给定一个矩阵arr[][]和一个整数K ,任务是从给定矩阵的所有可能的大小为K 的方形子矩阵中找到最小的元素。
例子:
Input: K = 2, arr[][] ={ {1, 2, 3}, {4, 5, 6}, {7, 8, 9} }
Output:
1 2
4 5
Explanation:
Smallest elements from all possible square submatrices of size 2 are as follows:
{ {1, 2}, {4, 5} } -> 1
{ {2, 3}, {5, 6} } -> 2
{ {4, 5}, {7, 8} } -> 4
{ {5, 6}, {8, 9} } -> 5
Input: K = 3,
arr[][] = { {-1, 5, 4, 1, -3},
{4, 3, 1, 1, 6},
{2, -2, 5, 3, 1},
{8, 5, 1, 9, -4},
{12, 3, 5, 8, 1} }
Output:
-2 -2 -3
-2 -2 -4
-2 -2 -4
朴素方法:解决问题的最简单方法是从给定矩阵生成所有可能的大小为K 的方形子矩阵,并打印每个此类子矩阵中的最小元素。
时间复杂度: O(N * M * K 2 )
辅助空间: O(1)
高效方法:按照以下步骤优化上述方法:
- 遍历矩阵的每一行,对于每个arr[i][j],就地更新索引arr[i][j]到arr[i][j + K – 1] ( 0 < = j < M – K + 1) 。
- 类似地,遍历矩阵的每一列,对于每个arr[i][j],就地更新索引arr[i][j]到arr[i+K-1][j]之间存在的最小元素( 0 <= i < N – K + 1) 。
- 执行上述操作后,矩阵arr[][]大小为(N – K + 1)*(M – K + 1)的左上子矩阵由所有K x K子矩阵的所有最小元素组成。给定矩阵的矩阵。
- 因此,打印大小为(N – K + 1)*(M – K + 1)的子矩阵作为所需的输出。
下面是上述方法的实现:
C++
// C++ Program for
// the above approach
#include
using namespace std;
// Function to returns a smallest
// elements of all KxK submatrices
// of a given NxM matrix
vector > matrixMinimum(
vector > nums, int K)
{
// Stores the dimensions
// of the given matrix
int N = nums.size();
int M = nums[0].size();
// Stores the required
// smallest elements
vector > res(N - K + 1,
vector(M - K + 1));
// Update the smallest elements row-wise
for (int i = 0; i < N; i++)
{
for (int j = 0; j < M - K + 1; j++)
{
int mini = INT_MAX;
for (int k = j; k < j + K; k++)
{
mini = min(mini, nums[i][k]);
}
nums[i][j] = mini;
}
}
// Update the minimum column-wise
for (int j = 0; j < M; j++)
{
for (int i = 0; i < N - K + 1; i++)
{
int mini = INT_MAX;
for (int k = i; k < i + K; k++)
{
mini = min(mini, nums[k][j]);
}
nums[i][j] = mini;
}
}
// Store the final submatrix with
// required minimum values
for (int i = 0; i < N - K + 1; i++)
for (int j = 0; j < M - K + 1; j++)
res[i][j] = nums[i][j];
// Return the resultant matrix
return res;
}
void smallestinKsubmatrices(vector > arr,
int K)
{
// Function Call
vector > res = matrixMinimum(arr, K);
// Print resultant matrix with the
// minimum values of KxK sub-matrix
for (int i = 0; i < res.size(); i++)
{
for (int j = 0; j < res[0].size(); j++)
{
cout << res[i][j] << " ";
}
cout << endl;
}
}
// Driver Code
int main()
{
// Given matrix
vector > arr = {{-1, 5, 4, 1, -3},
{4, 3, 1, 1, 6},
{2, -2, 5, 3, 1},
{8, 5, 1, 9, -4},
{12, 3, 5, 8, 1}};
// Given K
int K = 3;
smallestinKsubmatrices(arr, K);
}
// This code is contributed by Chitranayal
Java
// Java Program for the above approach
import java.util.*;
import java.lang.*;
class GFG {
// Function to returns a smallest
// elements of all KxK submatrices
// of a given NxM matrix
public static int[][] matrixMinimum(
int[][] nums, int K)
{
// Stores the dimensions
// of the given matrix
int N = nums.length;
int M = nums[0].length;
// Stores the required
// smallest elements
int[][] res
= new int[N - K + 1][M - K + 1];
// Update the smallest elements row-wise
for (int i = 0; i < N; i++) {
for (int j = 0; j < M - K + 1; j++) {
int min = Integer.MAX_VALUE;
for (int k = j; k < j + K; k++) {
min = Math.min(min, nums[i][k]);
}
nums[i][j] = min;
}
}
// Update the minimum column-wise
for (int j = 0; j < M; j++) {
for (int i = 0; i < N - K + 1; i++) {
int min = Integer.MAX_VALUE;
for (int k = i; k < i + K; k++) {
min = Math.min(min, nums[k][j]);
}
nums[i][j] = min;
}
}
// Store the final submatrix with
// required minimum values
for (int i = 0; i < N - K + 1; i++)
for (int j = 0; j < M - K + 1; j++)
res[i][j] = nums[i][j];
// Return the resultant matrix
return res;
}
public static void smallestinKsubmatrices(
int arr[][], int K)
{
// Function Call
