#3418

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximum Amount of Money Robot Can Earn

Find the maximum amount of money a robot can collect while neutralizing robbers on its path in a grid.

Problem Statement

You are given an m x n grid, where each cell contains a value representing coins. The robot starts at the top-left corner (0, 0) and needs to reach the bottom-right corner (m - 1, n - 1), moving only right or down. The goal is to collect the maximum amount of coins along the way while avoiding negative coin values.

However, there's a twist: the robot can neutralize robbers in at most two cells on its path, preventing them from stealing coins in those cells. This gives the robot an advantage but must be used wisely to maximize the total coins collected by the time it reaches the end of the grid.

Examples

Example 1

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

Output: 8

An optimal path for maximum coins is:

Example 2

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

Output: 40

An optimal path for maximum coins is:

Constraints

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

Solution Approach

Dynamic Programming Approach

Use dynamic programming to keep track of the maximum coins the robot can collect at each cell. The state transition depends on the robot’s previous position, and whether it neutralized a robber or not. The decision-making process is updated by considering both available moves (right and down).

State Transition with Robber Neutralization

Introduce a state variable to track whether the robot has neutralized robbers and in how many cells. This will affect the decision-making process, as neutralizing a robber can help optimize the path for coin collection. Transition through states based on the robot's current position and robber neutralization state.

Space Optimization

To reduce the space complexity, optimize the dynamic programming table by keeping only the current and previous rows, as the state transition depends only on adjacent cells in the grid. This significantly improves the space efficiency, especially with large grids.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity depends on the final approach, but in the general case, it will be O(m * n) because we visit each cell once. Space complexity can be optimized to O(n) by using only the current and previous rows for the dynamic programming table.

What Interviewers Usually Probe

Candidate effectively utilizes dynamic programming to solve the problem.
Candidate identifies and handles the robber neutralization aspect appropriately.
Candidate demonstrates awareness of space optimization techniques in dynamic programming.

Common Pitfalls or Variants

Common pitfalls

Failing to correctly account for robber neutralization, leading to suboptimal paths.
Overcomplicating the dynamic programming solution and using unnecessary space.
Not considering edge cases, such as grids with only one row or one column.

Follow-up variants

What if the robot can neutralize more than two robbers?
How would the solution change if the robot can move diagonally?
What if negative coins are not allowed in the grid?

FAQ

What is the core pattern used to solve the "Maximum Amount of Money Robot Can Earn" problem?

The problem primarily utilizes state transition dynamic programming to track the robot's path and neutralize robbers efficiently.

How does dynamic programming optimize the solution for this problem?

Dynamic programming helps track the maximum coins collected at each cell, while factoring in robber neutralization and optimal path decisions.

What edge cases should be considered when solving this problem?

Edge cases include grids with only one row or one column, as well as grids with all negative coin values or no robbers.

Can the robot move diagonally in this problem?

No, the robot can only move right or down in this problem, limiting the possible moves to two directions.

How can GhostInterview help with similar dynamic programming problems?

GhostInterview provides problem-specific hints and guides to solve similar dynamic programming problems efficiently by breaking down the key concepts.

terminal

Solution

Solution 1: Memoized Search

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 $\textit{dfs}(i, j + 1, k)$.

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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])
        ans = dfs(0, 0, 2)
        dfs.cache_clear()
        return ans

Continue Practicing

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