[LeetCode] 1219. Path with Maximum Gold 黃金礦工

In a gold mine grid of size m x n, each cell in this mine has an integer representing the amount of gold in that cell, 0 if it is empty.

Return the maximum amount of gold you can collect under the conditions:

• Every time you are located in a cell you will collect all the gold in that cell.
• From your position, you can walk one step to the left, right, up, or down.
• You can't visit the same cell more than once.
• Never visit a cell with 0 gold.
• You can start and stop collecting gold from any position in the grid that has some gold.

Example 1:

Input: grid = [[0,6,0],[5,8,7],[0,9,0]]
Output: 24
Explanation:
[[0,6,0],
 [5,8,7],
 [0,9,0]]
Path to get the maximum gold, 9 -> 8 -> 7.

Example 2:

Input: grid = [[1,0,7],[2,0,6],[3,4,5],[0,3,0],[9,0,20]]
Output: 28
Explanation:
[[1,0,7],
 [2,0,6],
 [3,4,5],
 [0,3,0],
 [9,0,20]]
Path to get the maximum gold, 1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7.

Constraints:

• m == grid.length
• n == grid[i].length
• 1 <= m, n <= 15
• 0 <= grid[i][j] <= 100
• There are at most 25 cells containing gold.

這道題給了一個 m by n 的二維數組 grid，說是里面的數字代表金子的數量，0表示沒有金子。現在可以選一個任意的起點，可以朝四個方向走，條件的是不能越界，不能走重復的位置，以及不能走值為0的地方，現在問最多能獲得多少的金子。這道題雖然說也是一道迷宮遍歷的問題，也是求極值的問題，但並不是求最短步數，而是求路徑值之和最大，那么顯然 BFS 就不太適合了，因為這里嚴格限制了不能走重復路徑，並且要統計每一條路徑之和，用 DFS 是墜好的。這道題的時間卡的非常的嚴格，博主最開始寫的一個版本，由於用了 HashSet 來記錄訪問過的位置，都超時了，於是只能用 grid 數組本身來記錄，首先遍歷所有的位置，跳過所有為0的位置，對於有金子的位置，調用遞歸函數。

為了節省時間，這里的遞歸函數都加上了返回值，博主一般是不喜歡加返回值的。在遞歸函數中，首先判斷當前位置是否越界，且是否有金子，不滿足的話直接返回0。然后此時記錄當前位置的金子數到一個變量 val 中，然后將當前位置的值置為0，表示訪問過了，然后對其四個鄰居位置調用遞歸函數，將最大值取出來放到變量 mx 中，之后將當前位置恢復為 val 值，並返回 mx+val 即可。用所有非0位置為起點調用遞歸函數的返回值中取最大值就是所求結果，參見代碼如下：

class Solution {
public:
    int getMaximumGold(vector<vector<int>>& grid) {
        int res = 0, m = grid.size(), n = grid[0].size();
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                if (grid[i][j] == 0) continue;
                res = max(res, helper(grid, i, j));
            }
        }
        return res;
    }
    int helper(vector<vector<int>>& grid, int i, int j) {
        int m = grid.size(), n = grid[0].size();
        if (i < 0 || i >= m || j < 0 || j >= n || grid[i][j] == 0) return 0;
        int val = grid[i][j], mx = 0;
        grid[i][j] = 0;
        mx = max({helper(grid, i + 1, j), helper(grid, i - 1, j), helper(grid, i, j + 1), helper(grid, i, j - 1)});
        grid[i][j] = val;
        return mx + val;
    }
};

Github 同步地址:

https://github.com/grandyang/leetcode/issues/1219

參考資料：

https://leetcode.com/problems/path-with-maximum-gold/

https://leetcode.com/problems/path-with-maximum-gold/discuss/398388/C%2B%2BJavaPython-DFS-Backtracking-Clean-code-O(3k)

https://leetcode.com/problems/path-with-maximum-gold/discuss/398282/JavaPython-3-DFS-and-BFS-w-comment-brief-explanation-and-analysis.

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