找到一个总和最大的 m*m (2<=m<n) 的子数组;来自 n*n int 数组(具有 +ve, -ve, 0s)
Find a subarray of m*m (2<=m<n) having largest sum; out of an n*n int array(having +ve, -ve, 0s)
我已经为上述问题写了一个解决方案,但有人可以建议一个优化的方式。我已经遍历了count(2到n)的数组,其中count是查找大小为count*count的子数组。
int n = 5; //Size of array, you may take a dynamic array as well
int a[5][5] = {{1,2,3,4,5},{2,4,7,-2,1},{4,3,9,9,1},{5,2,6,8,0},{5,4,3,2,1}};
int max = 0;
int **tempStore, size;
for(int count = 2; count < n; count++)
{
for(int i = 0; i <= (n-count); i++)
{
for(int j = 0; j <= (n-count); j++)
{
int **temp = new int*[count];
for(int i = 0; i < count; ++i) {
temp[i] = new int[count];
}
for(int k = 0; k < count; k++)
{
for(int l = 0; l <count; l++)
{
temp[k][l] = a[i+k][j+l];
}
}
//printing fetched array
int sum = 0;
for(int k = 0; k < count; k++)
{
for(int l = 0; l <count; l++)
{
sum += temp[k][l];
cout<<temp[k][l]<<" ";
}cout<<endl;
}cout<<"Sum = "<<sum<<endl;
if(sum > max)
{
max = sum;
size = count;
tempStore = new int*[count];
for(int i = 0; i < count; ++i) {
tempStore[i] = new int[count];
}
//Locking the max sum array
for(int k = 0; k < count; k++)
{
for(int l = 0; l <count; l++)
{
tempStore[k][l] = temp[k][l];
}
}
}
//printing finished
cout<<"------------------n";
//Clear temp memory
for(int i = 0; i < size; ++i) {
delete[] temp[i];
}
delete[] temp;
}
}
}
cout<<"Max sum is = "<<max<<endl;
for(int k = 0; k < size; k++)
{
for(int l = 0; l <size; l++)
{
cout<<tempStore[k][l]<<" ";
}cout<<endl;
}cout<<"-------------------------";
//Clear tempStore memory
for(int i = 0; i < size; ++i) {
delete[] tempStore[i];
}
delete[] tempStore;
的例子:
1 2 3 4 5
2 4 7 -2 1
4 3 9 9 1
5 2 6 8 0
5 4 3 2 1
输出:最大和= 71
2 4 7 -2
4 3 9 9
5 2 6 8
这个问题最好使用动态规划(DP)或记忆法来解决。
假设n非常大,您会发现重新计算矩阵的所有可能组合的和将花费太长时间,因此如果您可以重用以前的计算,将使一切都快得多。
这个想法是从较小的矩阵开始,再利用预先计算的较小矩阵的值计算较大矩阵的和。
long long *sub_solutions = new long long[n*n*m];
#define at(r,c,i) sub_solutions[((i)*n + (r))*n + (c)]
// Winner:
unsigned int w_row = 0, w_col = 0, w_size = 0;
// Fill first layer:
for ( int row = 0; row < n; row++) {
for (int col = 0; col < n; col++) {
at(r, c, 0) = data[r][c];
if (data[r][c] > data[w_row][w_col]) {
w_row = r;
w_col = c;
}
}
}
// Fill remaining layers.
for ( int size = 1; size < m; size++) {
for ( int row = 0; row < n-size; row++) {
for (int col = 0; col < n-size; col++) {
long long sum = data[row+size][col+size];
for (int i = 0; i < size; i++) {
sum += data[row+size][col+i];
sum += data[row+i][col+size];
}
sum += at(row, col, size-1); // Reuse previous solution.
at(row, col, size) = sum;
if (sum > at(w_row, w_col, w_size)) { // Could optimize this part if you only need the sum.
w_row = row;
w_col = col;
w_size = size;
}
}
}
}
// The largest sum is of the sub_matrix starting a w_row, w_col, and has dimensions w_size+1.
long long largest = at(w_row, w_col, w_size);
delete [] sub_solutions;
该算法的复杂度为:O(n*n*m*m)
,更准确地说是:0.5*n*(n-1)*m*(m-1)
。(现在我还没有测试这个,所以请让我知道如果有任何错误。)
试试这个(使用朴素的方法,会更容易理解):
#include <iostream>
#include<vector>
using namespace std;
int main( )
{
int n = 5; //Size of array, you may take a dynamic array as well
int a[5][5] =
{{2,1,8,9,0},{2,4,7,-2,1},{5,4,3,2,1},{3,4,9,9,2},{5,2,6,8,0}};
int sum, partsum;
int i, j, k, m;
sum = -999999; // presume minimum part sum
for (i = 0; i < n; i++) {
partsum = 0;
m = sizeof(a[i])/sizeof(int);
for (j = 0; j < m; j++) {
partsum += a[i][j];
}
if (partsum > sum) {
k = i;
sum = partsum;
}
}
// print subarray having largest sum
m = sizeof(a[k])/sizeof(int); // m needs to be recomputed
for (j = 0; j < m - 1; j++) {
cout << a[k][j] << ", ";
}
cout << a[k][m - 1] <<"nmax part sum = " << sum << endl;
return 0;
}
对于累积和,您可以在常量时间内计算部分和
std::vector<std::vector<int>>
compute_cumulative(const std::vector<std::vector<int>>& m)
{
std::vector<std::vector<int>> res(m.size() + 1, std::vector<int>(m.size() + 1));
for (std::size_t i = 0; i != m.size(); ++i) {
for (std::size_t j = 0; j != m.size(); ++j) {
res[i + 1][j + 1] = m[i][j] - res[i][j]
+ res[i + 1][j] + res[i][j + 1];
}
}
return res;
}
int compute_partial_sum(const std::vector<std::vector<int>>& cumulative, std::size_t i, std::size_t j, std::size_t size)
{
return cumulative[i][j] + cumulative[i + size][j + size]
- cumulative[i][j + size] - cumulative[i + size][j];
}
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