int[][] res = matrixMinimum(arr, K);
// Print resultant matrix with the
// minimum values of KxK sub-matrix
for (int i = 0; i < res.length; i++) {
for (int j = 0; j < res[0].length; j++) {
System.out.print(res[i][j] + " ");
}
System.out.println();
}
}
// Driver Code
public static void main(String[] args)
{
// Given matrix
int[][] arr = { { -1, 5, 4, 1, -3 },
{ 4, 3, 1, 1, 6 },
{ 2, -2, 5, 3, 1 },
{ 8, 5, 1, 9, -4 },
{ 12, 3, 5, 8, 1 } };
// Given K
int K = 3;
smallestinKsubmatrices(arr, K);
}
}
Python3
# Python3 program for the above approach
import sys
# Function to returns a smallest
# elements of all KxK submatrices
# of a given NxM matrix
def matrixMinimum(nums, K):
# Stores the dimensions
# of the given matrix
N = len(nums)
M = len(nums[0])
# Stores the required
# smallest elements
res = [[0 for x in range(M - K + 1)]
for y in range(N - K + 1)]
# Update the smallest elements row-wise
for i in range(N):
for j in range(M - K + 1):
mn = sys.maxsize
for k in range(j, j + K):
mn = min(mn, nums[i][k])
nums[i][j] = mn
# Update the minimum column-wise
for j in range(M):
for i in range(N - K + 1):
mn = sys.maxsize
for k in range(i, i + K):
mn = min(mn, nums[k][j])
nums[i][j] = mn
# Store the final submatrix with
# required minimum values
for i in range(N - K + 1):
for j in range(M - K + 1):
res[i][j] = nums[i][j]
# Return the resultant matrix
return res
def smallestinKsubmatrices(arr, K):
# Function call
res = matrixMinimum(arr, K)
# Print resultant matrix with the
# minimum values of KxK sub-matrix
for i in range(len(res)):
for j in range(len(res[0])):
print(res[i][j], end = " ")
print()
# Driver Code
# Given matrix
arr = [ [ -1, 5, 4, 1, -3 ],
[ 4, 3, 1, 1, 6 ],
[ 2, -2, 5, 3, 1 ],
[ 8, 5, 1, 9, -4 ],
[ 12, 3, 5, 8, 1 ] ]
# Given K
K = 3
# Function call
smallestinKsubmatrices(arr, K)
# This code is contributed by Shivam Singh
C#
// C# program for the above approach
using System;
class GFG{
// Function to returns a smallest
// elements of all KxK submatrices
// of a given NxM matrix
public static int[,] matrixMinimum(int[,] nums,
int K)
{
// Stores the dimensions
// of the given matrix
int N = nums.GetLength(0);
int M = nums.GetLength(1);
// Stores the required
// smallest elements
int[,] res = new int[N - K + 1,
M - K + 1];
// Update the smallest elements row-wise
for(int i = 0; i < N; i++)
{
for(int j = 0; j < M - K + 1; j++)
{
int min = int.MaxValue;
for(int k = j; k < j + K; k++)
{
min = Math.Min(min, nums[i, k]);
}
nums[i, j] = min;
}
}
// Update the minimum column-wise
for(int j = 0; j < M; j++)
{
for(int i = 0; i < N - K + 1; i++)
{
int min = int.MaxValue;
for(int k = i; k < i + K; k++)
{
min = Math.Min(min, nums[k, j]);
}
nums[i, j] = min;
}
}
// Store the readonly submatrix with
// required minimum values
for(int i = 0; i < N - K + 1; i++)
for(int j = 0; j < M - K + 1; j++)
res[i, j] = nums[i, j];
// Return the resultant matrix
return res;
}
public static void smallestinKsubmatrices(int [,]arr,
int K)
{
// Function call
int[,] res = matrixMinimum(arr, K);
// Print resultant matrix with the
// minimum values of KxK sub-matrix
for(int i = 0; i < res.GetLength(0); i++)
{
for(int j = 0; j < res.GetLength(1); j++)
{
Console.Write(res[i, j] + " ");
}
Console.WriteLine();
}
}
// Driver Code
public static void Main(String[] args)
{
// Given matrix
int[,] arr = { { -1, 5, 4, 1, -3 },
{ 4, 3, 1, 1, 6 },
{ 2, -2, 5, 3, 1 },
{ 8, 5, 1, 9, -4 },
{ 12, 3, 5, 8, 1 } };
// Given K
int K = 3;
smallestinKsubmatrices(arr, K);
}
}
// This code is contributed by Amit Katiyar
Javascript
-2 -2 -3
-2 -2 -4
-2 -2 -4
时间复杂度: O( max(N, M) 3 )
辅助空间: O((N-K+1)*(M-K+1))
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