows = len(m)
    if (rows == 0):
        return False
    columns = len(m[0])
 
    # scanning the matrix
    for y1 in range(rows):
 
        for x1 in range(columns):
 
            # if any index found 1 then
            # try for all rectangles
            if (m[y1][x1] == 1):
                for y2 in range(y1 + 1, rows):
                    for x2 in range(x1 + 1, columns):
                        if (m[y1][x2] == 1 and
                            m[y2][x1] == 1 and
                            m[y2][x2] == 1):
                            return True
    return False
 
# Driver code
mat = [[1, 0, 0, 1, 0],
       [0, 0, 1, 0, 1],
       [0, 0, 0, 1, 0],
       [1, 0, 1, 0, 1]]
 
if (isRectangle(mat)):
    print("Yes")
else:
    print("No")
 
# This code is contributed
# by mohit kumar 29

C#

// A brute force approach based C# program to
// find if there is a rectangle with 1 as corners.
using System;
 
public class FindRectangle {
    // Returns true if there is a rectangle with
    // 1 as corners.
    static Boolean isRectangle(int[, ] m)
    {
        // finding row and column size
        int rows = m.GetLength(0);
        if (rows == 0)
            return false;
        int columns = m.GetLength(1);
 
        // scanning the matrix
        for (int y1 = 0; y1 < rows; y1++)
          for (int x1 = 0; x1 < columns; x1++)
 
            // if any index found 1 then try
            // for all rectangles
            if (m[y1, x1] == 1)
              for (int y2 = y1 + 1; y2 < rows; y2++)
                for (int x2 = x1 + 1; x2 < columns; x2++)
                  if (m[y1, x2] == 1 && m[y2, x1] == 1 &&
                                          m[y2, x2] == 1)
                    return true;
        return false;
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        int[, ] mat = { { 1, 0, 0, 1, 0 },
                        { 0, 0, 1, 0, 1 },
                        { 0, 0, 0, 1, 0 },
                        { 1, 0, 1, 0, 1 } };
        if (isRectangle(mat))
            Console.Write("Yes");
        else
            Console.Write("No");
    }
}
 
// This code contributed by Rajput-Ji

Javascript

 
// A brute force approach based Javascript program to
// find if there is a rectangle with 1 as corners.
     
     
    // Returns true if there is a rectangle with
    // 1 as corners.
    function isRectangle(m)
    {
        // finding row and column size
        let rows = m.length;
        if (rows == 0)
            return false;
        let columns = m[0].length;
  
        // scanning the matrix
        for (let y1 = 0; y1 < rows; y1++)
          for (let x1 = 0; x1 < columns; x1++)
            // if any index found 1 then try
            // for all rectangles
            if (m[y1][x1] == 1)
              for (let y2 = y1 + 1; y2 < rows; y2++)
                for (let x2 = x1 + 1; x2 < columns; x2++)
                  if (m[y1][x2] == 1 && m[y2][x1] == 1 &&
                                           m[y2][x2] == 1)
                    return true;
        return false;
    }
     
    let mat = [[1, 0, 0, 1, 0],
       [0, 0, 1, 0, 1],
       [0, 0, 0, 1, 0],
       [1, 0, 1, 0, 1]];
     
    if (isRectangle(mat))
        document.write("Yes");
    else
        document.write("No");
     
 
// This code is contributed by patel2127
 

输出

Yes

时间复杂度： O(m ² * n ² )

有效的方法

1. 从上到下逐行扫描
2. 对于每一行，记住 2 个 1 的每个组合并将其推入哈希集
3. 如果我们在后面的一行中再次找到那个组合，我们就会得到我们的矩形

下面是上述方法的实现：

C++

// An efficient approach based CPP program to
// find if there is a rectangle with 1 as
// corners.
#include 
using namespace std;
 
// Returns true if there is a rectangle with
// 1 as corners.
bool isRectangle(const vector >& matrix)
{
    // finding row and column size
    int rows = matrix.size();
    if (rows == 0)
        return false;
 
    int columns = matrix[0].size();
 
