给定一个矩阵mat[][]和两个整数K和S ，任务是计算所有K x K子矩阵，使得子矩阵中所有元素的总和大于或等于S 。
例子：
Input: K = 2, S = 15
mat[][] = {{1, 2, 3},
{4, 5, 6},
{7, 8, 9}}
Output: 3
Explanation:
In the given matrix there are 3 sub-matrix
Sub-Matrix 1: (0, 1) to (1, 2)
Sum = 2 + 3 + 5 + 6 = 16
Sub-matrix 2: (1, 0) to (2, 1)
Sum = 4 + 5 + 7 + 8 = 24
Sub-matrix 3: (1, 1) to (2, 2)
Sum = 5 + 6 + 8 + 9 = 28
Input: K = 3, S = 35
arr[][] = {{1, 7, 1, 1, 1},
{2, 2, 2, 2, 2},
{3, 9, 6, 7, 3},
{4, 3, 2, 4, 5},
{5, 1, 5, 3, 1}}
Output: 5
朴素的方法：迭代所有可能的大小为K x K 的子矩阵并找到每个矩阵的总和。如果任何子矩阵的总和大于给定的总和S ，则将计数加 1。
高效方法：思想是预先计算矩阵的前缀和，使得每个子矩阵和的总和可以在 O(1) 时间内计算出来。最后，迭代矩阵的所有可能位置，并根据包含和排除原理检查来自这些位置的大小为K x K的子矩阵的总和。如果总和大于给定的总和，则将此类子矩阵的计数增加 1。
下面是上述方法的实现：
C++
// C++ programm to count total number
// of k x k sub matrix whose sum is
// greater than the given number S
#include
using namespace std;
#define dim 5
// Function to create Prefix sum
// matrix from the given matrix
void createTable(int mtrx[][dim],
int k, int p, int dp[][dim])
{
// Store first value in table
dp[0][0] = mtrx[0][0];
// Initialize first row of matrix
for (int j = 1; j < dim; j++) {
dp[0][j] = mtrx[0][j] +
dp[0][j - 1];
}
// Initialize first coloumn of matrix
for (int i = 1; i < dim; i++) {
dp[i][0] = mtrx[i][0] +
dp[i - 1][0];
}
// Initialize rest table with sum
for (int i = 1; i < dim; i++) {
for (int j = 1; j < dim; j++) {
dp[i][j] = mtrx[i][j] +
dp[i - 1][j] + dp[i][j - 1] -
dp[i - 1][j - 1];
}
}
}
// Utility Function to count the submatrix
// whose sum is greater than the S
int countSubMatrixUtil(int dp[][dim],
int k, int p)
{
int count = 0;
int subMatSum = 0;
// Loop to iterate over all the
// possible positions of the
// given matrix mat[][]
for (int i = k - 1; i < dim; i++) {
for (int j = k - 1; j < dim;
j++) {
if (i == (k - 1) || j == (k - 1)) {
// Condition to check, if K x K
// is first sub matrix
if (i == (k - 1) && j == (k - 1)) {
subMatSum = dp[i][j];
}
// Condition to check sub-matrix
// has no margin at top
else if (i == (k - 1)) {
subMatSum = dp[i][j] - dp[i][j - k];
}
// Condition when sub matrix
// has no margin at left
else {
subMatSum = dp[i][j] - dp[i - k][j];
}
}
// Condtion when submatrix has
// margin at top and left
else {
subMatSum = dp[i][j] - dp[i - k][j] -
dp[i][j - k] + dp[i - k][j - k];
}
// Increament count, If sub matrix
// sum is greater than S
if (subMatSum >= p) {
count++;
}
}
}
return count;
}
// Function to count submatrix of
// size k x k such that sum if
// greater than or equal to S
int countSubMatrix(int mtrx[][dim], int k, int p)
{
int dp[dim][dim];
// For loop to initialize prefix sum
// matrix with zero, initially
for (int i = 0; i < dim; i++) {
for (int j = 0; j < dim; j++) {
dp[i][j] = 0;
}
}
// Function to create the
// prefix sum matrix
createTable(mtrx, k, p, dp);
return countSubMatrixUtil(dp, k, p);
}
// Driver Code
int main()
{
int mtrx[dim][dim] = {
{ 1, 7, 1, 1, 1 },
{ 2, 2, 2, 2, 2 },
{ 3, 9, 6, 7, 3 },
{ 4, 3, 2, 4, 5 },
{ 5, 1, 5, 3, 1 }
};
int k = 3;
int p = 35;
// Print total number of sub matrix
cout << countSubMatrix(mtrx, k, p);
return 0;
} Java
// Java program to count total number
// of k x k sub matrix whose sum is
// greater than the given number S
import java.util.*;
class GFG{
static final int dim = 5;
// Function to create prefix sum
// matrix from the given matrix
static void createTable(int mtrx[][],
int k, int p,
int dp[][])
{
// Store first value in table
dp[0][0] = mtrx[0][0];
