📜  在Q查询的给定范围内翻转子矩阵后的二进制矩阵

📅  最后修改于: 2021-05-25 04:37:26             🧑  作者: Mango

给定大小为M x N的Q和二进制查询形式为( x1,y1,x2,y2)的二进制矩阵arr [] [] ,其中( x1,y1 )和( x2,y2 )分别表示左上和下子矩阵的右索引分别需要翻转(将0s转换为1s,反之亦然)。任务是打印执行给定的Q查询后获得的最终矩阵
例子:

天真的方法:解决每个查询问题的最简单方法是遍历给定的子矩阵,并针对每个元素检查其是否为0或1,然后进行相应翻转。在完成所有查询的这些操作之后,打印获得的最终矩阵。

下面是上述方法的实现:

C++
// C++ program to implement
// the above approach
#include
using namespace std;
 
// Function to flip a submatrices
void manipulation(vector> &matrix,
                         vector &q)
{
   
  // Boundaries of the submatrix
  int x1 = q[0], y1 = q[1],
      x2 = q[2], y2 = q[3];
 
  // Iterate over the submatrix
  for(int i = x1 - 1; i < x2; i++)
  {
    for(int j = y1 - 1; j < y2; j++)
    {
       
      // Check for 1 or 0
      // and flip accordingly
      if (matrix[i][j] == 1)
        matrix[i][j] = 0;
      else
        matrix[i][j] = 1;
    }
  }
}
 
// Function to perform the queries
void queries_fxn(vector> &matrix,
                 vector> &queries)
{
  for(auto q : queries)
    manipulation(matrix, q);       
}
 
// Driver code
int main()
{
  vector> matrix = { { 0, 1, 0 },
                                 { 1, 1, 0 } };
  vector> queries = { { 1, 1, 2, 3 },
                                  { 1, 1, 1, 1 },
                                  { 1, 2, 2, 3 } };
   
  // Function call
  queries_fxn(matrix, queries);
  cout << "[";
 
  for(int i = 0; i < matrix.size(); i++)
  {
    cout << "["; 
    for(int j = 0; j < matrix[i].size(); j++)
      cout << matrix[i][j] << " ";
 
    if (i == matrix.size() - 1)
      cout << "]";
    else
      cout << "], ";
  }
   cout << "]";
}
 
// This code is contributed by bgangwar59


Java
// Java program to implement
// the above approach
import java.util.*;
import java.lang.*;
class GFG{
 
// Function to flip a submatrices
static void manipulation(int[][] matrix,
                         int[] q)
{
  // Boundaries of the submatrix
  int x1 = q[0], y1 = q[1],
      x2 = q[2], y2 = q[3];
 
  // Iterate over the submatrix
  for(int i = x1 - 1; i < x2; i++)
  {
    for(int j = y1 - 1; j < y2; j++)
    {
      // Check for 1 or 0
      // and flip accordingly
      if (matrix[i][j] == 1)
        matrix[i][j] = 0;
      else
        matrix[i][j] = 1;
    }
  }
}
 
// Function to perform the queries
static void queries_fxn(int[][] matrix,
                        int[][] queries)
{
  for(int[] q : queries)
    manipulation(matrix, q);       
}
 
// Driver code
public static void main (String[] args)
{
  int[][] matrix = {{0, 1, 0}, {1, 1, 0}};
  int[][] queries = {{1, 1, 2, 3},
                     {1, 1, 1, 1},
                     {1, 2, 2, 3}};
 
  // Function call
  queries_fxn(matrix, queries);
  System.out.print("[");
 
  for(int i = 0; i < matrix.length; i++)
  {
    System.out.print("["); 
    for(int j = 0; j < matrix[i].length; j++)
      System.out.print(matrix[i][j] + " ");
 
    if(i == matrix.length - 1)
      System.out.print("]");
    else
      System.out.print("], ");
  }
  System.out.print("]");
}
}
 
// This code is contributed by offbeat


Python3
# Python3 Program to implement
# the above approach
 
# Function to flip a submatrices
def manipulation(matrix, q):
 
    # Boundaries of the submatrix
    x1, y1, x2, y2 = q
 
    # Iterate over the submatrix
    for i in range(x1-1, x2):
        for j in range(y1-1, y2):
 
            # Check for 1 or 0
            # and flip accordingly
            if matrix[i][j]:
                matrix[i][j] = 0
            else:
                matrix[i][j] = 1
 
# Function to perform the queries
def queries_fxn(matrix, queries):
    for q in queries:
        manipulation(matrix, q)
 
 
# Driver Code
matrix = [[0, 1, 0], [1, 1, 0]]
queries = [[1, 1, 2, 3], \
           [1, 1, 1, 1], \
           [1, 2, 2, 3]]
 
# Function call
queries_fxn(matrix, queries)
print(matrix)


C#
// C# program to implement
// the above approach
using System;
 
class GFG{
 
// Function to flip a submatrices
static void manipulation(int[,] matrix,
                         int[] q)
{
     
    // Boundaries of the submatrix
    int x1 = q[0], y1 = q[1],
        x2 = q[2], y2 = q[3];
     
    // Iterate over the submatrix
    for(int i = x1 - 1; i < x2; i++)
    {
        for(int j = y1 - 1; j < y2; j++)
        {
             