    // map for storing the index of combination of 2 1's
    unordered_map > table;
 
    // scanning from top to bottom line by line
    for (int i = 0; i < rows; ++i) {
 
        for (int j = 0; j < columns - 1; ++j) {
            for (int k = j + 1; k < columns; ++k) {
 
                // if found two 1's in a column
                if (matrix[i][j] == 1 && matrix[i][k] == 1) {
 
                    // check if there exists 1's in same
                    // row previously then return true
                    // we don't need to check (j, k) pair
                    // and again (k, j) pair because we always
                    // store pair in ascending order and similarly
                    // check in ascending order, i.e. j always less
                    // than k.
                    if (table.find(j) != table.end()
                        && table[j].find(k) != table[j].end())
                        return true;
 
                    // store the indexes in hashset
                    table[j].insert(k);
                }
            }
        }
    }
    return false;
}
 
// Driver code
int main()
{
    vector > mat = { { 1, 0, 0, 1, 0 },
                                 { 0, 1, 1, 1, 1 },
                                 { 0, 0, 0, 1, 0 },
                                 { 1, 1, 1, 1, 0 } };
    if (isRectangle(mat))
        cout << "Yes";
    else
        cout << "No";
}
// This code is improved by Gautam Agrawal

Java

// An efficient approach based Java program to
// find if there is a rectangle with 1 as
// corners.
import java.util.HashMap;
import java.util.HashSet;
public class FindRectangle {
    // Returns true if there is a rectangle with
    // 1 as corners.
    static boolean isRectangle(int matrix[][])
    {
        // finding row and column size
        int rows = matrix.length;
        if (rows == 0)
            return false;
        int columns = matrix[0].length;
 
        // map for storing the index of combination of 2 1's
        HashMap > table = new HashMap<>();
 
        // scanning from top to bottom line by line
        for (int i = 0; i < rows; i++) {
            for (int j = 0; j < columns - 1; j++) {
                for (int k = j + 1; k < columns; k++) {
                    // if found two 1's in a column
                    if (matrix[i][j] == 1 && matrix[i][k] == 1) {
                        // check if there exists 1's in same
                        // row previously then return true
                        if (table.containsKey(j) && table.get(j).contains(k)) {
                            return true;
                        }
                        if (table.containsKey(k) && table.get(k).contains(j)) {
                            return true;
                        }
 
                        // store the indexes in hashset
                        if (!table.containsKey(j)) {
                            HashSet x = new HashSet<>();
                            x.add(k);
                            table.put(j, x);
                        }
                        else {
                            table.get(j).add(k);
                        }
                        if (!table.containsKey(k)) {
                            HashSet x = new HashSet<>();
                            x.add(j);
                            table.put(k, x);
                        }
                        else {
                            table.get(k).add(j);
                        }
                    }
                }
            }
        }
        return false;
    }
 
    public static void main(String args[])
    {
        int mat[][] = { { 1, 0, 0, 1, 0 },
                        { 0, 0, 1, 0, 1 },
                        { 0, 0, 0, 1, 0 },
                        { 1, 0, 1, 0, 1 } };
        if (isRectangle(mat))
            System.out.print("Yes");
        else
            System.out.print("No");
    }
}
// This code is contributed by Gaurav Tiwari

蟒蛇3

# An efficient approach based Python program
# to find if there is a rectangle with 1 as
# corners.
 
# Returns true if there is a rectangle
# with 1 as corners.
def isRectangle(matrix):
 
    # finding row and column size
    rows = len(matrix)
    if (rows == 0):
        return False
 
    columns = len(matrix[0])
 
    # map for storing the index of
    # combination of 2 1's
    table = {}
 
    # scanning from top to bottom
    # line by line
    for i in range(rows):
        for j in range(columns - 1):
            for k in range(j + 1, columns):
 
                # if found two 1's in a column
                if (matrix[i][j] == 1 and
                    matrix[i][k] == 1):
 
                    # check if there exists 1's in same
                    # row previously then return true
                    if (j in table and k in table[j]):
                        return True
 
                    if (k in table and j in table[k]):
                        return True
 
                    # store the indexes in hashset
                    if j not in table:
                        table[j] = set()
                    if k not in table:
                        table[k] = set()
                    table[j].add(k)
                    table[k].add(j)
    return False
 