// Initialize first row of matrix
for(int j = 1; j < dim; j++)
{
dp[0][j] = mtrx[0][j] +
dp[0][j - 1];
}
// Initialize first coloumn of matrix
for(int i = 1; i < dim; i++)
{
dp[i][0] = mtrx[i][0] +
dp[i - 1][0];
}
// Initialize rest table with sum
for(int i = 1; i < dim; i++)
{
for(int j = 1; j < dim; j++)
{
dp[i][j] = mtrx[i][j] +
dp[i - 1][j] +
dp[i][j - 1] -
dp[i - 1][j - 1];
}
}
}
// Utility Function to count the submatrix
// whose sum is greater than the S
static int countSubMatrixUtil(int dp[][],
int k, int p)
{
int count = 0;
int subMatSum = 0;
// Loop to iterate over all the
// possible positions of the
// given matrix mat[][]
for(int i = k - 1; i < dim; i++)
{
for(int j = k - 1; j < dim; j++)
{
if (i == (k - 1) || j == (k - 1))
{
// Condition to check, if K x K
// is first sub matrix
if (i == (k - 1) && j == (k - 1))
{
subMatSum = dp[i][j];
}
// Condition to check sub-matrix
// has no margin at top
else if (i == (k - 1))
{
subMatSum = dp[i][j] -
dp[i][j - k];
}
// Condition when sub matrix
// has no margin at left
else
{
subMatSum = dp[i][j] -
dp[i - k][j];
}
}
// Condtion when submatrix has
// margin at top and left
else
{
subMatSum = dp[i][j] - dp[i - k][j] -
dp[i][j - k] +
dp[i - k][j - k];
}
// Increament count, If sub matrix
// sum is greater than S
if (subMatSum >= p)
{
count++;
}
}
}
return count;
}
// Function to count submatrix of
// size k x k such that sum if
// greater than or equal to S
static int countSubMatrix(int mtrx[][],
int k, int p)
{
int [][]dp = new int[dim][dim];
// For loop to initialize prefix sum
// matrix with zero, initially
for(int i = 0; i < dim; i++)
{
for(int j = 0; j < dim; j++)
{
dp[i][j] = 0;
}
}
// Function to create the
// prefix sum matrix
createTable(mtrx, k, p, dp);
return countSubMatrixUtil(dp, k, p);
}
// Driver Code
public static void main(String[] args)
{
int mtrx[][] = { { 1, 7, 1, 1, 1 },
{ 2, 2, 2, 2, 2 },
{ 3, 9, 6, 7, 3 },
{ 4, 3, 2, 4, 5 },
{ 5, 1, 5, 3, 1 } };
int k = 3;
int p = 35;
// Print total number of sub matrix
System.out.print(countSubMatrix(mtrx, k, p));
}
}
// This code is contributed by Rohit_ranjanPython3
# Python3 programm to count total number
# of k x k sub matrix whose sum is
# greater than the given number S
dim = 5
# Function to create prefix sum
# matrix from the given matrix
def createTable(mtrx, k, p, dp):
# Store first value in table
dp[0][0] = mtrx[0][0]
# Initialize first row of matrix
for j in range(1, dim):
dp[0][j] = mtrx[0][j] + dp[0][j - 1]
# Initialize first coloumn of matrix
for i in range(1, dim):
dp[i][0] = mtrx[i][0] + dp[i - 1][0]
# Initialize rest table with sum
for i in range(1, dim):
for j in range(1, dim):
dp[i][j] = (mtrx[i][j] +
dp[i - 1][j] +
dp[i][j - 1] -
dp[i - 1][j - 1])
# Utility function to count the submatrix
# whose sum is greater than the S
def countSubMatrixUtil(dp, k, p):
count = 0
subMatSum = 0
# Loop to iterate over all the
# possible positions of the
# given matrix mat[][]
for i in range(k - 1, dim):
for j in range(k - 1, dim, 1):
if (i == (k - 1) or j == (k - 1)):
# Condition to check, if K x K
# is first sub matrix
if (i == (k - 1) and j == (k - 1)):
subMatSum = dp[i][j]
# Condition to check sub-matrix
# has no margin at top
elif (i == (k - 1)):
subMatSum = dp[i][j] - dp[i][j - k]
# Condition when sub matrix
# has no margin at left
else:
subMatSum = dp[i][j] - dp[i - k][j]
# Condtion when submatrix has
# margin at top and left
else:
