Given a m x n grid filled with non-negative numbers, find a path from top left to bottom right which minimizes the sum of all numbers along its path.

Note: You can only move either down or right at any point in time.

Summary: DP, calucate the minimun sum from start point to every cell unitil we traverse every cell, then just output the last cell.

    void goRight(int row_index, int column_index, int **grid_sum, vector<vector<int> > &grid) {
column_index ++ ;
int sum = grid_sum[row_index][column_index - ] + grid[row_index][column_index];
if(grid_sum[row_index][column_index] == - || sum < grid_sum[row_index][column_index])
grid_sum[row_index][column_index] = sum;
} void goDown(int row_index, int column_index, int **grid_sum, vector<vector<int> > &grid ) {
row_index ++;
int sum = grid_sum[row_index -][column_index] + grid[row_index][column_index];
if(grid_sum[row_index][column_index] == - || sum < grid_sum[row_index][column_index])
grid_sum[row_index][column_index] = sum;
} int minPathSum(vector<vector<int> > &grid) {
if( grid.size() <= || grid[].size() <=){
return ;
}else if ( grid.size() == ){
int sum = ;
for (auto item : grid[])
sum += item;
return sum;
}else if (grid.size() > && grid[].size() == ) {
int sum = ;
for ( auto item : grid)
sum += item[];
return sum;
}else {
int row_size = grid.size();
int column_size = grid[].size(); int ** grid_sum = new int *[row_size];
for( int i=; i< row_size; i++ )
{
grid_sum[i] = new int [column_size] ;
} for(int i = ; i< row_size; i ++){
for(int j =; j < column_size; j++)
grid_sum[i][j] = -;
} for(int i = ; i <= (row_size - + column_size -); i++) {
if (i == ) {
int row_index = ;
int column_index = ;
grid_sum[][] = grid[][];
if(row_index + < row_size)
goDown(row_index, column_index, grid_sum, grid);
if(column_index + < column_size)
goRight(row_index, column_index, grid_sum, grid);
}else {
int row_index = ;
int column_index = ;
for(row_index = ; row_index <= i; row_index ++ ){
if(row_index >= row_size )
continue;
column_index = i - row_index;
if(column_index >= column_size)
continue; if(row_index + < row_size)
goDown(row_index, column_index, grid_sum, grid);
if(column_index + < column_size)
goRight(row_index, column_index, grid_sum, grid);
}
}
} int sum = grid_sum[row_size - ][column_size - ];
for( int i=; i< row_size; i++ ) {
delete [] grid_sum[i];
}
delete grid_sum;
return sum;
}
}

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