# Driver Code
if __name__ == '__main__':
    mat = [[ 1, 0, 0, 1, 0 ],
           [ 0, 0, 1, 0, 1 ],
           [ 0, 0, 0, 1, 0 ],
           [ 1, 0, 1, 0, 1 ]]
    if (isRectangle(mat)):
        print("Yes")
    else:
        print("No")
     
# This code is contributed
# by SHUBHAMSINGH10

C＃

// An efficient approach based C# program to
// find if there is a rectangle with 1 as
// corners.
using System;
using System.Collections.Generic;
 
public class FindRectangle {
    // Returns true if there is a rectangle with
    // 1 as corners.
    static bool isRectangle(int[, ] matrix)
    {
        // finding row and column size
        int rows = matrix.GetLength(0);
        if (rows == 0)
            return false;
        int columns = matrix.GetLength(1);
 
        // map for storing the index of combination of 2 1's
        Dictionary > table = new Dictionary >();
 
        // scanning from top to bottom line by line
        for (int i = 0; i < rows; i++) {
            for (int j = 0; j < columns - 1; j++) {
                for (int k = j + 1; k < columns; k++) {
                    // if found two 1's in a column
                    if (matrix[i, j] == 1 && matrix[i, k] == 1) {
                        // check if there exists 1's in same
                        // row previously then return true
                        if (table.ContainsKey(j) && table[j].Contains(k)) {
                            return true;
                        }
                        if (table.ContainsKey(k) && table[k].Contains(j)) {
                            return true;
                        }
 
                        // store the indexes in hashset
                        if (!table.ContainsKey(j)) {
                            HashSet x = new HashSet();
                            x.Add(k);
                            table.Add(j, x);
                        }
                        else {
                            table[j].Add(k);
                        }
                        if (!table.ContainsKey(k)) {
                            HashSet x = new HashSet();
                            x.Add(j);
                            table.Add(k, x);
                        }
                        else {
                            table[k].Add(j);
                        }
                    }
                }
            }
        }
        return false;
    }
 
    public static void Main(String[] args)
    {
        int[, ] mat = { { 1, 0, 0, 1, 0 },
                        { 0, 0, 1, 0, 1 },
                        { 0, 0, 0, 1, 0 },
                        { 1, 0, 1, 0, 1 } };
        if (isRectangle(mat))
            Console.Write("Yes");
        else
            Console.Write("No");
    }
}
 
// This code is contributed by PrinciRaj1992

Javascript

 
// An efficient approach based Javascript program to
// find if there is a rectangle with 1 as
// corners.
     
    // Returns true if there is a rectangle with
    // 1 as corners.
    function isRectangle(matrix)
    {
        // finding row and column size
        let rows = matrix.length;
        if (rows == 0)
            return false;
        let columns = matrix[0].length;
  
        // map for storing the index of
        // combination of 2 1's
        let table = new Map();
  
        // scanning from top to bottom line by line
        for (let i = 0; i < rows; i++) {
            for (let j = 0; j < columns - 1; j++) {
                for (let k = j + 1; k < columns; k++) {
                    // if found two 1's in a column
                    if (matrix[i][j] == 1 &&
                    matrix[i][k] == 1) {
                        // check if there
                        // exists 1's in same
                        // row previously then
                        // return true
                        if (table.has(j) &&
                        table.get(j).has(k)) {
                            return true;
                        }
                        if (table.has(k) &&
                        table.get(k).has(j)) {
                            return true;
                        }
  