subMatSum = (dp[i][j] -
dp[i - k][j] -
dp[i][j - k] +
dp[i - k][j - k])
# Increament count, If sub matrix
# sum is greater than S
if (subMatSum >= p):
count += 1
return count
# Function to count submatrix of
# size k x k such that sum if
# greater than or equal to S
def countSubMatrix(mtrx, k, p):
dp = [[0 for i in range(dim)]
for j in range(dim)]
# For loop to initialize prefix sum
# matrix with zero, initially
for i in range(dim):
for j in range(dim):
dp[i][j] = 0
# Function to create the
# prefix sum matrix
createTable(mtrx, k, p, dp)
return countSubMatrixUtil(dp, k, p)
# Driver Code
if __name__ == '__main__':
mtrx = [ [ 1, 7, 1, 1, 1 ],
[ 2, 2, 2, 2, 2 ],
[ 3, 9, 6, 7, 3 ],
[ 4, 3, 2, 4, 5 ],
[ 5, 1, 5, 3, 1 ] ]
k = 3
p = 35
# Print total number of sub matrix
print(countSubMatrix(mtrx, k, p))
# This code is contributed by Bhupendra_SinghC#
// C# program to count total number
// of k x k sub matrix whose sum is
// greater than the given number S
using System;
class GFG{
static readonly int dim = 5;
// Function to create prefix sum
// matrix from the given matrix
static void createTable(int [,]mtrx,
int k, int p,
int [,]dp)
{
// Store first value in table
dp[0, 0] = mtrx[0, 0];
// Initialize first row of matrix
for(int j = 1; j < dim; j++)
{
dp[0, j] = mtrx[0, j] +
dp[0, j - 1];
}
// Initialize first coloumn of matrix
for(int i = 1; i < dim; i++)
{
dp[i, 0] = mtrx[i, 0] +
dp[i - 1, 0];
}
// Initialize rest table with sum
for(int i = 1; i < dim; i++)
{
for(int j = 1; j < dim; j++)
{
dp[i, j] = mtrx[i, j] +
dp[i - 1, j] +
dp[i, j - 1] -
dp[i - 1, j - 1];
}
}
}
// Utility Function to count the submatrix
// whose sum is greater than the S
static int countSubMatrixUtil(int [,]dp,
int k, int p)
{
int count = 0;
int subMatSum = 0;
// Loop to iterate over all the
// possible positions of the
// given matrix [,]mat
for(int i = k - 1; i < dim; i++)
{
for(int j = k - 1; j < dim; j++)
{
if (i == (k - 1) || j == (k - 1))
{
// Condition to check, if K x K
// is first sub matrix
if (i == (k - 1) && j == (k - 1))
{
subMatSum = dp[i, j];
}
// Condition to check sub-matrix
// has no margin at top
else if (i == (k - 1))
{
subMatSum = dp[i, j] -
dp[i, j - k];
}
// Condition when sub matrix
// has no margin at left
else
{
subMatSum = dp[i, j] -
dp[i - k, j];
}
}
// Condtion when submatrix has
// margin at top and left
else
{
subMatSum = dp[i, j] - dp[i - k, j] -
dp[i, j - k] +
dp[i - k, j - k];
}
// Increament count, If sub matrix
// sum is greater than S
if (subMatSum >= p)
{
count++;
}
}
}
return count;
}
// Function to count submatrix of
// size k x k such that sum if
// greater than or equal to S
static int countSubMatrix(int [,]mtrx,
int k, int p)
{
int [,]dp = new int[dim, dim];
// For loop to initialize prefix sum
// matrix with zero, initially
for(int i = 0; i < dim; i++)
{
for(int j = 0; j < dim; j++)
{
dp[i, j] = 0;
}
}
// Function to create the
// prefix sum matrix
createTable(mtrx, k, p, dp);
return countSubMatrixUtil(dp, k, p);
}
// Driver Code
public static void Main(String[] args)
{
int [,]mtrx = { { 1, 7, 1, 1, 1 },
{ 2, 2, 2, 2, 2 },
{ 3, 9, 6, 7, 3 },
{ 4, 3, 2, 4, 5 },
{ 5, 1, 5, 3, 1 } };
int k = 3;
int p = 35;
// Print total number of sub matrix
Console.Write(countSubMatrix(mtrx, k, p));
}
}
// This code is contributed by Rajput-Ji输出：
5
时间复杂度： O(M * N)
辅助空间： O(M * N)
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