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3418. Maximum Amount of Money Robot Can Earn

Description

You are given an m x n grid. A robot starts at the top-left corner of the grid (0, 0) and wants to reach the bottom-right corner (m - 1, n - 1). The robot can move either right or down at any point in time.

The grid contains a value coins[i][j] in each cell:

  • If coins[i][j] >= 0, the robot gains that many coins.
  • If coins[i][j] < 0, the robot encounters a robber, and the robber steals the absolute value of coins[i][j] coins.

The robot has a special ability to neutralize robbers in at most 2 cells on its path, preventing them from stealing coins in those cells.

Note: The robot's total coins can be negative.

Return the maximum profit the robot can gain on the route.

 

Example 1:

Input: coins = [[0,1,-1],[1,-2,3],[2,-3,4]]

Output: 8

Explanation:

An optimal path for maximum coins is:

  1. Start at (0, 0) with 0 coins (total coins = 0).
  2. Move to (0, 1), gaining 1 coin (total coins = 0 + 1 = 1).
  3. Move to (1, 1), where there's a robber stealing 2 coins. The robot uses one neutralization here, avoiding the robbery (total coins = 1).
  4. Move to (1, 2), gaining 3 coins (total coins = 1 + 3 = 4).
  5. Move to (2, 2), gaining 4 coins (total coins = 4 + 4 = 8).

Example 2:

Input: coins = [[10,10,10],[10,10,10]]

Output: 40

Explanation:

An optimal path for maximum coins is:

  1. Start at (0, 0) with 10 coins (total coins = 10).
  2. Move to (0, 1), gaining 10 coins (total coins = 10 + 10 = 20).
  3. Move to (0, 2), gaining another 10 coins (total coins = 20 + 10 = 30).
  4. Move to (1, 2), gaining the final 10 coins (total coins = 30 + 10 = 40).

 

Constraints:

  • m == coins.length
  • n == coins[i].length
  • 1 <= m, n <= 500
  • -1000 <= coins[i][j] <= 1000

Solutions

We design a function $\textit{dfs}(i, j, k)$, which represents the maximum amount of coins the robot can collect starting from $(i, j)$ with $k$ conversion opportunities left. The robot can only move right or down, so the value of $\textit{dfs}(i, j, k)$ depends only on $\textit{dfs}(i + 1, j, k)$ and $\textit{dfs}(i, j + 1, k)$.

  • If $i \geq m$ or $j \geq n$, it means the robot has moved out of the grid, so we return a very small value.
  • If $i = m - 1$ and $j = n - 1$, it means the robot has reached the bottom-right corner of the grid. If $k > 0$, the robot can choose to convert the bandit at the current position, so we return $\max(0, \textit{coins}[i][j])$. If $k = 0$, the robot cannot convert the bandit at the current position, so we return $\textit{coins}[i][j]$.
  • If $\textit{coins}[i][j] < 0$, it means there is a bandit at the current position. If $k > 0$, the robot can choose to convert the bandit at the current position, so we return $\textit{coins}[i][j] + \max(\textit{dfs}(i + 1, j, k), \textit{dfs}(i, j + 1, k))$. If $k = 0$, the robot cannot convert the bandit at the current position, so we return $\textit{coins}[i][j] + \max(\textit{dfs}(i + 1, j, k), \textit{dfs}(i, j + 1, k))$.

Based on the above analysis, we can write the code for memoized search.

The time complexity is $O(m \times n \times k)$, and the space complexity is $O(m \times n \times k)$. Here, $m$ and $n$ are the number of rows and columns of the 2D array $\textit{coins}$, and $k$ is the number of conversion opportunities, which is $3$ in this problem.