                        // store the indexes in hashset
                        if (!table.has(j)) {
                            let x = new Set();
                            x.add(k);
                            table.set(j, x);
                        }
                        else {
                            table.get(j).add(k);
                        }
                        if (!table.has(k)) {
                            let x = new Set();
                            x.add(j);
                            table.set(k, x);
                        }
                        else {
                            table.get(k).add(j);
                        }
                    }
                }
            }
        }
        return false;
    }
     
    let mat = [[ 1, 0, 0, 1, 0 ],
           [ 0, 0, 1, 0, 1 ],
           [ 0, 0, 0, 1, 0 ],
           [ 1, 0, 1, 0, 1 ]];
            
    if (isRectangle(mat))
        document.write("Yes");
    else
        document.write("No");
     
     
// This code is contributed by unknown2108
 

 
    

输出

Yes

时间复杂度： O(n*m ² )

更有效的方法

前面的方法扫描每对列索引以找到 2 个 1 的每个组合。

• 为了更有效地找到 2 个 1 的每个组合，请将每一行转换为一组列索引。
• 然后，从行集中选择列索引对以快速获得 2 个 1 的每个组合。
• 如果一对列索引出现不止一次，则存在一个角为 1 的矩形。
• 运行时间变为 O(m*n+n*n*log(n*n))。这是因为矩阵中有 m*n 个单元格，并且列索引最多有 O(n^2) 个组合，我们使用的映射将在 log(n) 时间内存储每个条目。
• 此外，如果 n > m，则通过首先转置矩阵，运行时间变为 O(m*n+m*m*log(m*m))。

注意 min{m*n+n*n*log(n*n), m*n+m*m*log(m*m)} 是 O(m*n + p*p*log(p*p) ))，p=max(n,m)。

下面是上述方法的实现：

C++

// C++ implementation comes from:
// https://github.com/MichaelWehar/FourCornersProblem
// Written by Niteesh Kumar and Michael Wehar
// References:
//   [1] F. Mráz, D. Prusa, and M. Wehar.
//   Two-dimensional Pattern Matching against
//    Basic Picture Languages. CIAA 2019.
//   [2] D. Prusa and M. Wehar. Complexity of
//    Searching for 2 by 2 Submatrices in Boolean
//    Matrices. DLT 2020.
 
#include 
using namespace std;
 
bool searchForRectangle(int rows, int cols,
                        vector> mat)
{
    // Make sure that matrix is non-trivial
    if (rows < 2 || cols < 2)
    {
        return false;
    }
 
    // Create map
    int num_of_keys;
    map> adjsList;
    if (rows >= cols)
    {
        // Row-wise
        num_of_keys = rows;
         
        // Convert each row into vector of col indexes
        for (int i = 0; i < rows; i++)
        {
            for (int j = 0; j < cols; j++)
            {
                if (mat[i][j])
                {
                    adjsList[i].push_back(j);
                }
            }
        }
    }
     
    else
    {
        // Col-wise
        num_of_keys = cols;
         
        // Convert each col into vector of row indexes
        for (int i = 0; i < rows; i++)
        {
            for (int j = 0; j < cols; j++)
            {
                if (mat[i][j])
                {
                    adjsList[j].push_back(i);
                }
            }
        }
    }
 
    // Search for a rectangle whose four corners are 1's
    map, int> pairs;
    for (int i = 0; i < num_of_keys; i++)
    {
        vector values = adjsList[i];
        int size = values.size();
        for (int j = 0; j < size - 1; j++)
        {
            for (int k = j + 1; k < size; k++)
            {
                pair temp
                        = make_pair(values[j],
                                     values[k]);
                if (pairs.find(temp)
                    != pairs.end())
                {
                    return true;
                } else {
                    pairs[temp] = i;
                }
            }
        }
    }
    return false;
}
 
// Driver code
int main()
{
    vector > mat = { { 1, 0, 0, 1, 0 },
                                 { 0, 1, 1, 1, 1 },
                                 { 0, 0, 0, 1, 0 },
                                 { 1, 1, 1, 1, 0 } };
    if (searchForRectangle(4, 5, mat))
        cout << "Yes";
    else
        cout << "No";
}

输出

Yes

时间复杂度： O(m*n + p*p*log(p*p))，p=max(n,m)。

如果您希望与专家一起参加现场课程，请参阅DSA 现场工作专业课程和学生竞争性编程现场课程。