使用两次遍历收集网格中的最大点
每个单元格中都有一个点矩阵，该矩阵如何使用两次遍历从该网格中获取最大点。
有一些条件要满足-
第一次遍历从网格的左上角单元格开始，并且应该到达左下角。在第二次遍历中，从右上角到右下角
我们只能从一个单元格移到当前单元格的底部，左下角和当前单元格的右下角。
如果一个遍历已经从一个单元格中获得了一些点，则在下一次遍历中将不会从该单元格中获得任何点。
输入输出
Input: A grid with points. 3 6 8 2 5 2 4 3 1 1 20 10 1 1 20 10 1 1 20 10 Output: Maximum points collected by two traversals is 73. From the first traversal, it gains: 3 + 2 + 20 + 1 + 1 = 27 From the second traversal, it gains: 2 + 4 + 10 + 20 + 10 = 46
算法
findMaxVal(mTable, x, y1, y2)
输入-一个3D数组作为存储表，x值和y1，y2。
输出-最大值。
Begin
if x, y1 and y2 is not valid, then
return - ∞
if both traversal is complete, then
if y1 = y2, then
return grid[x, y1]
else
return grid[x, y1] + grid[x, y2]
if both traversal are at last row, then
return - ∞
if subProblem is solved, then
return mTable[x, y1, y2]
set res := - ∞
if y1 = y2, then
temp := grid[x, y1]
else
temp := grid[x, y1] + grid[x, y2]
res := max of res and (temp + findMaxVal(mTable, x+1, y1, y2-1))
res := max of res and (temp + findMaxVal(mTable, x+1, y1, y2+1))
res := max of res and (temp + findMaxVal(mTable, x+1, y1, y2))
res := max of res and (temp + findMaxVal(mTable, x+1, y1-1, y2))
res := max of res and (temp + findMaxVal(mTable, x+1, y1-1, y2-1))
res := max of res and (temp + findMaxVal(mTable, x+1, y1-1, y2+1))
res := max of res and (temp + findMaxVal(mTable, x+1, y1+1, y2))
res := max of res and (temp + findMaxVal(mTable, x+1, y1+1, y2-1))
res := max of res and (temp + findMaxVal(mTable, x+1, y1+1, y2+1))
return true if mTable[x, y1, y2] = res
End示例
#include<iostream>
#define ROW 5
#define COL 4
using namespace std;
int grid[ROW][COL] = {
{3, 6, 8, 2},
{5, 2, 4, 3},
{1, 1, 20, 10},
{1, 1, 20, 10},
{1, 1, 20, 10},
};
bool isValidInput(int x, int y1, int y2) {
return (x >= 0 && x < ROW && y1 >=0 && y1 < COL && y2 >=0 && y2 < COL);
}
int max(int a, int b) {
return (a>b)?a:b;
}
int findMaxVal(int mTable[ROW][COL][COL], int x, int y1, int y2) {
if (!isValidInput(x, y1, y2)) //when in invalid cell, return -ve infinity
return INT_MIN;
if (x == ROW-1 && y1 == 0 && y2 == COL-1) //when both traversal is complete
return (y1 == y2)? grid[x][y1]: grid[x][y1] + grid[x][y2];
if (x == ROW-1) //both traversal are at last row but not completed
return INT_MIN;
if (mTable[x][y1][y2] != -1) //when subproblem is solved
return mTable[x][y1][y2];
int answer = INT_MIN; //initially the answer is -ve infinity
int temp = (y1 == y2)? grid[x][y1]: grid[x][y1] + grid[x][y2]; //store gain of the current room
//找到所有可能值的答案并使用最大值
answer = max(answer, temp + findMaxVal(mTable, x+1, y1, y2-1));
answer = max(answer, temp + findMaxVal(mTable, x+1, y1, y2+1));
answer = max(answer, temp + findMaxVal(mTable, x+1, y1, y2));
answer = max(answer, temp + findMaxVal(mTable, x+1, y1-1, y2));
answer = max(answer, temp + findMaxVal(mTable, x+1, y1-1, y2-1));
answer = max(answer, temp + findMaxVal(mTable, x+1, y1-1, y2+1));
answer = max(answer, temp + findMaxVal(mTable, x+1, y1+1, y2));
answer = max(answer, temp + findMaxVal(mTable, x+1, y1+1, y2-1));
answer = max(answer, temp + findMaxVal(mTable, x+1, y1+1, y2+1));
return (mTable[x][y1][y2] = answer); //store the answer in the mTable and return.
}
int findMaxCollection() {
//创建一个备忘录表并将所有值设置为-1-
int mTable[ROW][COL][COL];
for(int i = 0; i<ROW; i++)
for(int j = 0; j<COL; j++)
for(int k = 0; k<COL; k++)
mTable[i][j][k] = -1;
return findMaxVal(mTable, 0, 0, COL-1);
}
int main() {
cout << "Maximum collection is " << findMaxCollection();
return 0;
}输出结果
Maximum collection is 73