            // Check for 1 or 0
            // and flip accordingly
            if (matrix[i, j] == 1)
                matrix[i, j] = 0;
            else
                matrix[i, j] = 1;
        }
    }
}
 
public static int[] GetRow(int[,] matrix, int row)
{
    var rowLength = matrix.GetLength(1);
    var rowVector = new int[rowLength];
     
    for(var i = 0; i < rowLength; i++)
        rowVector[i] = matrix[row, i];
     
    return rowVector;
}
 
// Function to perform the queries
static void queries_fxn(int[,] matrix,
                        int[,] queries)
{
    for(int i = 0; i < queries.GetLength(0); i++)
        manipulation(matrix, GetRow(queries, i));       
}
 
// Driver code
public static void Main(String[] args)
{
    int[,] matrix = { { 0, 1, 0 },
                      { 1, 1, 0 } };
    int[,] queries = { { 1, 1, 2, 3 },
                       { 1, 1, 1, 1 },
                       { 1, 2, 2, 3 } };
     
    // Function call
    queries_fxn(matrix, queries);
    Console.Write("[");
     
    for(int i = 0; i < matrix.GetLength(0); i++)
    {
        Console.Write("["); 
        for(int j = 0; j < matrix.GetLength(1); j++)
            Console.Write(matrix[i, j] + ", ");
         
        if (i == matrix.Length - 1)
            Console.Write("]");
        else
            Console.Write("], ");
    }
    Console.Write("]");
}
}
 
// This code is contributed by Princi Singh


Python3
# Python3 program to implement
# the above approach
 
# Function to modify dp[][] array by
# generating prefix sum
def modifyDP(matrix, dp):
     
    for j in range(1, len(matrix)+1):
         
        for k in range(1, len(matrix[0])+1):
             
            # Update the tabular data
            dp[j][k] = dp[j][k] + dp[j-1][k] \
            + dp[j][k-1]-dp[j-1][k-1]
             
            # If the count of flips is even
            if dp[j][k] % 2 != 0:
                matrix[j-1][k-1] = int(matrix[j-1][k-1]) ^ 1
 
# Function to update dp[][] matrix
# for each query
def queries_fxn(matrix, queries, dp):
    for q in queries:
        x1, y1, x2, y2 = q
         
        # Update the table
        dp[x1][y1] += 1
        dp[x2 + 1][y2 + 1] += 1
        dp[x1][y2 + 1] -= 1
        dp[x2 + 1][y1] -= 1
         
    modifyDP(matrix, dp)
 
 
# Driver Code
matrix = [[0, 1, 0], [1, 1, 0]]
queries = [[1, 1, 2, 3], \
           [1, 1, 1, 1], \
           [1, 2, 2, 3]]
 
# Initialize dp table
dp = [[0 for i in range(len(matrix[0])+2)] \
for j in range(len(matrix)+2)]
 
queries_fxn(matrix, queries, dp)
print(matrix)


输出:
[[0, 1, 0], [0, 1, 0]]









时间复杂度: O(N * M * Q)
辅助空间: O(1)

高效方法:可以使用动态编程和Prefix Sum技术优化上述方法。标记每个查询中涉及的子矩阵的边界,然后计算矩阵中涉及的运算的前缀和,并相应地更新矩阵。请按照以下步骤解决问题:

  • 初始化一个二维状态空间表dp [] [],以将翻转的次数存储在矩阵的各个索引处
  • 对于每个查询{x1,y1,x2,y2,K},通过以下操作更新dp [] []矩阵:
    • dp [x1] [y1] + = 1
    • dp [x2 + 1] [y1]-= 1
    • dp [x2 +1] [y2 +1] + = 1
    • dp [x1] [y2 + 1]-= 1
  • 现在,遍历dp [] []矩阵,并通过以下关系计算行和列以及对角线的前缀和,从而更新dp [i] [j]:
  • 如果发现dp [i] [j]为奇数,则将mat [i – 1] [j – 1]减1。
  • 最后,打印更新后的矩阵mat [] []作为结果。

下面是上述方法的实现:

Python3

# Python3 program to implement
# the above approach
 
# Function to modify dp[][] array by
# generating prefix sum
def modifyDP(matrix, dp):
     
    for j in range(1, len(matrix)+1):
         
        for k in range(1, len(matrix[0])+1):
             
            # Update the tabular data
            dp[j][k] = dp[j][k] + dp[j-1][k] \
            + dp[j][k-1]-dp[j-1][k-1]
             
            # If the count of flips is even
            if dp[j][k] % 2 != 0:
                matrix[j-1][k-1] = int(matrix[j-1][k-1]) ^ 1
 
# Function to update dp[][] matrix
# for each query
def queries_fxn(matrix, queries, dp):
    for q in queries:
        x1, y1, x2, y2 = q
         
        # Update the table
        dp[x1][y1] += 1
        dp[x2 + 1][y2 + 1] += 1
        dp[x1][y2 + 1] -= 1
        dp[x2 + 1][y1] -= 1
         
    modifyDP(matrix, dp)
 
 
# Driver Code
matrix = [[0, 1, 0], [1, 1, 0]]
queries = [[1, 1, 2, 3], \
           [1, 1, 1, 1], \
           [1, 2, 2, 3]]
 
# Initialize dp table
dp = [[0 for i in range(len(matrix[0])+2)] \
for j in range(len(matrix)+2)]
 
queries_fxn(matrix, queries, dp)
print(matrix)
输出:
[[0, 1, 0], [0, 1, 0]]









时间复杂度: O(N * M)
辅助空间: O(N * M)