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class Solution:
    def maximumAmount(self, coins: List[List[int]]) -> int:
        @cache
        def dfs(i: int, j: int, k: int) -> int:
            if i >= m or j >= n:
                return -inf
            if i == m - 1 and j == n - 1:
                return max(coins[i][j], 0) if k else coins[i][j]
            ans = coins[i][j] + max(dfs(i + 1, j, k), dfs(i, j + 1, k))
            if coins[i][j] < 0 and k:
                ans = max(ans, dfs(i + 1, j, k - 1), dfs(i, j + 1, k - 1))
            return ans

        m, n = len(coins), len(coins[0])
        return dfs(0, 0, 2)

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class Solution {
    private Integer[][][] f;
    private int[][] coins;
    private int m;
    private int n;

    public int maximumAmount(int[][] coins) {
        m = coins.length;
        n = coins[0].length;
        this.coins = coins;
        f = new Integer[m][n][3];
        return dfs(0, 0, 2);
    }

    private int dfs(int i, int j, int k) {
        if (i >= m || j >= n) {
            return Integer.MIN_VALUE / 2;
        }
        if (f[i][j][k] != null) {
            return f[i][j][k];
        }
        if (i == m - 1 && j == n - 1) {
            return k > 0 ? Math.max(0, coins[i][j]) : coins[i][j];
        }
        int ans = coins[i][j] + Math.max(dfs(i + 1, j, k), dfs(i, j + 1, k));
        if (coins[i][j] < 0 && k > 0) {
            ans = Math.max(ans, Math.max(dfs(i + 1, j, k - 1), dfs(i, j + 1, k - 1)));
        }
        return f[i][j][k] = ans;
    }
}

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class Solution {
public:
    int maximumAmount(vector<vector<int>>& coins) {
        int m = coins.size(), n = coins[0].size();
        vector<vector<vector<int>>> f(m, vector<vector<int>>(n, vector<int>(3, -1)));
        auto dfs = [&](this auto&& dfs, int i, int j, int k) -> int {
            if (i >= m || j >= n) {
                return INT_MIN / 2;
            }
            if (f[i][j][k] != -1) {
                return f[i][j][k];
            }
            if (i == m - 1 && j == n - 1) {
                return k > 0 ? max(0, coins[i][j]) : coins[i][j];
            }
            int ans = coins[i][j] + max(dfs(i + 1, j, k), dfs(i, j + 1, k));
            if (coins[i][j] < 0 && k > 0) {
                ans = max({ans, dfs(i + 1, j, k - 1), dfs(i, j + 1, k - 1)});
            }
            return f[i][j][k] = ans;
        };
        return dfs(0, 0, 2);
    }
};

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func maximumAmount(coins [][]int) int {
    m, n := len(coins), len(coins[0])
    f := make([][][]int, m)
    for i := range f {
        f[i] = make([][]int, n)
        for j := range f[i] {
            f[i][j] = make([]int, 3)
            for k := range f[i][j] {
                f[i][j][k] = math.MinInt32
            }
        }
    }
    var dfs func(i, j, k int) int
    dfs = func(i, j, k int) int {
        if i >= m || j >= n {
            return math.MinInt32 / 2
        }
        if f[i][j][k] != math.MinInt32 {
            return f[i][j][k]
        }
        if i == m-1 && j == n-1 {
            if k > 0 {
                return max(0, coins[i][j])
            }
            return coins[i][j]
        }
        ans := coins[i][j] + max(dfs(i+1, j, k), dfs(i, j+1, k))
        if coins[i][j] < 0 && k > 0 {
            ans = max(ans, max(dfs(i+1, j, k-1), dfs(i, j+1, k-1)))
        }
        f[i][j][k] = ans
        return ans
    }
    return dfs(0, 0, 2)
}

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function maximumAmount(coins: number[][]): number {
    const [m, n] = [coins.length, coins[0].length];
    const f = Array.from({ length: m }, () =>
        Array.from({ length: n }, () => Array(3).fill(-Infinity)),
    );
    const dfs = (i: number, j: number, k: number): number => {
        if (i >= m || j >= n) {
            return -Infinity;
        }
        if (f[i][j][k] !== -Infinity) {
            return f[i][j][k];
        }
        if (i === m - 1 && j === n - 1) {
            return k > 0 ? Math.max(0, coins[i][j]) : coins[i][j];
        }
        let ans = coins[i][j] + Math.max(dfs(i + 1, j, k), dfs(i, j + 1, k));
        if (coins[i][j] < 0 && k > 0) {
            ans = Math.max(ans, dfs(i + 1, j, k - 1), dfs(i, j + 1, k - 1));
        }
        return (f[i][j][k] = ans);
    };
    return dfs(0, 0, 2);
}

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